Hindawi Publishing CorporationJournal of Function Spaces and ApplicationsVolume 2013 Article ID 692879 12 pageshttpdxdoiorg1011552013692879

Research ArticleSchauder-Tychonoff Fixed-Point Theorem inTheory of Superconductivity

Mariusz Gil and StanisBaw Wwdrychowicz

Department of Mathematics Rzeszow University of Technology alPowstancow Warszawy 6 35-959 Rzeszow Poland

Correspondence should be addressed to Stanisław Wędrychowicz swedrychprzedupl

Received 25 April 2013 Accepted 5 June 2013

Academic Editor Jozef Banas

Copyright copy 2013 M Gil and S WędrychowiczThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

We study the existence of mild solutions to the time-dependent Ginzburg-Landau ((TDGL) for short) equations on an unboundedinterval The rapidity of the growth of those solutions is characterized We investigate the local and global attractivity of solutionsof TDGL equations and we describe their asymptotic behaviourThe TDGL equations model the state of a superconducting samplein a magnetic field near critical temperature This paper is based on the theory of Banach space Frechet space and Sobolew space

1 Introduction

The objective of the paper is to investigate the existenceand asymptotic behaviour of mild solutions on an unbound-ed interval of time-dependent Ginzburg-Landau equations(TDGL for short) in superconductivity

In the Ginzburg-Landau theory of phase transitions [1]the state of a superconducting material near the critical tem-perature is described by a complex-valued order parameter120595 a real vector-valued vector potential A and when thesystem changes with time a real-valued scalar potential 120601The latter is a diagnostic variable 120595 and A are prognosticvariables whose evolution is governed by a system of coupleddifferential equations

120578 (120597

120597119905+ 119894120581120601)120595 = minus(

119894

120581nabla + A)

2

120595 + (1 minus100381610038161003816100381612059510038161003816100381610038162

) 120595 (1)

120597A120597119905

+ nabla120601 = minusnabla times nabla times A + J119904+ nabla timesH (2)

The supercurrent density J119904is a nonlinear function of 120595 and

A

J119904equiv J119904(120595A) = 1

2119894120581(120595lowastnabla120595 minus 120595nabla120595

lowast) minus

100381610038161003816100381612059510038161003816100381610038162A

= minusRe [120595lowast ( 119894120581nabla + A)120595]

(3)

The system of (1)ndash(3) must be satisfied everywhere in Ω theregion occupied by the superconducting material and at alltimes 119905 gt 0 The boundary conditions associated with thedifferential equations have the form

n sdot ( 119894120581nabla + A)120595 + 119894

120581120574120595 = 0 (4)

n times (nabla times A minusH) = 0 (5)

on 120597Ω where 120597Ω is the boundary ofΩ and n is the local outerunit normal to 120597Ω They must be satisfied at all times 119905 gt 0

We prove that the systems of (1)ndash(5) can be reducedto a semilinear equation to use the appropriate theoremwe investigate the local and global attractivity of solutionsof equations in question and describe their asymptoticbehaviour

In this paper we consider the existence and asymptoticbehaviour of mild solutions on an unbounded interval of thesemilinear evolution equation of the following form

119889119906

119889119905+A119906 = F (119905 119906 (119905)) 119905 isin R

+= [0 +infin) (6)

119906 (0) = 1199060isin X (7)

where the operator A 119863(A) sub X rarr X generates a 1198620-

semigroup 119890minusA(119905minus119904)119905ge0

andX is a real Banach spaceRecently a lot of papers have appeared that deal with the

same or similar equations on a bounded interval (see [2ndash21])

2 Journal of Function Spaces and Applications

However only in a few papers problem (6)-(7) was con-sidered on an unbounded interval [10 22] Additionallyin assumptions concerning the semigroup 119890

minusA(119905minus119904)119905ge0

orthe function F(119905 119906) rather restrictive conditions have beenimposedwhich frequently require the compactness ofF(119905 119906)or 119890minusA(119905minus119904)

119905ge0or equicontinuity of semigroup 119890minusA(119905minus119904)

119905ge0

[2ndash11 13ndash22] It is worthwhile mentioning that only a fewpapers have discussed asymptotic behaviour of solutionsmostly without the formulation of existence theorems [10 2324]

In this paper we present conditions guaranteeing theexistence of mild solutions on an unbounded interval ofproblem (6)-(7) We dispense with assumptions on thecompactness ofF(119905 119906) or 119890minusA(119905minus119904)

119905ge0

Moreover we formulate theorems about asymptotic prop-erties and both local and global attractivity of solutions ofproblem (6)-(7) The existence theorems concerning thatproblem will be proved with the help of the technique of afamily of measures of noncompactness in the Frechet space119862(R+X) and Schauder-Tychonoff fixed-point principle

The approach applied here was introduced and developedin [20 25ndash35] for instance

The paper is organized as follows In Section 2 there aregiven notation and auxiliary facts that are needed further onIn Section 3 we formulate and prove a theorem on the exis-tence of mild solution of (6) with condition (7) Moreoverthe rate of the growth of those solutions is characterized

Section 4 contains a theorem on local and global attrac-tivity of solutions of problem (6)-(7) In Section 5 we give atheorem describing the asymptotic behaviour of solutions of(6)-(7)

Finally in Section 6 we formulate the gauged TDGLequation as an abstract evolution equation in a Hilbertspace Moreover this section is devoted to present examplesof application of previously obtained theorems for TDGLequations

In order to convert the systems equations (1)ndash(5) to theCauchy problem (6)-(7) we are based on the papers andmonographs (see [1 30 36ndash69])

2 Preliminaries

Let (X sdot ) be a real Banach space with the zero element 120579Denote by 119861(119909 119903) the closed ball inX centered at 119909 and withradius 119903 If119883 is a subset of a linear topological space then thesymbols 119883 and Conv119883 stand for the closure and the convexclosure of119883 respectively

Let 120594 denote the Hausdorff measure of noncompactnessinX defined on bounded subsets119883 ofX in the followingway(see [27])

120594 (119883) = inf 119903 gt 0 119883 can be covered with finite

numbers of balls of radius 119903 (8)

Further denote by 119862(R+X) the Frechet space consisting

of all functions defined and continuous on R+with values in

a Banach space X The space 119862(R+X) is furnished with the

family of seminorms

119906119899 = sup 119906 (119905) 119905 isin [0 119899]

for 119899 = 1 2 119906 isin 119862 (R+X)

(9)

Let us recall two facts

(lowast) a sequence 119906119899infin

119899=1is convergent to 119906 in 119862(R

+X) if

and only if 119906119899infin

119899=1is uniformly convergent to 119909 on

compact subsets of R+

(lowastlowast) a subset 119880 sub 119862(R+X) is relatively compact if and

only if the restrictions to [0 119879] of all functions from119880

form an equicontinuous set for each119879 gt 0 and119880(119905) isrelatively compact inX for each 119905 isin R

+ where119880(119905) =

119906(119905) 119906 isin 119880

Moreover we recall that a nonempty subset 119880 sub

119862(R+X) is said to be bounded if

sup 119906119899 119906 isin 119880 lt infin for 119899 = 1 2 (10)

Further the family of all nonempty and bounded subsetsof 119862(R

+X) will be denoted by M

119862 while the family of

all nonempty and relatively compact subsets of 119862(R+X) is

denoted byN119862 ObviouslyN

119862subM119862

We will use a family of measures of noncompactness120583119879119879ge0

in the Frechet space 119862(R+X) which was introduced

in [20 35] In order to define these measures recall somequantities [25ndash27] Let us fix a nonempty bounded subset 119880of the space 119862(R

+X) For 119906 isin 119880 120576 ge 0 and 119879 ge 0 denote

by 120596119879(119906 120576) the modulus of continuity of the function 119906 on theinterval [0 119879] that is

120596119879(119906 120576) = sup 1003817100381710038171003817119906 (1199052) minus 119906 (1199051)

1003817100381710038171003817

1199051 1199052isin [0 119879]

10038161003816100381610038161199052 minus 11990511003816100381610038161003816 le 120576

(11)

Further let us put

120596119879(119880 120576) = sup 120596119879 (119906 120576) 119906 isin 119880

120596119879

0(119880) = lim

120576rarr0+

120596119879(119880 120576)

(12)

Obviously the set119880 is equicontinuous on [0 119879] if and only if120596119879

0(119880) = 0Next let us define the functions 120583

119879119879ge0

on the family ofbounded subsets of 119862(R

+X) by the following formula

120583119879(119880) = 120596

119879

0(119880) + sup 120594 (119880 (119905)) 119905 isin [0 119879] (13)

It can be shown that the family 120583119879119879ge0

has the followingproperties

1119900 the family ker120583

119879 = 119880 isin M

119862 120583119879(119880) = 0 for 119879 ge

0 is nonempty and ker120583119879 = N

119862

2119900119880 sub 119881 rArr 120583

119879(119880) le 120583

119879(119881) for 119879 ge 0

3119900120583119879(Conv119880) = 120583

119879(119880) for 119879 ge 0

Journal of Function Spaces and Applications 3

4119900 if 119880

119899infin

119899=1is a sequence of closed sets from M

119862such

that 119880119899+1

sub 119880119899(119899 = 1 2 ) and if lim

119899rarrinfin120583119879(119880119899) =

0 for each 119879 ge 0 then the intersection set 119880infin

=

⋂infin

119899=1119880119899is nonempty

Remark 1 Observe that in contrast to the definition of theconcept of a measure of noncompactness given in [27] ourmapping 120583

119879may take the valueinfin Moreover a single map-

ping 120583119879is not the measure of noncompactness in 119862(R

+X)

but the whole family 120583119879119879ge0

can be called family of measuresof noncompactness

Remark 2 Let us notice that the intersection set 119880infin

de-scribed in axiom 4

119900 is a member of the kernel of the familyof measure of noncompactness 120583

119879119879ge0

and therefore 119880infin

is compact in 119862(R+X) In fact the inequality 120583

119879(119880infin) le

120583119879(119880119899) for 119899 = 1 2 and 119879 ge 0 implies that 120583

119879(119880infin) = 0

Hence 119880infinisin ker120583

119879 This property of the set 119880

infinwill be

very important in our further investigations

Definition 3 A set 119890minusA119905119905ge0

of bounded linear operators onX is called 119862

0-semigroup if

(i) 119890minusA0 = 119868 119890minusA(119905+119904) = 119890minusA119905119890minusA119904 for 119905 119904 ge 0(ii) for all 119909 isin X the function R

+ni 119905 997891rarr 119890

minusA119905119909 is

continuous in R+

Further denote

119873(119879) = sup 10038171003817100381710038171003817119890minusA11990510038171003817100381710038171003817

119905 le 119879 for 119879 ge 0 (14)

Definition 4 A function 119909 isin 119862(R+X) is said to be a mild

solution of the nonlocal initial-value problem (6)-(7) if forevery 119905 isin R

+

119906 (119905) = 119890minusA1199051199060+ int

119905

0

119890minusA(119905minus119904)

F (119904 119906 (119904)) 119889119904 (15)

To prove the existence results in this paper we need thefollowing lemmas

Lemma 5 (see [21]) If 119884 is a bounded subset of Banach spaceX then for each 120576 gt 0 there is a sequence 119910

119896infin

119896=1sub 119884 such that

120594 (119884) le 2120594 (119910119896infin

119896=1) + 120576 (16)

We call a set119880 sub 1198711([0 119879]X)uniformly integrable if there

exists 120578 isin 1198711([0 119879]R+) such that 119906(119904) le 120578(119904) for 119906 isin 119880 and

ae 119904 isin [0 119879]

Lemma 6 (see [70]) If 119906119896infin

119896=1sub 1198711([0 119879]X) is uniformly

integrable then 120594(119906119896(119905)infin

119896=1) is measurable and

120594(int

119905

0

119906119896(119904) 119889119904

infin

119896=1

)le2int

119905

0

120594 (119906119896(119904)infin

119896=1) 119889119904 119891119900119903 119905isin[0 119879]

(17)

Lemma 7 (see [34]) Assume that a set 119880 sub 119862(R+X) is

bounded Then

(i) sup119905isin[0119879]

120594(119880(119905)) le 120594(119880([0 119879])) le 120596119879

0(119880) + sup

119905isin[0119879]

120594(119880(119905))(ii) sup

119905isin[0119879]120594(119880(119905)) le 120594

119862(119880) le 120596

119879

0(119880) + sup

119905isin[0119879]

120594(119880(119905))

for 119879 ge 0 where 119880([0 119879]) = 119906(119905) 119905 isin [0 119879] 119906 isin 119880

Lemma 8 (Gronwallrsquos inequality [71]) Assume that the func-tion 120601

1 R+rarr R

+is measurable and the function 120601

2

R+rarr R

+is locally integrable Moreover we assume that

int119905

01206011(119904)1206012(119904)119889119904 lt infin for 119905 ge 0 and a measurable function

120595 R+rarr R+satisfies the following inequality

120595 (119905) le 1206011(119905) + int

119905

0

1206012(119904) 120595 (119904) 119889119904 119891119900119903 119905 ge 0 (18)

Then

120595 (119905) le 1206011(119905) + int

119905

0

1206011(119904) 1206012(119904) exp(int

119905

119904

1206012(120591) 119889120591) 119889119904

119891119900119903 119905 ge 0

(19)

Next we consider the operator 119865 119862(R+X) rarr

119862(R+X) defined by the following formula

(119865119906) (119905) = int

119905

0

119890minusA(119905minus119904)

F (119904 119906 (119904)) 119889119904 119905 isin R+

119906 isin 119862 (R+X)

(20)

Lemma 9 (see [20]) Assume that conditions (1198671) (1198672) and

(1198674) are satisfied (see Section 3) a set 119880 sub 119862(R

+X) is

bounded 119879 gt 0 a function ℎ [0 119879] rarr R+is measurable

and

2119873 (119879) 120594 (F (119904 119880 (119904))) le ℎ (119904) (21)

for ae 119904 isin [0 119879] Then

120596119879

0(119865119880) le int

119879

0

ℎ (119904) 119889119904 (22)

3 Main Result

In this section we give an existence result for the semilinearequation of evolution (6)-(7) and we describe the asymptoticbehaviour of those solutions

First we will assume that the functions involved in (6)satisfy the following conditions

(1198671) A is a linear operator acting from 119863(A) sub X and A

generates 1198620-semigroup 119890minusA(119905minus119904)

119905ge0

(1198672) the mapping F R

+times X rarr X satisfies the

Caratheodory condition that isF(sdot 119906) is measurablefor 119906 isin X andF(119905 sdot) is continuous for ae 119905 isin R

+

(1198673) 119886 R

+rarr R

+and 119887 R

+rarr R

+are locally

integrable functions such that

F (119905 119906) le 119886 (119905) + 119887 (119905) 119906 (23)

for any 119906 isin X and ae 119905 isin R+

4 Journal of Function Spaces and Applications

(1198674) There exists a locally integrable function 119871 R

+rarr

R+such that for any bounded119863 sub X

120594 (F (119905 119863)) le 119871 (119905) 120594 (119863) (24)

for ae 119905 isin R+

Theorem 10 Under assumptions (1198671)ndash(1198674) (6) with initial

condition (7) has for every 1199060isin X at least one mild solution

119906 = 119906(119905) which satisfied the following estimate

119906 (119905) le 119872 (119905)

+ 119873 (119905) int

119905

0

119887 (119904)119872 (119904) exp(int119905

119904

119887 (120591)119873 (120591) 119889120591) 119889119904

(25)

where

119872(119905) = 119873 (119905) (100381710038171003817100381711990601003817100381710038171003817 + int

119905

0

119886 (119904) 119889119904) (26)

Proof Consider the operator 119866 119862(R+X) rarr 119862(R

+X)

defined by the following formula

119866119906 (119905) = 119890minusA1199051199060+ int

119905

0

119890minusA(119905minus119904)

F (119904 119906 (119904)) 119889119904 (27)

Now let us observe that for any continuous function 119906

R+rarr X in view of condition (119867

3) we get the following

estimate

119866119906 (119905) le 119873 (119905)100381710038171003817100381711990601003817100381710038171003817 + 119873 (119905) int

119905

0

(119886 (119904) + 119887 (119904) 119906 (119904)) 119889119904

(28)

which yields

119866119906 (119905) le 119872 (119905) + 119873 (119905) int

119905

0

119887 (119904) 119906 (119904) 119889119904 (29)

Next consider the following integral equation

P (119905) = 119872 (119905) + 119873 (119905) int

119905

0

119887 (119904)P (119904) 119889119904 (30)

Solving this equation by standard methods we get

P (119905)=119872 (119905)+119873 (119905)int

119905

0

119887 (119904)119872 (119904) exp(int119905

119904

119887 (120591)119873 (120591) 119889120591) 119889119904

(31)

The function P(119905) is continuous nonnegative and nonde-creasing Observe that the following implication is true

119906 (119905) le P (119905) 997904rArr 119866119906 (119905) le P (119905) for 119905 ge 0 (32)

Indeed linking (29) and (32) we have

119866119906 (119905) le 119872 (119905) + 119873 (119905) int

119905

0

119887 (119904) 119906 (119904) 119889119904

le 119872 (119905) + 119873 (119905) int

119905

0

119887 (119904)P (119904) 119889119904 = P (119905)

(33)

In the space 119862(R+X) let us consider the following set

Γ = 119906 isin 119862 (R+X) 119906 (119905) le P (119905) for 119905 ge 0 (34)

Obviously the set Γ is convex and closed Moreover in viewof (32) we have that 119866 is a self-mapping of Γ

Using the criterion of convergence (lowast) in 119862(R+X) and

standard techniques (see [31ndash33 35]) we can show that theoperator 119866 is continuous on 119862(R

+X)

Now we consider the sequence of sets Γ119899 defined by

induction as follows

Γ0= Γ Γ

119899+1= Conv (119866Γ

119899) for 119899 = 0 1 (35)

This sequence is decreasing that is Γ119899sup Γ119899+1

for 119899 =

0 1 2 Further let us fix 119879 gt 0 and for 119905 isin [0 119879] let us put

119892119899(119905) = sup 120594 (Γ

119899(119904)) 119904 isin [0 119905] (36)

119908119899= 120596119879

0(Γ119899) (37)

The sequences 119908119899 and 119892

119899(119905) are nonincreasing for all 119905 isin

[0 119879] so they have limits

119892infin(119905) = lim

119899rarrinfin119892119899(119905) 119905 isin [0 119879]

119908infin= lim119899rarrinfin

119908119899

(38)

Moreover each function 119892119899is nondecreasing therefore 119892

119899

and 119906infin

are measurable on [0 119879] for 119899 = 1 2 Now we apply the family of measures of noncompactness

120583119879119879ge0

defined in 119862(R+X) by formula (13) In view of the

above notation we have

120583119879(Γ119899) = 119908119899+ 119892119899(119879) (39)

We show that

119908infin= 119892infin(119879) = 0 (40)

To fix a number 119905 isin [0 119879] 119899 isin N and take an arbitrarynumber 119904 isin [0 119905]

We know from Lemma 5 that for any 120576 gt 0 there is asequence 119910

119896infin

119896=1sub 119866Γ119899(119904) such that

120594 (119866Γ119899(119904)) le 2120594 (119910

119896infin

119896=1) + 120576 (41)

This implies that there is a sequence 119906119896infin

119896=1sub Γ119899 such that

119910119896= (119866119906

119896) (119904) for 119896 = 1 2 (42)

Journal of Function Spaces and Applications 5

Hence in view of Lemma 6 (1198674) and (36) we obtain

120594 (119866Γ119899(119904)) le 2120594 (119910

119896infin

119896=1) + 120576

= 2120594 ((119866119906119896) (119904)infin

119896=1) + 120576 le 2120594 (119890

minusA1199041199060)

+ 2120594(int

119904

0

119890minusA(119904minus120591)

F (120591 119906119896(120591)) 119889120591

infin

119896=1

)

+ 120576 le 4int

119904

0

120594 (119890minusA(119904minus120591)

F (120591 119906119896(120591)infin

119896=1)) 119889120591 + 120576

le 4119873 (119879)int

119904

0

119871 (120591) 120594 (119906119896(120591)infin

119896=1) 119889120591 + 120576

le 4119873 (119879)int

119905

0

119871 (120591) 119892119899(120591) 119889120591 + 120576

(43)

Since 120576 gt 0 is arbitrary it follows from the above inequalitiesthat

120594 (119866Γ119899(119904)) le 4119873 (119879)int

119905

0

119871 (120591) 119892119899(120591) 119889120591 119904 isin [0 119905] (44)

Using (36) we have

119892119899+1

(119905) = sup 120594 (119866Γ119899(119904)) 119904 isin [0 119905]

le 4119873 (119879)int

119905

0

119871 (120591) 119892119899(120591) 119889120591 119905 isin [0 119879]

(45)

Letting 119899 rarr infin we derive the following inequality

119892infin(119905) le 4119873 (119879)int

119905

0

119871 (120591) 119892infin(120591) 119889120591 119905 isin [0 119879] (46)

This inequality together with Gronwallrsquos Lemma 8 impliesthat

119892infin(119905) = 0 119905 isin [0 119879] (47)

Next let us notice that in view of (1198674) and (36) we have

120594 (F (119904 Γ119899(119904))) le 119871 (119904) 120594 (Γ

119899(119904)) le 119871 (119904) 119892

119899(119904) (48)

for ae 119904 isin [0 119879] and the function

[0 119879] ni 119904 997891997888rarr 2119873 (119879) 119871 (119904) 119892119899(119904) (49)

is measurable Then in virtue of equality 1205961198790(119866Γ119899) = 120596119879

0(119865Γ119899)

and Lemma 9 we get

120596119879

0(119866Γ119899) le 2119873 (119879)int

119879

0

119871 (119904) 119892119899(119904) 119889119904 (50)

Hence we derive

119908119899+1

= 120596119879

0(Γ119899+1) = 120596119879

0(119866Γ119899)

le 2119873 (119879)int

119879

0

119871 (119904) 119892119899(119904) 119889119904

(51)

Letting 119899 rarr infin we get

119908infinle 2119873 (119879)int

119879

0

119871 (119904) 119892infin(119904) 119889119904 (52)

Keeping in mind (47) we deduce that

119908infin= 0 (53)

This together with (39) and (47) yields

lim119899rarrinfin

120583119879(Γ119899) = lim119899rarrinfin

(119908119899+ 119892119899(119879)) = 119908

infin+ 119892infin= 0 (54)

Finally using Remark 2 for the measure 120583119879 we deduce

that the set Γinfin= ⋂infin

119899=0Γ119899is nonempty convex and compact

Then by the Schauder-Tychonoff theorem we conclude thatoperator 119865 Γ

infinrarr Γinfin

has at least one fixed-point 119906 = 119906(119905)Obviously the function 119906 = 119906(119905) is a solution of problem (6)-(7) and in view of the definition of the set Γ

infin the estimate

119906(119905) le P(119905) holds to be trueThis completes the proof

4 Local and Global Attractivity

Following the concepts introduced in [36] we introduce firsta few definitions of various kinds of the concept of attractivityof mild solution of (6)

Definition 11 The mild solution 119906 = 119906(119905) of (6) with initialcondition (7) is said to be globally attractive if for each mildsolution V = V(119905) of (6) with initial condition V(0) = V

0we

have that

lim119905rarrinfin

(119906 (119905) minus V (119905)) = 120579 (55)

In other words we may say that solutions of (6) are globallyattractive if for arbitrary solutions 119906(119905) and V(119905) of thisequation condition (55) are satisfied

Definition 12 We say that mild solution 119906 = 119906(119905) of (6) withinitial condition (7) is locally attractive if there exists a ball119861(119906(0) 119903) in the space X such that for arbitrary solution V(119905)of (1) with initial-value V(0) isin 119861(119906(0) 119903) condition (55) doeshold

In the case when the limit (55) is uniform with respect toall solutions V(119905) that is when for each 120576 gt 0 there exist119879 gt 0such that

119906 (119905) minus V (119905) le 120576 (56)

for all V(119905) being solutions of (6) with initial-value V(0) isin119861(119906(0) 119903) and for 119905 ge 119879 we will say that solution 119906 = 119906(119905)

is uniformly locally attractive on R+

Now we formulate themain result of this sectionWe willconsider (6) under the following conditions

(1198671015840

1) A is the infinitesimal generator of an exponentiallystable119862

0-semigroup 119890minusA119905

119905ge0 that is there exist119872 gt

0 120588 gt 0 such that 119890minusA119905 le 119872119890minus120588119905 for all 119905 ge 0

6 Journal of Function Spaces and Applications

(1198675) there exist locally integrable functions119898 R

+rarr R+

such that

F (119905 119906) minusF (119905 V) le 119898 (119905) 119906 minus V (57)

for 119905 ge 0 and 119906 V isin X Moreover we assume that

lim119905rarrinfin

(int

119905

0

119898(119904) 119889119904 minus120588

119872119905) = minusinfin (58)

Remark 13 The property (11986710158401) is generally satisfied in diffu-

sion problem A necessary and sufficient condition for (11986710158401)

is presented in [72]

The main result of this section is shown in the giventheorem below

Theorem 14 Under assumptions (11986710158401) and (119867

2)ndash(1198675) prob-

lem (6)-(7) has amild solution 119906 = 119906(119905) for each 1199060isin X which

is globally attractive and locally uniformly attractive

Proof Existence of a solution 119906 = 119906(119905) is a consequence ofTheorem 10 Let us fix 119903 gt 0 and V

0isin 119861(119906(0) 119903)

Let V = V(119905) denote a mild solution of (6) with the initialcondition V(0) = V

0 Using (1198671015840

1) and (119867

3) we get

V (119905) le10038171003817100381710038171003817119890minusA119905V0

10038171003817100381710038171003817+ int

119905

0

10038171003817100381710038171003817119890minusA(119905minus119904)10038171003817100381710038171003817

sdot F (119904 V (119904)) 119889119904

le 119872(100381710038171003817100381711990601003817100381710038171003817 + 119903) +119872int

119905

0

119890120588119904119886 (119904) 119889119904

+ int

119905

0

119872119890120588119904119887 (119904) V (119904) 119889119904

(59)

Now let us put

P1(119905) = 119872 (

1003817100381710038171003817V01003817100381710038171003817 + 119903) +119872int

119905

0

119890120588119904119886 (119904) 119889119904

P2(119905) = 119872119890

120588119904119887 (119905)

(60)

Taking into account Lemma 5 we obtain

V (119905) le P1(119905) + int

119905

0

P2(119904)P1(119904) exp(int

119904

0

P2(120591) 119889120591) 119889119904

(61)

Further let 119878 be the set of all mild solutions V(119905) of (6) withthe initial-value V

0= V(0) isin 119861(119906(0) 119903)

Put

119892 (119905) = sup 119906 (119905) minus V (119905) V isin 119878 (62)

The estimate (61) implies that the function119892(119905) is well definedApplying (119867

5) and (1198671015840

1) we get

119906 (119905) minus V (119905) le10038171003817100381710038171003817119890minusA11990510038171003817100381710038171003817

sdot10038171003817100381710038171199060 minus V

0

1003817100381710038171003817

+ int

119905

0

10038171003817100381710038171003817119890minusA(119905minus119904)10038171003817100381710038171003817

119898 (119904) 119906 (119904) minus V (119904) 119889119904

le 119872119890minus120588119905119903 + int

119905

0

119872119890minus119892(119905minus119904)

119898(119904) 119892 (119904) 119889119904

(63)

or equivalently

119890120588119905119892 (119905) le 119872119903 + int

119905

0

119872119898(119904) 119890120588119904119892 (119904) 119889119904 (64)

Using again Lemma 5 for the above estimate (where ℎ(119905) =119890120588119905119892(119905)) we obtain

119890120588119905119892 (119905) le 119872119903

+ int

119905

0

1198722119903119898 (119904) exp(int

119905

119904

119872119898(120591) 119889120591) 119889119904

(65)

Elementary calculations lead to the following equality

int

119905

0

1198722119903119898 (119904) exp(int

119905

119904

119872119898(120591) 119889120591) 119889119904

= 119872119903(exp(int119905

0

119872119898(119904) 119889119904) minus 1)

(66)

Hence

119892 (119905) le 119872119903 exp (minus120588119905) + 119872119903 exp(int119905

0

119872119898(119904) 119889119904 minus 120588119905)

minus119872119903 exp (minus120588119905) (67)

Applying assumption (1198675) we derive

lim119905rarrinfin

119892 (119905) = 0 (68)

and this proves that 119906(119905) is locally attractive Finally thisequality together with definition of the function 119892(119905) impliesthat 119906(119905) is globally attractive The proof is complete

Remark 15 In the case when 119898(119905) equiv 119898 is constant the fol-lowing condition

lim119905rarrinfin

(int

119905

0

119898(119904) 119889119904 minus120588

119872119905) = minusinfin (69)

means that 119898 lt 120588119872 Observe that this condition cannotbe weakened This observation is illustrated by the followingexampled

Example 16 Let X = R 119898(119905) equiv 119898 F(119905 119906) = sin 119905 + 119898119906119872 = 1 and 119890minusA119905 = 119890minus120588119905 Then the equation

119906 (119905) = 119890minusA1199051199060+ int

119905

0

119890minusA(119905minus119904)

F (119904 119906 (119904)) 119889119904 (70)

(for any fixed 1199060isin R) has the solution 119906(119905) expressed by the

following formula

119906 (119905) =(120588 minus 119898) sin 119905 minus cos 119905

(120588 minus 119898)2

+ 1+ (1199060+

1

(120588 minus 119898)2

+ 1) 119890(119898minus120588)119905

(71)

Notice that for 119898 ge 120588 the solution 119906(119905) is neither globallyattractive nor locally uniformly attractive because for each

Journal of Function Spaces and Applications 7

other solution V(119905) with initial condition V(0) = V0 obviously

described by similar formula as 119906(119905) we would have acontradiction

|119906 (119905) minus V (119905)| =

10038161003816100381610038161199060 minus V0

1003816100381610038161003816 119890(119898minus120588)119905

997888rarr infin

or10038161003816100381610038161199060 minus V

0

1003816100381610038161003816

(72)

5 Asymptotic Behaviour

In this section we will give a theorem describing asymptoticbehaviour of mild solutions of (6) with condition (7) Thistheorem generalizes the result included in [72Theorem 44]First we formulate the assumptions

(1198671015840

3) This condition is almost identical with (119867

3) and the

only difference is that we assume the functions 119886 and119887 are locally essentially bounded on R

+

(1198671015840

5) There exists 119906

infinisin X such that there exists the limit

lim119905rarrinfin

F(119905 119906infin) and

lim119905rarrinfin

F (119905 119906infin) = A

minus1119906infin (73)

Moreover there exists a number119898 lt 120588119872 such that1003817100381710038171003817F (119905 119906) minusF (119905 119906

infin)1003817100381710038171003817 le 119898

1003817100381710038171003817119906 minus 119906infin1003817100381710038171003817 (74)

for 119905 ge 0 and 119906 isin X

Remark 17 The condition (11986710158405) in conjunction with (1198671015840

1)

ensures the existence ofAminus1 (see [72]) Clearly (11986710158403) implies

(1198673)

Theorem 18 Under assumptions (11986710158401) (1198672) (11986710158403) (1198674) and

(11986710158405) (6) with condition (7) has a mild solution 119906 = 119906(119905) for

each 1199060isin X such that lim

119905rarrinfin119906(119905) = 119906

infin

Proof The existence of a mild solution 119906(119905) is guaranteed byTheorem 10 Let us put

119892 (119905) =1003817100381710038171003817119906 (119905) minus 119906infin

1003817100381710038171003817

Finfin= lim119905rarrinfin

F (119905 119906infin)

(75)

We show that

lim119905rarrinfin

119892 (119905) = 0 (76)

Recall that if assumption (11986710158401) is fulfilled then for each 119911 isin X

we have (see [72])

int

infin

0

119890minusA119904

119911 119889119904 = minusAminus1119911 (77)

Using the above fact and (11986710158405) we get

119906infin= (minusA

minus1(minusAminus1119906infin)) = minusA

minus1Finfin

= int

infin

0

119890minusA119904

Finfin119889119904

(78)

Linking the above equality with (11986710158401) we obtain

119892 (119905)

=

10038171003817100381710038171003817100381710038171003817119890minusA1199051199060+ int

119905

0

119890minusA(119905minus119904)

F (119904 119906 (119904)) 119889119904 minus int

infin

0

119890minusA119904

Finfin119889119904

10038171003817100381710038171003817100381710038171003817

le 119872119890minus120588119905 10038171003817100381710038171199060

1003817100381710038171003817 +

10038171003817100381710038171003817100381710038171003817int

infin

119905

119890minusA119904

Finfin119889119904

10038171003817100381710038171003817100381710038171003817

+

10038171003817100381710038171003817100381710038171003817int

119905

0

119890minusA(119905minus119904)

F (119904 119906 (119904)) 119889119904 minus int

119905

0

119890minusA119904

Finfin119889119904

10038171003817100381710038171003817100381710038171003817

le 119872119890minus120588119905 10038171003817100381710038171199060

1003817100381710038171003817 +119872

119890minus1205881199051205881003817100381710038171003817Finfin

1003817100381710038171003817

+

10038171003817100381710038171003817100381710038171003817int

119905

0

119890minusA(119905minus119904)

F (119904 119906 (119904)) 119889119904 minus int

119905

0

119890minusA(119905minus119904)

Finfin119889119904

10038171003817100381710038171003817100381710038171003817

le 119872119890minus120588119905(100381710038171003817100381711990601003817100381710038171003817 +

1003817100381710038171003817Finfin1003817100381710038171003817

120588)

+ int

119905

0

119872119890minus120588(119905minus119904) 1003817100381710038171003817F (119904 119906 (119904)) minusF

infin

1003817100381710038171003817 119889119904

le 119872119890minus120588119905(100381710038171003817100381711990601003817100381710038171003817 +

1003817100381710038171003817Finfin1003817100381710038171003817

120588)

+119872119890minus120588119905int

119905

0

119890120588119904 1003817100381710038171003817F (119904 119906

infin) minusF

infin

1003817100381710038171003817 119889119904

+119872119890minus120588119905int

119905

0

119890120588119904 1003817100381710038171003817F (119904 119906 (119904)) minusF (119904 119906

infin)1003817100381710038171003817 119889119904

(79)

Next putting

119888 (119905) = 119872119890minus120588(119905)

(100381710038171003817100381711990601003817100381710038171003817 +

1003817100381710038171003817Finfin1003817100381710038171003817

120588)

+119872119890minus120588119905int

119905

0

119890120588119904 1003817100381710038171003817F (119904 119906

infin) minusF

infin

1003817100381710038171003817 119889119904

(80)

and applying (11986710158405) we derive the following inequality

119892 (119905) le 119888 (119905) + 119898119872119890minus120588119905int

119905

0

119890120588119904119892 (119904) 119889119904 (81)

Hence

119890120588119905119892 (119905) le 119890

120588(119905)119888 (119905) + 119898119872int

119905

0

119890120588119904119892 (119904) 119889119904 (82)

The above inequality in conjunction with Lemma 5 gives

119890120588119905119892 (119905) le 119890

120588119905119888 (119905)

+ int

119905

0

119890120588119904119888 (119904)119898119872 sdot exp (119898119872(119905 minus 119904)) 119889119904

= 119890120588119905119888 (119905)

+ 119898119872119890119898119872119905

int

119905

0

119888 (119904) exp ((120588 minus 119898119872) 119904) 119889119904

(83)

8 Journal of Function Spaces and Applications

Hence

119892 (119905) le 119888 (119905) +119898119872int

119905

0119888 (119904) 119890(119892minus119898119872)119904

119889119904

exp ((120588 minus 119898119872) 119905) (84)

Before proving (76) we first show that

lim119905rarrinfin

119888 (119905) = 0 (85)

To prove this equality it is sufficient to show that the lastcomponent in the formula expressing 119888(119905) that is 119872119890minus120588119905

int119905

0119890120588119904F(119904 119906

infin) minusF

infin119889119904 tends to zero as 119905 rarr infin

To this end let us fix 120576 gt 0 Assumption (11986710158405) implies that

there exists 1199050such that

sup 1003817100381710038171003817F (119904 119906infin) minusF

infin

1003817100381710038171003817 119904 ge 1199050 le120576120588

2119872 (86)

This inequality together with (11986710158403) implies

ess sup 1003817100381710038171003817F (119904 119906infin) minusF

infin

1003817100381710038171003817 119904 isin R lt infin (87)

Now fix 1199051isin R such that 119905

1ge 119905 and

ess sup 1003817100381710038171003817F (119904 119906infin) minusF

infin

1003817100381710038171003817 119904 isin R

le120576120588 exp (120588119905

1)

2119872 (exp (1205881199050) minus 1)

(88)

Then using (86) and (88) we conclude that for 119905 ge 1199051we have

119872119890minus120588119905int

119905

0

119890120588119904 1003817100381710038171003817F (119904 119906

infin) minusF

infin

1003817100381710038171003817 119889119904

le 119872119890minus120588119905int

1199050

0

119890120588119904 1003817100381710038171003817F (119904 119906

infin) minusF

infin

1003817100381710038171003817 119889119904

+119872119890minus120588119905int

119905

1199050

119890120588119904 1003817100381710038171003817F (119904 119906

infin) minusF

infin

1003817100381710038171003817 119889119904 le120576

2+120576

2= 120576

(89)

This fact proves (85)Further using (1198671015840

5) and (85) and employing de

LrsquoHospitalrsquos rule for the fraction on the right-hand sideof inequality (84) we obtain that condition (76) is satisfiedThis fact completes the proof

6 An Application to the Ginzburg-LandauEquations of Superconductivity

In this section we formulate the gauged time-dependentGinzburg-Landau (TDGL) equations as an abstract evolutionequation in a Hilbert space Moreover we show applicationsof the above theorems to TDGL equations

We assume that Ω is a bounded domain in R119899 withboundary 120597Ω of class11986211That isΩ is an open and connectedset whose boundary 120597Ω is a compact (119899 minus 1)-manifolddescribed by Lipschitz continuous differentiable charts Weconsider two- and three-dimensional problems (119899 = 2 and119899 = 3 resp) Assume that the vector potential A takes its

values in R119899 The vector H will represent the (externally)applied magnetic field which is a function of space andtime similarly to A it takes its values in R119899 The function120574 is defined and satisfies Lipschitz condition on 120597Ω and120574(119909) ge 0 for 119909 isin 120597Ω The parameters in the TDGLequations are 120578 a (dimensionless) friction coefficient and 120581the (dimensionless) Ginzburg-Landau parameter

The order parameter should be thought of as the wavefunction of the center-of-massmotion of the ldquosuperelectronsrdquo(Cooper pairs) whose density is 119899

119904= |120595|

2 and whose fluxis J119904 The vector potential A determines the electromagnetic

field E = minus120597119905A minus nabla120601 is the electric field and B = nabla times A

is the magnetic induction where J the total current is thesum of a ldquonormalrdquo current J

119899= E the supercurrent J

119904 and

the transport current J119905= nabla times H The normal current obeys

Ohmrsquos law J119899= 120590119899E the ldquonormal conductivityrdquo coefficient 120590

119899

is equal to one in the adopted system of units The differenceM = BminusH is known as themagnetizationThe trivial solution(120595 = 0 B = H E = 0) represents the normal state where allsuperconducting properties have been lost

Nowwe accept the following notion all Banach spaces arereal the (real) dual of a Banach space119883 is denoted by1198831015840Thesymbol 119871119901(Ω) for 1 le 119901 le infin denotes the usual Lebesguespace with norm sdot

119871119901 (sdot sdot) is the inner product in 1198712(Ω)

1198821198982(Ω) for nonnegative integer 119898 is the usual Sobolev

space with norm sdot 1198821198982 1198821198982(Ω) is a Hilbert space for the

inner product (sdot sdot)1198982

given by (119906 V)1198982

= sum|120572|le119898

(120597120572119906 120597120572V)

for 119906 V isin 1198821198982(Ω) Fractional Sobolev space1198821199042(Ω) with afractional 119904 is defined by interpolation ([40 Chap VII] and[41 49 50]) 119862]

(Ω) for ] ge 0 ] = 119898 + 120582 with 0 le 120582 lt 1 isthe space of 119898 times continuous differentiable functions onΩ those 119898th-order derivatives satisfy the Holder conditionwith exponent 120582 if ] is a proper fraction the norm sdot

119862] is

defined in the usual wayThe definitions extend to the space of vector-valued

functions in the standard way with the caveat that the innerproduct in [1198712(Ω)]119899 is defined by (119906 V) = int

Ω119906 sdot V where the

symbol (119906 V) indicates the scalar product in R119899 Complex-valued functions are interpreted as vector-valued functionswith two real components

Functions that vary in space and time like the orderparameter and the vector potential are considered as map-pings from the time domain which is a subinterval of [0infin)into spaces of complex- or vector-valued functions definedin Ω Let 119883 = (119883 sdot

119883) be a Banach space of functions

defined inΩ Then the functions are defined in Ω Then thefunctions of space and time defined on Ω times (0 119879) for 119879 gt 0may be considered as elements of 119871119901(0 119879119883) for 1 le 119901 le infinor1198822119898(0 119879119883) for nonnegative119898 or119862]

(0 119879119883) for ] ge 0] = 119898 + 120582 with 0 le 120582 lt 1 Detailed definitions can be foundfor example in [43]

Obviously function spaces of ordered pairs (120595A) where120595 Ω rarr R2 and A Ω rarr R119899 (119899 = 2 3) play an importantrole in the study of the gauged TDGL equationsWe thereforeadapt the following special notationH = [119883(Ω)]

2times[119883(Ω)]

119899for any Banach space for the order parameter 120595 and thevector potential A respectively A suitable framework forthe functional analysis of the gauged TDGL equations is the

Journal of Function Spaces and Applications 9

Cartesian product W1+1205722 = [1198821+1205722

(Ω)]2times [1198821+1205722

(Ω)]119899

where 120572 isin (12 1) This space is continuously imbedded inW12 capLinfin

Assume H = 0 and H isin [1198712(Ω)]119899 Let AH be a minimizer

of the convex quadratic form 119869120596equiv J120596[119860]

119869120596 [A] = int

Ω

[120596(nabla sdot A)2 + |nabla times A minusH|2] 119889119909 (90)

on the domain

D (119869120596) = A isin [119882

12(Ω)]119899

n sdot A = 0 on 120597Ω (91)

We now introduce the reduced vector potential A1015840

A1015840 = A minus AH (92)

In terms of 120595 and A1015840 the gauged TDGL equations have thefollowing form

120597120595

120597119905minus

1

1205781205812Δ120595 = 120593 in Ω times (0infin)

120597A1015840

120597119905+ nabla times nabla times A1015840 minus 120596nabla (nabla sdot A1015840) = F in Ω times (0infin)

n sdot nabla + 120574120595 = 0 n sdot A1015840 = 0

n times (nabla sdot A1015840) = 0 on 120597Ω times (0infin)

(93)

Here 120593 and F are nonlinear functions of 120595 and A1015840

120593 equiv 120593 (119905 120595A1015840) = 1

120578[minus2119894

120581(nabla120595) sdot (A1015840 + AH)

minus119894

120581(1 minus 120578120581

2120596)120595 (nabla sdot A1015840)

minus12059510038161003816100381610038161003816A1015840 + AH

10038161003816100381610038161003816

2

+ (1 minus100381610038161003816100381612059510038161003816100381610038162

) 120595]

(94)

F equiv F (119905 120595A1015840) = 1

2119894119896(120595lowastnabla120595 minus 120595nabla120595

lowast)

minus100381610038161003816100381612059510038161003816100381610038162

(A1015840 + AH) minus120597AH120597119905

(95)

The equations are supplemented by initial data which is inthe followimg form

120595 = 1205950 A = A

0on Ω times 0 (96)

where 1205950and A

0are given and by (92) we have

120595 = 1205950 A1015840 = A

0minus AH (0) on Ω times 0 (97)

In the next part we connect the evolution of the solution(120595A1015840) of the system of (93) with the initial data (120595

0A0minus

AH(0)) with the dynamics of a vector 119906 in a Hilbert space

L2= [119871119901(Ω)]2

times [119871119901(Ω)]119899

(98)

The following analysis is restricted to the case 120596 gt 0 andthe case 120596 = 0 (see [73])

Let vector 119906 [0infin) rarr L2 represent the pair (120595A1015840)

119906 = (120595A1015840) equiv (120595A minus AH) (99)

and letA be the linear self-adjoint operator inL2 associatedwith the quadratic form 119876

120596equiv 119876120596[119906] given by the following

formula

119876120596 [119906] = int

Ω

[1

12057812058121003816100381610038161003816nabla120595

10038161003816100381610038162

+ 120596(nabla sdot A1015840)2

+10038161003816100381610038161003816nabla times A101584010038161003816100381610038161003816

2

] 119889119909

+ int120597Ω

120574

1205781205812100381610038161003816100381612059510038161003816100381610038162

119889120590 (119909)

(100)

on the domain

D (119876120596) = D (A

12)

= 119906 = (120595A1015840) isinW12 n sdot A1015840 = 0 on 120597Ω

(101)

The quadratic form 119876120596is nonnegative Furthermore since

120596 gt 0119876120596[120595A1015840]+119888 120595

1198712 is coercive onW12 for any constant

119888 gt 0 Hence A is positively definite in L2 ([44 Chap 1]equation (545)) If it does not lead to confusion we use thesame symbol A for the restriction A

120595and AA of A to the

respective linear subspace [1198712(Ω)]2 = [1198712(Ω)]2 times 0 (for 120595)and [1198712(Ω)]119899 equiv 0 times [1198712(Ω)]119899 (for A) ofL2

Now consider the initial-value problems (6) and (7) inL2 whereF(119905 119906) = (120593 F) 120593 and F is given by (94) and (95)and 119906

0= (1205950A0minus AH(0))

With 120582 isin (12 1) and 1199060isin W1+1205822 we say that 119906 are a

solution of (6) and (7) on the interval [0 119879] for some 119879 gt 0if 119906 [0 119879] rarr W1+1205822 is continuous and

119906 (119905) = 119890minusA(119905)

+ int

119905

0

119890minusA(119905minus119904)

F (119904 119906 (119904)) 119889119904 for 119905 isin [0 119879]

(102)

in L2 A mild solution of the initial-value problems (6) and(7) defines a weak solution (120595A1015840) of the boundary valueproblem (93) which in turn defines a weak solution (120595A)of the gauged TDGL equations provided AH is sufficientlyregular

Namely let us assume that X = L2 and 1199060isin L2 and

119906 R+times R119899 rarr R119899 for 119899 isin N is an unknown function 119906 =

119906(119905 119909) In order to applyTheorems 10 and 14 we are not goingto consider 119906 as a function of 119905 and 119909 together but rather asa mapping 119906 of variable 119905 into the spaceX =L2 of functions119909 that is 119906 R

+rarr L2 119906(119905)(119909) = 119906(119905 119909) 119909 isin R119899 119905 isin R

+

Remark 19 Since A is the linear self-adjoint operator inL2

associated with the quadratic form (100) thenA generates asemigroup 119890minusA119905

119905ge0(see [74])

Below we formulate the principal theorem of this paperThis theorem is a simple consequence of Theorem 10

10 Journal of Function Spaces and Applications

Theorem 20 Under assumptions (1198671)ndash(1198674) problem (6)-(7)

is equivalent to the system of (1)ndash(5) (TDGL equations) andhas for each 119906

0isin L2 at least one mild solution in sense of

Definition 4 119906 = 119906(119905) Moreover each solution 119906(119905) satisfiesestimate (25)

We observe that in the case when 119890minusA119905119905ge0

is the semi-group of contractions and 119886(119905) equiv 119886 119887(119905) equiv 119887 for 119905 isin

R+are constant after simple calculations based on estimate

(25) we get that the solution 119906 = 119906(119905) has the asymptoticcharacterization

119906 (119905) = 119874 (1199053) as 119905 997888rarr infin (103)

For further purposes let us formulate the followingassumption

(A) A is the infinitesimal generator of an exponentiallystable 119862

0-semigroup 119890minusA119905

119905ge0

The next result of this chapter is shown in the giventheorem below

Theorem21 Under assumptions (A) and (1198672)ndash(1198674) problem

(6)-(7) which is equivalent to the systems equations (1)ndash(5)(TDGL equations) has a mild solution 119906 = 119906(119905) for each1199060isin L2 which is globally attractive and locally uniformly

attractive

References

[1] V L Ginzburg and L D Landau ldquoOn the theory of supercon-ductivityrdquo Zhurnal Eksperimentalrsquonoi i Teoreticheskoi Fiziki vol20 pp 1064ndash1082 1950 English translation in L D LandauMen of Physics D ter Haar ed Vol I Pergamon Press OxfordUK pp 138ndash167 1965

[2] S Aizicovici and H Lee ldquoNonlinear nonlocal Cauchy problemsin Banach spacesrdquo Applied Mathematics Letters vol 18 no 4pp 401ndash407 2005

[3] S Aizicovici and M McKibben ldquoExistence results for a classof abstract nonlocal Cauchy problemsrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 39 no 5 pp 649ndash6682000

[4] MMcKibbenDiscovering Evolution Equation with ApplicationCRC Press Taylor amp Francis Group 2011

[5] S Aizicovici and V Staicu ldquoMultivalued evolution equationswith nonlocal initial conditions in Banach spacesrdquo NonlinearDifferential Equations and Applications vol 14 no 3-4 pp 361ndash376 2007

[6] R P Agarwal and D OrsquoRegan Infinite Interval Problems for Dif-ferential Difference and Integral Equations Kluwer AcademicPublishers Dordrecht The Netherlands 2001

[7] Z Fan ldquoExistence of nondensely defined evolution equationswith nonlocal conditionsrdquo Nonlinear Analysis Theory Methodsamp Applications A vol 70 no 11 pp 3829ndash3836 2009

[8] Z Fan ldquoImpulsive problems for semilinear differential equa-tions with nonlocal conditionsrdquo Nonlinear Analysis TheoryMethods amp Applications A vol 72 no 2 pp 1104ndash1109 2010

[9] Z Fan and G Li ldquoExistence results for semilinear differentialequations with nonlocal and impulsive conditionsrdquo Journal ofFunctional Analysis vol 258 no 5 pp 1709ndash1727 2010

[10] J Garcıa-Falset ldquoExistence results and asymptotic behavior fornonlocal abstract Cauchy problemsrdquo Journal of MathematicalAnalysis and Applications vol 338 no 1 pp 639ndash652 2008

[11] J Liang J H Liu and T-J Xiao ldquoNonlocal impulsive problemsfor nonlinear differential equations in Banach spacesrdquo Mathe-matical and Computer Modelling vol 49 no 3-4 pp 798ndash8042009

[12] L Zhu and G Li ldquoOn a nonlocal problem for semilinear differ-ential equations with upper semicontinuous nonlinearities ingeneral Banach spacesrdquo Journal of Mathematical Analysis andApplications vol 341 no 1 pp 660ndash675 2008

[13] Q Liu and R Yuan ldquoExistence of mild solutions for semilinearevolution equations with non-local initial conditionsrdquo Nonlin-ear AnalysisTheoryMethodsampApplications A vol 71 no 9 pp4177ndash4184 2009

[14] C Xie W Zhong and Z Fan ldquoExistence results for nonlinearnonlocal problems in Banach spacesrdquo Applied MathematicsLetters vol 22 no 7 pp 998ndash1002 2009

[15] T-J Xiao and J Liang ldquoExistence of classical solutions tononautonomous nonlocal parabolic problemsrdquoNonlinear Anal-ysisTheoryMethods and Applications vol 63 no 5-7 pp e225ndashe232 2005

[16] X Xue ldquo119871119901 theory for semilinear nonlocal problems with mea-sure of noncompactness in separable Banach spacesrdquo Journal ofFixed Point Theory and Applications vol 5 no 1 pp 129ndash1442009

[17] X Xue ldquoNonlinear differential equations with nonlocal condi-tions in Banach spacesrdquo Nonlinear Analysis Theory Methods ampApplications A vol 63 no 4 pp 575ndash586 2005

[18] X Xue ldquoExistence of solutions for semilinear nonlocal Cauchyproblems in Banach spacesrdquo Electronic Journal of DifferentialEquations vol 64 p 7 2005

[19] X Xue ldquoNonlocal nonlinear differential equations with a mea-sure of noncompactness in Banach spacesrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 70 no 7 pp 2593ndash26012009

[20] L Olszowy ldquoExistence ofmild solutions for the semilinear non-local problem in Banach spacesrdquo Nonlinear Analysis TheoryMethods amp Applications A vol 81 pp 211ndash223 2013

[21] D Bothe ldquoMultivalued perturbations of119898-accretive differentialinclusionsrdquo Israel Journal of Mathematics vol 108 pp 109ndash1381998

[22] H-S Ding J Liang G M NrsquoGuerekata and T-J XiaoldquoMild pseudo-almost periodic solutions of nonautonomoussemilinear evolution equationsrdquo Mathematical and ComputerModelling vol 45 no 5-6 pp 579ndash584 2007

[23] J Garcıa-Falset ldquoThe asymptotic behavior of the solutions of theCauchy problem generated by 120593-accretive operatorsrdquo Journal ofMathematical Analysis and Applications vol 310 no 2 pp 594ndash608 2005

[24] Z Fan and G Li ldquoAsymptotic behavior of the solutions of non-autonomous systems in Banach spacesrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 68 no 12 pp 3733ndash37412008

[25] J Banas ldquoOn existence theorems for differential equations inBanach spacesrdquo Bulletin of the Australian Mathematical Societyvol 32 no 1 pp 73ndash82 1985

[26] J Banas ldquoMeasures of noncompactness in the space of contin-uous tempered functionsrdquo Demonstratio Mathematica vol 14no 1 pp 127ndash133 1981

Journal of Function Spaces and Applications 11

[27] J Banas and K GoebelMeasures of Noncompactness in BanachSpaces vol 60 ofLectureNotes in Pure andAppliedMathematicsMarcel Dekker New York NY USA 1980

[28] J Banas and M Lecko ldquoSolvability of infinite systems ofdifferential equations in Banach sequence spacesrdquo Journal ofComputational andAppliedMathematics vol 137 no 2 pp 363ndash375 2001

[29] J Banas and K Sadarangani ldquoCompactness conditions inthe study of functional differential and integral equationsrdquoAbstract and Applied Analysis vol 2013 Article ID 819315 14pages 2013

[30] M Mursaleen and S A Mohiuddine ldquoApplications of mea-sures of noncompactness to the infinite system of differentialequations in ℓ

119901spacesrdquo Nonlinear Analysis Theory Methods amp

Applications A vol 75 no 4 pp 2111ndash2115 2012[31] L Olszowy ldquoSolvability of some functional integral equationrdquo

Dynamic Systems and Applications vol 18 no 3-4 pp 667ndash6762009

[32] L Olszowy ldquoOn some measures of noncompactness in theFrechet spaces of continuous functionsrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 72 pp 2119ndash2126 2010

[33] L Olszowy ldquoOn existence of solutions of a quadratic Urysohnintegral equation on an unbounded intervalrdquo CommentationesMathematicae vol 48 no 1 pp 103ndash112 2008

[34] L Olszowy ldquoExistence ofmild solutions for semilinear nonlocalCauchy problems in separable Banach spacesrdquo Journal of Anal-ysis and its Applications vol 32 no 2 pp 215ndash232 2013

[35] L Olszowy ldquoFixed point theorems in the Frechet space 119862(119877+)

and functional integral equations on an unbounded intervalrdquoApplied Mathematics and Computation vol 218 no 18 pp9066ndash9074 2012

[36] J Banas and D OrsquoRegan ldquoOn existence and local attractivity ofsolutions of a quadratic Volterra integral equation of fractionalorderrdquo Journal of Mathematical Analysis and Applications vol345 no 1 pp 573ndash582 2008

[37] L D Landau EM Lifshitz and L P Pitaevski Electrodynamicsof Continuous Media vol 8 Butterworth-Heinemann 2ndedition 1984

[38] L D Landau and E M Lifshitz Fluid Mechanics vol bButterworth-Heinemann 2nd edition 1987

[39] R Temam Infinite-Dimensional Dynamical Systems inMechan-ics and Physics vol 68 Springer New York NY USA 2ndedition 1997

[40] J K Hale Asymptotic Behavior of Dissipative Systems vol 25 ofMathematical Surveys and Monographs American Mathemati-cal Society Providence RI USA 1988

[41] R A Adams and J J F Fournier Sobolev Spaces vol 140Elsevier 2nd edition 2003

[42] V Mazrsquoya Sobolev Spaces with Applications to Elliptic PartialDifferential Equations Springer Heidelberg Germany 2ndedition 2011

[43] V Georgescu ldquoSome boundary value problems for differentialforms on compact Riemannian manifoldsrdquo Annali di Matemat-ica Pura ed Applicata vol 122 pp 159ndash198 1979

[44] D Henry Geometric Theory of Semilinear Parabolic Equationsvol 840 of Lecture Notes in Mathematics Springer New YorkNY USA 1981

[45] V Girault and P-A Raviart Finite Element Methods for Navier-Stokes Equations Springer New York NY USA 1986

[46] J Bardeen L N Cooper and J R Schrieffer ldquoTheory ofsuperconductivityrdquo Physical Review vol 108 pp 1175ndash12041957

[47] A A Abrikosov Fundamental of the Theory of Metals North-Holland Amstedram The Netherlands 1988

[48] M Tinkham Introduction To Superconductivity McGraw-HillNew York NY USA 2nd edition 1996

[49] P DeGennes Superconductivity inMetals and Alloys BenjaminNew York NY USA 1966

[50] H Brezis Functional Analysis Sobolev Spaces and PartialDifferential Equations Springer New York NY USA 2011

[51] E Zeidler Nonlinear Functional Analysis and its Applicationsvol 2 Springer New York NY USA 1990

[52] A Schmid ldquoA time dependent Ginzburg-Landau equation andits application to the problem of resistivity in the mixed staterdquoPhysik der KondensiertenMaterie vol 5 no 4 pp 302ndash317 1966

[53] C M Elliot and Q Tang Existence Theorems For an Evolution-ary Superconductivity Model University of Sussex BrightonUK 1992

[54] Q Tang ldquoOn an evolutionary system of Ginzburg-Landauequations with fixed total magnetic fluxrdquo Communications inPartial Differential Equations vol 20 no 1-2 pp 1ndash36 1995

[55] Q Du ldquoGlobal existence and uniqueness of solutions of thetime-dependent Ginzburg-Landau model for superconductiv-ityrdquo Applicable Analysis vol 53 no 1-2 pp 1ndash18 1994

[56] Z M Chen K-H Hoffmann and J Liang ldquoOn a nonstation-ary Ginzburg-Landau superconductivity modelrdquoMathematicalMethods in the Applied Sciences vol 16 no 12 pp 855ndash875 1993

[57] J Liang and Q Tang ldquoAsymptotic behaviour of the solutionsof an evolutionaryGinzburg-Landau superconductivitymodelrdquoJournal of Mathematical Analysis and Applications vol 195 no1 pp 92ndash107 1995

[58] L P Gorrsquokov and G M Eliasherberg ldquoGeneralization of theGinzburg-Landau equations for an nonstationary problemsin the case of alloys with paramagnetic impartiesrdquo ZhurnalEksperimentalrsquonoi i Teoreticheskoi Fiziki vol 54 pp 612ndash6261968 Soviet Physics vol 27 pp 328ndash334 1968

[59] J Fleckinger-Pelle and H G Kaper ldquoGauges for the Ginzburg-Landau equations of superconductivityrdquo Zeitschrift fur Ange-wandte Mathematik und Mechanik vol 76 no 2 pp 345ndash3481996

[60] P Takac ldquoOn the dynamical process generated by supercon-ductivity modelrdquo Zeitschrift fur Angewandte Mathematik undMechanik vol 76 no 52 pp 349ndash352 1995

[61] W D Gropp H G Kaper DM Levine M Palumbo and VMVinokur ldquoNumerical simulation of vortex dynamics in type-IIsuperconductorsrdquo Journal of Computational Physics vol 123 no2 pp 254ndash266 1996

[62] D W Braun G W Crabtree H G Kaper et al ldquoStructure of amoving vortex latticerdquo Physical Review Letters vol 76 no 5 pp831ndash834 1996

[63] G W Crabtree G K Leaf H G Kaper et al ldquoTime-dependent Ginzburg-Landau simulations of vortex guidance bytwin boundariesrdquo Physica C vol 263 no 1ndash4 pp 401ndash408 1996

[64] G W Carabtree G K Leaf H G Kaper D W Braun V MVinokur andA E Koshelev ldquoDynamics vortex phases in super-conductors with correlated disorderrdquo Preprint ANLMCS-P590-0496 Mathematics and Computer Science DivisionAragonne National Laboratory 1996

[65] Th Gallay ldquoA center-stable manifold theorem for differentialequations in Banach spacesrdquo Communications in MathematicalPhysics vol 152 no 2 pp 249ndash268 1993

[66] J-M Ghidaglia and B Heron ldquoDimension of the attractorsassociated to the Ginzburg-Landau partial differential equa-tionrdquo Physica D vol 28 no 3 pp 282ndash304 1987

12 Journal of Function Spaces and Applications

[67] P Mattila Geometry of Sets and Measures in Euclidean Spacesvol 44 Cambridge University Press Cambridge UK 1995

[68] L HormanderThe Analysis of Linear Partial Differential Oper-ators vol 1 Springer Berlin Germany 2nd edition 1990

[69] H Amann Linear and Quasilinear Parabolic ProblemBikhauser Basel Switzerland 1995

[70] H-P Heinz ldquoOn the behaviour ofmeasures of noncompactnesswith respect to differentiation and integration of vector-valuedfunctionsrdquo Nonlinear Analysis Theory Methods amp ApplicationsA vol 7 no 12 pp 1351ndash1371 1983

[71] R P Agarwal M Meehan and D OrsquoRegan Fixed Point Theoryand Applications vol 141 of Cambridge Tracts in MathematicsCambridge University Press Cambridge UK 2001

[72] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer New York NY USA1983

[73] J Fleckinger-Pelle H G Kaper and P Takac ldquoDynamics of theGinzburg-Landau equations of superconductivityrdquo NonlinearAnalysis Theory Methods amp Applications A vol 32 no 5 pp647ndash665 1998

[74] H G Kaper and P Takac ldquoAn equivalence relation for theGinzburg-Landau equations of superconductivityrdquo Journal ofApplied Mathematics and Physics vol 48 no 4 pp 665ndash6751997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Function Spaces and Applications

However only in a few papers problem (6)-(7) was con-sidered on an unbounded interval [10 22] Additionallyin assumptions concerning the semigroup 119890

minusA(119905minus119904)119905ge0

orthe function F(119905 119906) rather restrictive conditions have beenimposedwhich frequently require the compactness ofF(119905 119906)or 119890minusA(119905minus119904)

119905ge0or equicontinuity of semigroup 119890minusA(119905minus119904)

119905ge0

[2ndash11 13ndash22] It is worthwhile mentioning that only a fewpapers have discussed asymptotic behaviour of solutionsmostly without the formulation of existence theorems [10 2324]

In this paper we present conditions guaranteeing theexistence of mild solutions on an unbounded interval ofproblem (6)-(7) We dispense with assumptions on thecompactness ofF(119905 119906) or 119890minusA(119905minus119904)

119905ge0

Moreover we formulate theorems about asymptotic prop-erties and both local and global attractivity of solutions ofproblem (6)-(7) The existence theorems concerning thatproblem will be proved with the help of the technique of afamily of measures of noncompactness in the Frechet space119862(R+X) and Schauder-Tychonoff fixed-point principle

The approach applied here was introduced and developedin [20 25ndash35] for instance

The paper is organized as follows In Section 2 there aregiven notation and auxiliary facts that are needed further onIn Section 3 we formulate and prove a theorem on the exis-tence of mild solution of (6) with condition (7) Moreoverthe rate of the growth of those solutions is characterized

Section 4 contains a theorem on local and global attrac-tivity of solutions of problem (6)-(7) In Section 5 we give atheorem describing the asymptotic behaviour of solutions of(6)-(7)

Finally in Section 6 we formulate the gauged TDGLequation as an abstract evolution equation in a Hilbertspace Moreover this section is devoted to present examplesof application of previously obtained theorems for TDGLequations

In order to convert the systems equations (1)ndash(5) to theCauchy problem (6)-(7) we are based on the papers andmonographs (see [1 30 36ndash69])

2 Preliminaries

Let (X sdot ) be a real Banach space with the zero element 120579Denote by 119861(119909 119903) the closed ball inX centered at 119909 and withradius 119903 If119883 is a subset of a linear topological space then thesymbols 119883 and Conv119883 stand for the closure and the convexclosure of119883 respectively

Let 120594 denote the Hausdorff measure of noncompactnessinX defined on bounded subsets119883 ofX in the followingway(see [27])

120594 (119883) = inf 119903 gt 0 119883 can be covered with finite

numbers of balls of radius 119903 (8)

Further denote by 119862(R+X) the Frechet space consisting

of all functions defined and continuous on R+with values in

a Banach space X The space 119862(R+X) is furnished with the

family of seminorms

119906119899 = sup 119906 (119905) 119905 isin [0 119899]

for 119899 = 1 2 119906 isin 119862 (R+X)

(9)

Let us recall two facts

(lowast) a sequence 119906119899infin

119899=1is convergent to 119906 in 119862(R

+X) if

and only if 119906119899infin

119899=1is uniformly convergent to 119909 on

compact subsets of R+

(lowastlowast) a subset 119880 sub 119862(R+X) is relatively compact if and

only if the restrictions to [0 119879] of all functions from119880

form an equicontinuous set for each119879 gt 0 and119880(119905) isrelatively compact inX for each 119905 isin R

+ where119880(119905) =

119906(119905) 119906 isin 119880

Moreover we recall that a nonempty subset 119880 sub

119862(R+X) is said to be bounded if

sup 119906119899 119906 isin 119880 lt infin for 119899 = 1 2 (10)

Further the family of all nonempty and bounded subsetsof 119862(R

+X) will be denoted by M

119862 while the family of

all nonempty and relatively compact subsets of 119862(R+X) is

denoted byN119862 ObviouslyN

119862subM119862

We will use a family of measures of noncompactness120583119879119879ge0

in the Frechet space 119862(R+X) which was introduced

in [20 35] In order to define these measures recall somequantities [25ndash27] Let us fix a nonempty bounded subset 119880of the space 119862(R

+X) For 119906 isin 119880 120576 ge 0 and 119879 ge 0 denote

by 120596119879(119906 120576) the modulus of continuity of the function 119906 on theinterval [0 119879] that is

120596119879(119906 120576) = sup 1003817100381710038171003817119906 (1199052) minus 119906 (1199051)

1003817100381710038171003817

1199051 1199052isin [0 119879]

10038161003816100381610038161199052 minus 11990511003816100381610038161003816 le 120576

(11)

Further let us put

120596119879(119880 120576) = sup 120596119879 (119906 120576) 119906 isin 119880

120596119879

0(119880) = lim

120576rarr0+

120596119879(119880 120576)

(12)

Obviously the set119880 is equicontinuous on [0 119879] if and only if120596119879

0(119880) = 0Next let us define the functions 120583

119879119879ge0

on the family ofbounded subsets of 119862(R

+X) by the following formula

120583119879(119880) = 120596

119879

0(119880) + sup 120594 (119880 (119905)) 119905 isin [0 119879] (13)

It can be shown that the family 120583119879119879ge0

has the followingproperties

1119900 the family ker120583

119879 = 119880 isin M

119862 120583119879(119880) = 0 for 119879 ge

0 is nonempty and ker120583119879 = N

119862

2119900119880 sub 119881 rArr 120583

119879(119880) le 120583

119879(119881) for 119879 ge 0

3119900120583119879(Conv119880) = 120583

119879(119880) for 119879 ge 0

Journal of Function Spaces and Applications 3

4119900 if 119880

119899infin

119899=1is a sequence of closed sets from M

119862such

that 119880119899+1

sub 119880119899(119899 = 1 2 ) and if lim

119899rarrinfin120583119879(119880119899) =

0 for each 119879 ge 0 then the intersection set 119880infin

=

⋂infin

119899=1119880119899is nonempty

Remark 1 Observe that in contrast to the definition of theconcept of a measure of noncompactness given in [27] ourmapping 120583

119879may take the valueinfin Moreover a single map-

ping 120583119879is not the measure of noncompactness in 119862(R

+X)

but the whole family 120583119879119879ge0

can be called family of measuresof noncompactness

Remark 2 Let us notice that the intersection set 119880infin

de-scribed in axiom 4

119900 is a member of the kernel of the familyof measure of noncompactness 120583

119879119879ge0

and therefore 119880infin

is compact in 119862(R+X) In fact the inequality 120583

119879(119880infin) le

120583119879(119880119899) for 119899 = 1 2 and 119879 ge 0 implies that 120583

119879(119880infin) = 0

Hence 119880infinisin ker120583

119879 This property of the set 119880

infinwill be

very important in our further investigations

Definition 3 A set 119890minusA119905119905ge0

of bounded linear operators onX is called 119862

0-semigroup if

(i) 119890minusA0 = 119868 119890minusA(119905+119904) = 119890minusA119905119890minusA119904 for 119905 119904 ge 0(ii) for all 119909 isin X the function R

+ni 119905 997891rarr 119890

minusA119905119909 is

continuous in R+

Further denote

119873(119879) = sup 10038171003817100381710038171003817119890minusA11990510038171003817100381710038171003817

119905 le 119879 for 119879 ge 0 (14)

Definition 4 A function 119909 isin 119862(R+X) is said to be a mild

solution of the nonlocal initial-value problem (6)-(7) if forevery 119905 isin R

+

119906 (119905) = 119890minusA1199051199060+ int

119905

0

119890minusA(119905minus119904)

F (119904 119906 (119904)) 119889119904 (15)

To prove the existence results in this paper we need thefollowing lemmas

Lemma 5 (see [21]) If 119884 is a bounded subset of Banach spaceX then for each 120576 gt 0 there is a sequence 119910

119896infin

119896=1sub 119884 such that

120594 (119884) le 2120594 (119910119896infin

119896=1) + 120576 (16)

We call a set119880 sub 1198711([0 119879]X)uniformly integrable if there

exists 120578 isin 1198711([0 119879]R+) such that 119906(119904) le 120578(119904) for 119906 isin 119880 and

ae 119904 isin [0 119879]

Lemma 6 (see [70]) If 119906119896infin

119896=1sub 1198711([0 119879]X) is uniformly

integrable then 120594(119906119896(119905)infin

119896=1) is measurable and

120594(int

119905

0

119906119896(119904) 119889119904

infin

119896=1

)le2int

119905

0

120594 (119906119896(119904)infin

119896=1) 119889119904 119891119900119903 119905isin[0 119879]

(17)

Lemma 7 (see [34]) Assume that a set 119880 sub 119862(R+X) is

bounded Then

(i) sup119905isin[0119879]

120594(119880(119905)) le 120594(119880([0 119879])) le 120596119879

0(119880) + sup

119905isin[0119879]

120594(119880(119905))(ii) sup

119905isin[0119879]120594(119880(119905)) le 120594

119862(119880) le 120596

119879

0(119880) + sup

119905isin[0119879]

120594(119880(119905))

for 119879 ge 0 where 119880([0 119879]) = 119906(119905) 119905 isin [0 119879] 119906 isin 119880

Lemma 8 (Gronwallrsquos inequality [71]) Assume that the func-tion 120601

1 R+rarr R

+is measurable and the function 120601

2

R+rarr R

+is locally integrable Moreover we assume that

int119905

01206011(119904)1206012(119904)119889119904 lt infin for 119905 ge 0 and a measurable function

120595 R+rarr R+satisfies the following inequality

120595 (119905) le 1206011(119905) + int

119905

0

1206012(119904) 120595 (119904) 119889119904 119891119900119903 119905 ge 0 (18)

Then

120595 (119905) le 1206011(119905) + int

119905

0

1206011(119904) 1206012(119904) exp(int

119905

119904

1206012(120591) 119889120591) 119889119904

119891119900119903 119905 ge 0

(19)

Next we consider the operator 119865 119862(R+X) rarr

119862(R+X) defined by the following formula

(119865119906) (119905) = int

119905

0

119890minusA(119905minus119904)

F (119904 119906 (119904)) 119889119904 119905 isin R+

119906 isin 119862 (R+X)

(20)

Lemma 9 (see [20]) Assume that conditions (1198671) (1198672) and

(1198674) are satisfied (see Section 3) a set 119880 sub 119862(R

+X) is

bounded 119879 gt 0 a function ℎ [0 119879] rarr R+is measurable

and

2119873 (119879) 120594 (F (119904 119880 (119904))) le ℎ (119904) (21)

for ae 119904 isin [0 119879] Then

120596119879

0(119865119880) le int

119879

0

ℎ (119904) 119889119904 (22)

3 Main Result

In this section we give an existence result for the semilinearequation of evolution (6)-(7) and we describe the asymptoticbehaviour of those solutions

First we will assume that the functions involved in (6)satisfy the following conditions

(1198671) A is a linear operator acting from 119863(A) sub X and A

generates 1198620-semigroup 119890minusA(119905minus119904)

119905ge0

(1198672) the mapping F R

+times X rarr X satisfies the

Caratheodory condition that isF(sdot 119906) is measurablefor 119906 isin X andF(119905 sdot) is continuous for ae 119905 isin R

+

(1198673) 119886 R

+rarr R

+and 119887 R

+rarr R

+are locally

integrable functions such that

F (119905 119906) le 119886 (119905) + 119887 (119905) 119906 (23)

for any 119906 isin X and ae 119905 isin R+

4 Journal of Function Spaces and Applications

(1198674) There exists a locally integrable function 119871 R

+rarr

R+such that for any bounded119863 sub X

120594 (F (119905 119863)) le 119871 (119905) 120594 (119863) (24)

for ae 119905 isin R+

Theorem 10 Under assumptions (1198671)ndash(1198674) (6) with initial

condition (7) has for every 1199060isin X at least one mild solution

119906 = 119906(119905) which satisfied the following estimate

119906 (119905) le 119872 (119905)

+ 119873 (119905) int

119905

0

119887 (119904)119872 (119904) exp(int119905

119904

119887 (120591)119873 (120591) 119889120591) 119889119904

(25)

where

119872(119905) = 119873 (119905) (100381710038171003817100381711990601003817100381710038171003817 + int

119905

0

119886 (119904) 119889119904) (26)

Proof Consider the operator 119866 119862(R+X) rarr 119862(R

+X)

defined by the following formula

119866119906 (119905) = 119890minusA1199051199060+ int

119905

0

119890minusA(119905minus119904)

F (119904 119906 (119904)) 119889119904 (27)

Now let us observe that for any continuous function 119906

R+rarr X in view of condition (119867

3) we get the following

estimate

119866119906 (119905) le 119873 (119905)100381710038171003817100381711990601003817100381710038171003817 + 119873 (119905) int

119905

0

(119886 (119904) + 119887 (119904) 119906 (119904)) 119889119904

(28)

which yields

119866119906 (119905) le 119872 (119905) + 119873 (119905) int

119905

0

119887 (119904) 119906 (119904) 119889119904 (29)

Next consider the following integral equation

P (119905) = 119872 (119905) + 119873 (119905) int

119905

0

119887 (119904)P (119904) 119889119904 (30)

Solving this equation by standard methods we get

P (119905)=119872 (119905)+119873 (119905)int

119905

0

119887 (119904)119872 (119904) exp(int119905

119904

119887 (120591)119873 (120591) 119889120591) 119889119904

(31)

The function P(119905) is continuous nonnegative and nonde-creasing Observe that the following implication is true

119906 (119905) le P (119905) 997904rArr 119866119906 (119905) le P (119905) for 119905 ge 0 (32)

Indeed linking (29) and (32) we have

119866119906 (119905) le 119872 (119905) + 119873 (119905) int

119905

0

119887 (119904) 119906 (119904) 119889119904

le 119872 (119905) + 119873 (119905) int

119905

0

119887 (119904)P (119904) 119889119904 = P (119905)

(33)

In the space 119862(R+X) let us consider the following set

Γ = 119906 isin 119862 (R+X) 119906 (119905) le P (119905) for 119905 ge 0 (34)

Obviously the set Γ is convex and closed Moreover in viewof (32) we have that 119866 is a self-mapping of Γ

Using the criterion of convergence (lowast) in 119862(R+X) and

standard techniques (see [31ndash33 35]) we can show that theoperator 119866 is continuous on 119862(R

+X)

Now we consider the sequence of sets Γ119899 defined by

induction as follows

Γ0= Γ Γ

119899+1= Conv (119866Γ

119899) for 119899 = 0 1 (35)

This sequence is decreasing that is Γ119899sup Γ119899+1

for 119899 =

0 1 2 Further let us fix 119879 gt 0 and for 119905 isin [0 119879] let us put

119892119899(119905) = sup 120594 (Γ

119899(119904)) 119904 isin [0 119905] (36)

119908119899= 120596119879

0(Γ119899) (37)

The sequences 119908119899 and 119892

119899(119905) are nonincreasing for all 119905 isin

[0 119879] so they have limits

119892infin(119905) = lim

119899rarrinfin119892119899(119905) 119905 isin [0 119879]

119908infin= lim119899rarrinfin

119908119899

(38)

Moreover each function 119892119899is nondecreasing therefore 119892

119899

and 119906infin

are measurable on [0 119879] for 119899 = 1 2 Now we apply the family of measures of noncompactness

120583119879119879ge0

defined in 119862(R+X) by formula (13) In view of the

above notation we have

120583119879(Γ119899) = 119908119899+ 119892119899(119879) (39)

We show that

119908infin= 119892infin(119879) = 0 (40)

To fix a number 119905 isin [0 119879] 119899 isin N and take an arbitrarynumber 119904 isin [0 119905]

We know from Lemma 5 that for any 120576 gt 0 there is asequence 119910

119896infin

119896=1sub 119866Γ119899(119904) such that

120594 (119866Γ119899(119904)) le 2120594 (119910

119896infin

119896=1) + 120576 (41)

This implies that there is a sequence 119906119896infin

119896=1sub Γ119899 such that

119910119896= (119866119906

119896) (119904) for 119896 = 1 2 (42)

Journal of Function Spaces and Applications 5

Hence in view of Lemma 6 (1198674) and (36) we obtain

120594 (119866Γ119899(119904)) le 2120594 (119910

119896infin

119896=1) + 120576

= 2120594 ((119866119906119896) (119904)infin

119896=1) + 120576 le 2120594 (119890

minusA1199041199060)

+ 2120594(int

119904

0

119890minusA(119904minus120591)

F (120591 119906119896(120591)) 119889120591

infin

119896=1

)

+ 120576 le 4int

119904

0

120594 (119890minusA(119904minus120591)

F (120591 119906119896(120591)infin

119896=1)) 119889120591 + 120576

le 4119873 (119879)int

119904

0

119871 (120591) 120594 (119906119896(120591)infin

119896=1) 119889120591 + 120576

le 4119873 (119879)int

119905

0

119871 (120591) 119892119899(120591) 119889120591 + 120576

(43)

Since 120576 gt 0 is arbitrary it follows from the above inequalitiesthat

120594 (119866Γ119899(119904)) le 4119873 (119879)int

119905

0

119871 (120591) 119892119899(120591) 119889120591 119904 isin [0 119905] (44)

Using (36) we have

119892119899+1

(119905) = sup 120594 (119866Γ119899(119904)) 119904 isin [0 119905]

le 4119873 (119879)int

119905

0

119871 (120591) 119892119899(120591) 119889120591 119905 isin [0 119879]

(45)

Letting 119899 rarr infin we derive the following inequality

119892infin(119905) le 4119873 (119879)int

119905

0

119871 (120591) 119892infin(120591) 119889120591 119905 isin [0 119879] (46)

This inequality together with Gronwallrsquos Lemma 8 impliesthat

119892infin(119905) = 0 119905 isin [0 119879] (47)

Next let us notice that in view of (1198674) and (36) we have

120594 (F (119904 Γ119899(119904))) le 119871 (119904) 120594 (Γ

119899(119904)) le 119871 (119904) 119892

119899(119904) (48)

for ae 119904 isin [0 119879] and the function

[0 119879] ni 119904 997891997888rarr 2119873 (119879) 119871 (119904) 119892119899(119904) (49)

is measurable Then in virtue of equality 1205961198790(119866Γ119899) = 120596119879

0(119865Γ119899)

and Lemma 9 we get

120596119879

0(119866Γ119899) le 2119873 (119879)int

119879

0

119871 (119904) 119892119899(119904) 119889119904 (50)

Hence we derive

119908119899+1

= 120596119879

0(Γ119899+1) = 120596119879

0(119866Γ119899)

le 2119873 (119879)int

119879

0

119871 (119904) 119892119899(119904) 119889119904

(51)

Letting 119899 rarr infin we get

119908infinle 2119873 (119879)int

119879

0

119871 (119904) 119892infin(119904) 119889119904 (52)

Keeping in mind (47) we deduce that

119908infin= 0 (53)

This together with (39) and (47) yields

lim119899rarrinfin

120583119879(Γ119899) = lim119899rarrinfin

(119908119899+ 119892119899(119879)) = 119908

infin+ 119892infin= 0 (54)

Finally using Remark 2 for the measure 120583119879 we deduce

that the set Γinfin= ⋂infin

119899=0Γ119899is nonempty convex and compact

Then by the Schauder-Tychonoff theorem we conclude thatoperator 119865 Γ

infinrarr Γinfin

has at least one fixed-point 119906 = 119906(119905)Obviously the function 119906 = 119906(119905) is a solution of problem (6)-(7) and in view of the definition of the set Γ

infin the estimate

119906(119905) le P(119905) holds to be trueThis completes the proof

4 Local and Global Attractivity

Following the concepts introduced in [36] we introduce firsta few definitions of various kinds of the concept of attractivityof mild solution of (6)

Definition 11 The mild solution 119906 = 119906(119905) of (6) with initialcondition (7) is said to be globally attractive if for each mildsolution V = V(119905) of (6) with initial condition V(0) = V

0we

have that

lim119905rarrinfin

(119906 (119905) minus V (119905)) = 120579 (55)

In other words we may say that solutions of (6) are globallyattractive if for arbitrary solutions 119906(119905) and V(119905) of thisequation condition (55) are satisfied

Definition 12 We say that mild solution 119906 = 119906(119905) of (6) withinitial condition (7) is locally attractive if there exists a ball119861(119906(0) 119903) in the space X such that for arbitrary solution V(119905)of (1) with initial-value V(0) isin 119861(119906(0) 119903) condition (55) doeshold

In the case when the limit (55) is uniform with respect toall solutions V(119905) that is when for each 120576 gt 0 there exist119879 gt 0such that

119906 (119905) minus V (119905) le 120576 (56)

for all V(119905) being solutions of (6) with initial-value V(0) isin119861(119906(0) 119903) and for 119905 ge 119879 we will say that solution 119906 = 119906(119905)

is uniformly locally attractive on R+

Now we formulate themain result of this sectionWe willconsider (6) under the following conditions

(1198671015840

1) A is the infinitesimal generator of an exponentiallystable119862

0-semigroup 119890minusA119905

119905ge0 that is there exist119872 gt

0 120588 gt 0 such that 119890minusA119905 le 119872119890minus120588119905 for all 119905 ge 0

6 Journal of Function Spaces and Applications

(1198675) there exist locally integrable functions119898 R

+rarr R+

such that

F (119905 119906) minusF (119905 V) le 119898 (119905) 119906 minus V (57)

for 119905 ge 0 and 119906 V isin X Moreover we assume that

lim119905rarrinfin

(int

119905

0

119898(119904) 119889119904 minus120588

119872119905) = minusinfin (58)

Remark 13 The property (11986710158401) is generally satisfied in diffu-

sion problem A necessary and sufficient condition for (11986710158401)

is presented in [72]

The main result of this section is shown in the giventheorem below

Theorem 14 Under assumptions (11986710158401) and (119867

2)ndash(1198675) prob-

lem (6)-(7) has amild solution 119906 = 119906(119905) for each 1199060isin X which

is globally attractive and locally uniformly attractive

Proof Existence of a solution 119906 = 119906(119905) is a consequence ofTheorem 10 Let us fix 119903 gt 0 and V

0isin 119861(119906(0) 119903)

Let V = V(119905) denote a mild solution of (6) with the initialcondition V(0) = V

0 Using (1198671015840

1) and (119867

3) we get

V (119905) le10038171003817100381710038171003817119890minusA119905V0

10038171003817100381710038171003817+ int

119905

0

10038171003817100381710038171003817119890minusA(119905minus119904)10038171003817100381710038171003817

sdot F (119904 V (119904)) 119889119904

le 119872(100381710038171003817100381711990601003817100381710038171003817 + 119903) +119872int

119905

0

119890120588119904119886 (119904) 119889119904

+ int

119905

0

119872119890120588119904119887 (119904) V (119904) 119889119904

(59)

Now let us put

P1(119905) = 119872 (

1003817100381710038171003817V01003817100381710038171003817 + 119903) +119872int

119905

0

119890120588119904119886 (119904) 119889119904

P2(119905) = 119872119890

120588119904119887 (119905)

(60)

Taking into account Lemma 5 we obtain

V (119905) le P1(119905) + int

119905

0

P2(119904)P1(119904) exp(int

119904

0

P2(120591) 119889120591) 119889119904

(61)

Further let 119878 be the set of all mild solutions V(119905) of (6) withthe initial-value V

0= V(0) isin 119861(119906(0) 119903)

Put

119892 (119905) = sup 119906 (119905) minus V (119905) V isin 119878 (62)

The estimate (61) implies that the function119892(119905) is well definedApplying (119867

5) and (1198671015840

1) we get

119906 (119905) minus V (119905) le10038171003817100381710038171003817119890minusA11990510038171003817100381710038171003817

sdot10038171003817100381710038171199060 minus V

0

1003817100381710038171003817

+ int

119905

0

10038171003817100381710038171003817119890minusA(119905minus119904)10038171003817100381710038171003817

119898 (119904) 119906 (119904) minus V (119904) 119889119904

le 119872119890minus120588119905119903 + int

119905

0

119872119890minus119892(119905minus119904)

119898(119904) 119892 (119904) 119889119904

(63)

or equivalently

119890120588119905119892 (119905) le 119872119903 + int

119905

0

119872119898(119904) 119890120588119904119892 (119904) 119889119904 (64)

Using again Lemma 5 for the above estimate (where ℎ(119905) =119890120588119905119892(119905)) we obtain

119890120588119905119892 (119905) le 119872119903

+ int

119905

0

1198722119903119898 (119904) exp(int

119905

119904

119872119898(120591) 119889120591) 119889119904

(65)

Elementary calculations lead to the following equality

int

119905

0

1198722119903119898 (119904) exp(int

119905

119904

119872119898(120591) 119889120591) 119889119904

= 119872119903(exp(int119905

0

119872119898(119904) 119889119904) minus 1)

(66)

Hence

119892 (119905) le 119872119903 exp (minus120588119905) + 119872119903 exp(int119905

0

119872119898(119904) 119889119904 minus 120588119905)

minus119872119903 exp (minus120588119905) (67)

Applying assumption (1198675) we derive

lim119905rarrinfin

119892 (119905) = 0 (68)

and this proves that 119906(119905) is locally attractive Finally thisequality together with definition of the function 119892(119905) impliesthat 119906(119905) is globally attractive The proof is complete

Remark 15 In the case when 119898(119905) equiv 119898 is constant the fol-lowing condition

lim119905rarrinfin

(int

119905

0

119898(119904) 119889119904 minus120588

119872119905) = minusinfin (69)

means that 119898 lt 120588119872 Observe that this condition cannotbe weakened This observation is illustrated by the followingexampled

Example 16 Let X = R 119898(119905) equiv 119898 F(119905 119906) = sin 119905 + 119898119906119872 = 1 and 119890minusA119905 = 119890minus120588119905 Then the equation

119906 (119905) = 119890minusA1199051199060+ int

119905

0

119890minusA(119905minus119904)

F (119904 119906 (119904)) 119889119904 (70)

(for any fixed 1199060isin R) has the solution 119906(119905) expressed by the

following formula

119906 (119905) =(120588 minus 119898) sin 119905 minus cos 119905

(120588 minus 119898)2

+ 1+ (1199060+

1

(120588 minus 119898)2

+ 1) 119890(119898minus120588)119905

(71)

Notice that for 119898 ge 120588 the solution 119906(119905) is neither globallyattractive nor locally uniformly attractive because for each

Journal of Function Spaces and Applications 7

other solution V(119905) with initial condition V(0) = V0 obviously

described by similar formula as 119906(119905) we would have acontradiction

|119906 (119905) minus V (119905)| =

10038161003816100381610038161199060 minus V0

1003816100381610038161003816 119890(119898minus120588)119905

997888rarr infin

or10038161003816100381610038161199060 minus V

0

1003816100381610038161003816

(72)

5 Asymptotic Behaviour

In this section we will give a theorem describing asymptoticbehaviour of mild solutions of (6) with condition (7) Thistheorem generalizes the result included in [72Theorem 44]First we formulate the assumptions

(1198671015840

3) This condition is almost identical with (119867

3) and the

only difference is that we assume the functions 119886 and119887 are locally essentially bounded on R

+

(1198671015840

5) There exists 119906

infinisin X such that there exists the limit

lim119905rarrinfin

F(119905 119906infin) and

lim119905rarrinfin

F (119905 119906infin) = A

minus1119906infin (73)

Moreover there exists a number119898 lt 120588119872 such that1003817100381710038171003817F (119905 119906) minusF (119905 119906

infin)1003817100381710038171003817 le 119898

1003817100381710038171003817119906 minus 119906infin1003817100381710038171003817 (74)

for 119905 ge 0 and 119906 isin X

Remark 17 The condition (11986710158405) in conjunction with (1198671015840

1)

ensures the existence ofAminus1 (see [72]) Clearly (11986710158403) implies

(1198673)

Theorem 18 Under assumptions (11986710158401) (1198672) (11986710158403) (1198674) and

(11986710158405) (6) with condition (7) has a mild solution 119906 = 119906(119905) for

each 1199060isin X such that lim

119905rarrinfin119906(119905) = 119906

infin

Proof The existence of a mild solution 119906(119905) is guaranteed byTheorem 10 Let us put

119892 (119905) =1003817100381710038171003817119906 (119905) minus 119906infin

1003817100381710038171003817

Finfin= lim119905rarrinfin

F (119905 119906infin)

(75)

We show that

lim119905rarrinfin

119892 (119905) = 0 (76)

Recall that if assumption (11986710158401) is fulfilled then for each 119911 isin X

we have (see [72])

int

infin

0

119890minusA119904

119911 119889119904 = minusAminus1119911 (77)

Using the above fact and (11986710158405) we get

119906infin= (minusA

minus1(minusAminus1119906infin)) = minusA

minus1Finfin

= int

infin

0

119890minusA119904

Finfin119889119904

(78)

Linking the above equality with (11986710158401) we obtain

119892 (119905)

=

10038171003817100381710038171003817100381710038171003817119890minusA1199051199060+ int

119905

0

119890minusA(119905minus119904)

F (119904 119906 (119904)) 119889119904 minus int

infin

0

119890minusA119904

Finfin119889119904

10038171003817100381710038171003817100381710038171003817

le 119872119890minus120588119905 10038171003817100381710038171199060

1003817100381710038171003817 +

10038171003817100381710038171003817100381710038171003817int

infin

119905

119890minusA119904

Finfin119889119904

10038171003817100381710038171003817100381710038171003817

+

10038171003817100381710038171003817100381710038171003817int

119905

0

119890minusA(119905minus119904)

F (119904 119906 (119904)) 119889119904 minus int

119905

0

119890minusA119904

Finfin119889119904

10038171003817100381710038171003817100381710038171003817

le 119872119890minus120588119905 10038171003817100381710038171199060

1003817100381710038171003817 +119872

119890minus1205881199051205881003817100381710038171003817Finfin

1003817100381710038171003817

+

10038171003817100381710038171003817100381710038171003817int

119905

0

119890minusA(119905minus119904)

F (119904 119906 (119904)) 119889119904 minus int

119905

0

119890minusA(119905minus119904)

Finfin119889119904

10038171003817100381710038171003817100381710038171003817

le 119872119890minus120588119905(100381710038171003817100381711990601003817100381710038171003817 +

1003817100381710038171003817Finfin1003817100381710038171003817

120588)

+ int

119905

0

119872119890minus120588(119905minus119904) 1003817100381710038171003817F (119904 119906 (119904)) minusF

infin

1003817100381710038171003817 119889119904

le 119872119890minus120588119905(100381710038171003817100381711990601003817100381710038171003817 +

1003817100381710038171003817Finfin1003817100381710038171003817

120588)

+119872119890minus120588119905int

119905

0

119890120588119904 1003817100381710038171003817F (119904 119906

infin) minusF

infin

1003817100381710038171003817 119889119904

+119872119890minus120588119905int

119905

0

119890120588119904 1003817100381710038171003817F (119904 119906 (119904)) minusF (119904 119906

infin)1003817100381710038171003817 119889119904

(79)

Next putting

119888 (119905) = 119872119890minus120588(119905)

(100381710038171003817100381711990601003817100381710038171003817 +

1003817100381710038171003817Finfin1003817100381710038171003817

120588)

+119872119890minus120588119905int

119905

0

119890120588119904 1003817100381710038171003817F (119904 119906

infin) minusF

infin

1003817100381710038171003817 119889119904

(80)

and applying (11986710158405) we derive the following inequality

119892 (119905) le 119888 (119905) + 119898119872119890minus120588119905int

119905

0

119890120588119904119892 (119904) 119889119904 (81)

Hence

119890120588119905119892 (119905) le 119890

120588(119905)119888 (119905) + 119898119872int

119905

0

119890120588119904119892 (119904) 119889119904 (82)

The above inequality in conjunction with Lemma 5 gives

119890120588119905119892 (119905) le 119890

120588119905119888 (119905)

+ int

119905

0

119890120588119904119888 (119904)119898119872 sdot exp (119898119872(119905 minus 119904)) 119889119904

= 119890120588119905119888 (119905)

+ 119898119872119890119898119872119905

int

119905

0

119888 (119904) exp ((120588 minus 119898119872) 119904) 119889119904

(83)

8 Journal of Function Spaces and Applications

Hence

119892 (119905) le 119888 (119905) +119898119872int

119905

0119888 (119904) 119890(119892minus119898119872)119904

119889119904

exp ((120588 minus 119898119872) 119905) (84)

Before proving (76) we first show that

lim119905rarrinfin

119888 (119905) = 0 (85)

To prove this equality it is sufficient to show that the lastcomponent in the formula expressing 119888(119905) that is 119872119890minus120588119905

int119905

0119890120588119904F(119904 119906

infin) minusF

infin119889119904 tends to zero as 119905 rarr infin

To this end let us fix 120576 gt 0 Assumption (11986710158405) implies that

there exists 1199050such that

sup 1003817100381710038171003817F (119904 119906infin) minusF

infin

1003817100381710038171003817 119904 ge 1199050 le120576120588

2119872 (86)

This inequality together with (11986710158403) implies

ess sup 1003817100381710038171003817F (119904 119906infin) minusF

infin

1003817100381710038171003817 119904 isin R lt infin (87)

Now fix 1199051isin R such that 119905

1ge 119905 and

ess sup 1003817100381710038171003817F (119904 119906infin) minusF

infin

1003817100381710038171003817 119904 isin R

le120576120588 exp (120588119905

1)

2119872 (exp (1205881199050) minus 1)

(88)

Then using (86) and (88) we conclude that for 119905 ge 1199051we have

119872119890minus120588119905int

119905

0

119890120588119904 1003817100381710038171003817F (119904 119906

infin) minusF

infin

1003817100381710038171003817 119889119904

le 119872119890minus120588119905int

1199050

0

119890120588119904 1003817100381710038171003817F (119904 119906

infin) minusF

infin

1003817100381710038171003817 119889119904

+119872119890minus120588119905int

119905

1199050

119890120588119904 1003817100381710038171003817F (119904 119906

infin) minusF

infin

1003817100381710038171003817 119889119904 le120576

2+120576

2= 120576

(89)

This fact proves (85)Further using (1198671015840

5) and (85) and employing de

LrsquoHospitalrsquos rule for the fraction on the right-hand sideof inequality (84) we obtain that condition (76) is satisfiedThis fact completes the proof

6 An Application to the Ginzburg-LandauEquations of Superconductivity

In this section we formulate the gauged time-dependentGinzburg-Landau (TDGL) equations as an abstract evolutionequation in a Hilbert space Moreover we show applicationsof the above theorems to TDGL equations

We assume that Ω is a bounded domain in R119899 withboundary 120597Ω of class11986211That isΩ is an open and connectedset whose boundary 120597Ω is a compact (119899 minus 1)-manifolddescribed by Lipschitz continuous differentiable charts Weconsider two- and three-dimensional problems (119899 = 2 and119899 = 3 resp) Assume that the vector potential A takes its

values in R119899 The vector H will represent the (externally)applied magnetic field which is a function of space andtime similarly to A it takes its values in R119899 The function120574 is defined and satisfies Lipschitz condition on 120597Ω and120574(119909) ge 0 for 119909 isin 120597Ω The parameters in the TDGLequations are 120578 a (dimensionless) friction coefficient and 120581the (dimensionless) Ginzburg-Landau parameter

The order parameter should be thought of as the wavefunction of the center-of-massmotion of the ldquosuperelectronsrdquo(Cooper pairs) whose density is 119899

119904= |120595|

2 and whose fluxis J119904 The vector potential A determines the electromagnetic

field E = minus120597119905A minus nabla120601 is the electric field and B = nabla times A

is the magnetic induction where J the total current is thesum of a ldquonormalrdquo current J

119899= E the supercurrent J

119904 and

the transport current J119905= nabla times H The normal current obeys

Ohmrsquos law J119899= 120590119899E the ldquonormal conductivityrdquo coefficient 120590

119899

is equal to one in the adopted system of units The differenceM = BminusH is known as themagnetizationThe trivial solution(120595 = 0 B = H E = 0) represents the normal state where allsuperconducting properties have been lost

Nowwe accept the following notion all Banach spaces arereal the (real) dual of a Banach space119883 is denoted by1198831015840Thesymbol 119871119901(Ω) for 1 le 119901 le infin denotes the usual Lebesguespace with norm sdot

119871119901 (sdot sdot) is the inner product in 1198712(Ω)

1198821198982(Ω) for nonnegative integer 119898 is the usual Sobolev

space with norm sdot 1198821198982 1198821198982(Ω) is a Hilbert space for the

inner product (sdot sdot)1198982

given by (119906 V)1198982

= sum|120572|le119898

(120597120572119906 120597120572V)

for 119906 V isin 1198821198982(Ω) Fractional Sobolev space1198821199042(Ω) with afractional 119904 is defined by interpolation ([40 Chap VII] and[41 49 50]) 119862]

(Ω) for ] ge 0 ] = 119898 + 120582 with 0 le 120582 lt 1 isthe space of 119898 times continuous differentiable functions onΩ those 119898th-order derivatives satisfy the Holder conditionwith exponent 120582 if ] is a proper fraction the norm sdot

119862] is

defined in the usual wayThe definitions extend to the space of vector-valued

functions in the standard way with the caveat that the innerproduct in [1198712(Ω)]119899 is defined by (119906 V) = int

Ω119906 sdot V where the

symbol (119906 V) indicates the scalar product in R119899 Complex-valued functions are interpreted as vector-valued functionswith two real components

Functions that vary in space and time like the orderparameter and the vector potential are considered as map-pings from the time domain which is a subinterval of [0infin)into spaces of complex- or vector-valued functions definedin Ω Let 119883 = (119883 sdot

119883) be a Banach space of functions

defined inΩ Then the functions are defined in Ω Then thefunctions of space and time defined on Ω times (0 119879) for 119879 gt 0may be considered as elements of 119871119901(0 119879119883) for 1 le 119901 le infinor1198822119898(0 119879119883) for nonnegative119898 or119862]

(0 119879119883) for ] ge 0] = 119898 + 120582 with 0 le 120582 lt 1 Detailed definitions can be foundfor example in [43]

Obviously function spaces of ordered pairs (120595A) where120595 Ω rarr R2 and A Ω rarr R119899 (119899 = 2 3) play an importantrole in the study of the gauged TDGL equationsWe thereforeadapt the following special notationH = [119883(Ω)]

2times[119883(Ω)]

119899for any Banach space for the order parameter 120595 and thevector potential A respectively A suitable framework forthe functional analysis of the gauged TDGL equations is the

Journal of Function Spaces and Applications 9

Cartesian product W1+1205722 = [1198821+1205722

(Ω)]2times [1198821+1205722

(Ω)]119899

where 120572 isin (12 1) This space is continuously imbedded inW12 capLinfin

Assume H = 0 and H isin [1198712(Ω)]119899 Let AH be a minimizer

of the convex quadratic form 119869120596equiv J120596[119860]

119869120596 [A] = int

Ω

[120596(nabla sdot A)2 + |nabla times A minusH|2] 119889119909 (90)

on the domain

D (119869120596) = A isin [119882

12(Ω)]119899

n sdot A = 0 on 120597Ω (91)

We now introduce the reduced vector potential A1015840

A1015840 = A minus AH (92)

In terms of 120595 and A1015840 the gauged TDGL equations have thefollowing form

120597120595

120597119905minus

1

1205781205812Δ120595 = 120593 in Ω times (0infin)

120597A1015840

120597119905+ nabla times nabla times A1015840 minus 120596nabla (nabla sdot A1015840) = F in Ω times (0infin)

n sdot nabla + 120574120595 = 0 n sdot A1015840 = 0

n times (nabla sdot A1015840) = 0 on 120597Ω times (0infin)

(93)

Here 120593 and F are nonlinear functions of 120595 and A1015840

120593 equiv 120593 (119905 120595A1015840) = 1

120578[minus2119894

120581(nabla120595) sdot (A1015840 + AH)

minus119894

120581(1 minus 120578120581

2120596)120595 (nabla sdot A1015840)

minus12059510038161003816100381610038161003816A1015840 + AH

10038161003816100381610038161003816

2

+ (1 minus100381610038161003816100381612059510038161003816100381610038162

) 120595]

(94)

F equiv F (119905 120595A1015840) = 1

2119894119896(120595lowastnabla120595 minus 120595nabla120595

lowast)

minus100381610038161003816100381612059510038161003816100381610038162

(A1015840 + AH) minus120597AH120597119905

(95)

The equations are supplemented by initial data which is inthe followimg form

120595 = 1205950 A = A

0on Ω times 0 (96)

where 1205950and A

0are given and by (92) we have

120595 = 1205950 A1015840 = A

0minus AH (0) on Ω times 0 (97)

In the next part we connect the evolution of the solution(120595A1015840) of the system of (93) with the initial data (120595

0A0minus

AH(0)) with the dynamics of a vector 119906 in a Hilbert space

L2= [119871119901(Ω)]2

times [119871119901(Ω)]119899

(98)

The following analysis is restricted to the case 120596 gt 0 andthe case 120596 = 0 (see [73])

Let vector 119906 [0infin) rarr L2 represent the pair (120595A1015840)

119906 = (120595A1015840) equiv (120595A minus AH) (99)

and letA be the linear self-adjoint operator inL2 associatedwith the quadratic form 119876

120596equiv 119876120596[119906] given by the following

formula

119876120596 [119906] = int

Ω

[1

12057812058121003816100381610038161003816nabla120595

10038161003816100381610038162

+ 120596(nabla sdot A1015840)2

+10038161003816100381610038161003816nabla times A101584010038161003816100381610038161003816

2

] 119889119909

+ int120597Ω

120574

1205781205812100381610038161003816100381612059510038161003816100381610038162

119889120590 (119909)

(100)

on the domain

D (119876120596) = D (A

12)

= 119906 = (120595A1015840) isinW12 n sdot A1015840 = 0 on 120597Ω

(101)

The quadratic form 119876120596is nonnegative Furthermore since

120596 gt 0119876120596[120595A1015840]+119888 120595

1198712 is coercive onW12 for any constant

119888 gt 0 Hence A is positively definite in L2 ([44 Chap 1]equation (545)) If it does not lead to confusion we use thesame symbol A for the restriction A

120595and AA of A to the

respective linear subspace [1198712(Ω)]2 = [1198712(Ω)]2 times 0 (for 120595)and [1198712(Ω)]119899 equiv 0 times [1198712(Ω)]119899 (for A) ofL2

Now consider the initial-value problems (6) and (7) inL2 whereF(119905 119906) = (120593 F) 120593 and F is given by (94) and (95)and 119906

0= (1205950A0minus AH(0))

With 120582 isin (12 1) and 1199060isin W1+1205822 we say that 119906 are a

solution of (6) and (7) on the interval [0 119879] for some 119879 gt 0if 119906 [0 119879] rarr W1+1205822 is continuous and

119906 (119905) = 119890minusA(119905)

+ int

119905

0

119890minusA(119905minus119904)

F (119904 119906 (119904)) 119889119904 for 119905 isin [0 119879]

(102)

in L2 A mild solution of the initial-value problems (6) and(7) defines a weak solution (120595A1015840) of the boundary valueproblem (93) which in turn defines a weak solution (120595A)of the gauged TDGL equations provided AH is sufficientlyregular

Namely let us assume that X = L2 and 1199060isin L2 and

119906 R+times R119899 rarr R119899 for 119899 isin N is an unknown function 119906 =

119906(119905 119909) In order to applyTheorems 10 and 14 we are not goingto consider 119906 as a function of 119905 and 119909 together but rather asa mapping 119906 of variable 119905 into the spaceX =L2 of functions119909 that is 119906 R

+rarr L2 119906(119905)(119909) = 119906(119905 119909) 119909 isin R119899 119905 isin R

+

Remark 19 Since A is the linear self-adjoint operator inL2

associated with the quadratic form (100) thenA generates asemigroup 119890minusA119905

119905ge0(see [74])

Below we formulate the principal theorem of this paperThis theorem is a simple consequence of Theorem 10

10 Journal of Function Spaces and Applications

Theorem 20 Under assumptions (1198671)ndash(1198674) problem (6)-(7)

is equivalent to the system of (1)ndash(5) (TDGL equations) andhas for each 119906

0isin L2 at least one mild solution in sense of

Definition 4 119906 = 119906(119905) Moreover each solution 119906(119905) satisfiesestimate (25)

We observe that in the case when 119890minusA119905119905ge0

is the semi-group of contractions and 119886(119905) equiv 119886 119887(119905) equiv 119887 for 119905 isin

R+are constant after simple calculations based on estimate

(25) we get that the solution 119906 = 119906(119905) has the asymptoticcharacterization

119906 (119905) = 119874 (1199053) as 119905 997888rarr infin (103)

For further purposes let us formulate the followingassumption

(A) A is the infinitesimal generator of an exponentiallystable 119862

0-semigroup 119890minusA119905

119905ge0

The next result of this chapter is shown in the giventheorem below

Theorem21 Under assumptions (A) and (1198672)ndash(1198674) problem

(6)-(7) which is equivalent to the systems equations (1)ndash(5)(TDGL equations) has a mild solution 119906 = 119906(119905) for each1199060isin L2 which is globally attractive and locally uniformly

attractive

References

[1] V L Ginzburg and L D Landau ldquoOn the theory of supercon-ductivityrdquo Zhurnal Eksperimentalrsquonoi i Teoreticheskoi Fiziki vol20 pp 1064ndash1082 1950 English translation in L D LandauMen of Physics D ter Haar ed Vol I Pergamon Press OxfordUK pp 138ndash167 1965

[2] S Aizicovici and H Lee ldquoNonlinear nonlocal Cauchy problemsin Banach spacesrdquo Applied Mathematics Letters vol 18 no 4pp 401ndash407 2005

[3] S Aizicovici and M McKibben ldquoExistence results for a classof abstract nonlocal Cauchy problemsrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 39 no 5 pp 649ndash6682000

[4] MMcKibbenDiscovering Evolution Equation with ApplicationCRC Press Taylor amp Francis Group 2011

[5] S Aizicovici and V Staicu ldquoMultivalued evolution equationswith nonlocal initial conditions in Banach spacesrdquo NonlinearDifferential Equations and Applications vol 14 no 3-4 pp 361ndash376 2007

[6] R P Agarwal and D OrsquoRegan Infinite Interval Problems for Dif-ferential Difference and Integral Equations Kluwer AcademicPublishers Dordrecht The Netherlands 2001

[7] Z Fan ldquoExistence of nondensely defined evolution equationswith nonlocal conditionsrdquo Nonlinear Analysis Theory Methodsamp Applications A vol 70 no 11 pp 3829ndash3836 2009

[8] Z Fan ldquoImpulsive problems for semilinear differential equa-tions with nonlocal conditionsrdquo Nonlinear Analysis TheoryMethods amp Applications A vol 72 no 2 pp 1104ndash1109 2010

[9] Z Fan and G Li ldquoExistence results for semilinear differentialequations with nonlocal and impulsive conditionsrdquo Journal ofFunctional Analysis vol 258 no 5 pp 1709ndash1727 2010

[10] J Garcıa-Falset ldquoExistence results and asymptotic behavior fornonlocal abstract Cauchy problemsrdquo Journal of MathematicalAnalysis and Applications vol 338 no 1 pp 639ndash652 2008

[11] J Liang J H Liu and T-J Xiao ldquoNonlocal impulsive problemsfor nonlinear differential equations in Banach spacesrdquo Mathe-matical and Computer Modelling vol 49 no 3-4 pp 798ndash8042009

[12] L Zhu and G Li ldquoOn a nonlocal problem for semilinear differ-ential equations with upper semicontinuous nonlinearities ingeneral Banach spacesrdquo Journal of Mathematical Analysis andApplications vol 341 no 1 pp 660ndash675 2008

[13] Q Liu and R Yuan ldquoExistence of mild solutions for semilinearevolution equations with non-local initial conditionsrdquo Nonlin-ear AnalysisTheoryMethodsampApplications A vol 71 no 9 pp4177ndash4184 2009

[14] C Xie W Zhong and Z Fan ldquoExistence results for nonlinearnonlocal problems in Banach spacesrdquo Applied MathematicsLetters vol 22 no 7 pp 998ndash1002 2009

[15] T-J Xiao and J Liang ldquoExistence of classical solutions tononautonomous nonlocal parabolic problemsrdquoNonlinear Anal-ysisTheoryMethods and Applications vol 63 no 5-7 pp e225ndashe232 2005

[16] X Xue ldquo119871119901 theory for semilinear nonlocal problems with mea-sure of noncompactness in separable Banach spacesrdquo Journal ofFixed Point Theory and Applications vol 5 no 1 pp 129ndash1442009

[17] X Xue ldquoNonlinear differential equations with nonlocal condi-tions in Banach spacesrdquo Nonlinear Analysis Theory Methods ampApplications A vol 63 no 4 pp 575ndash586 2005

[18] X Xue ldquoExistence of solutions for semilinear nonlocal Cauchyproblems in Banach spacesrdquo Electronic Journal of DifferentialEquations vol 64 p 7 2005

[19] X Xue ldquoNonlocal nonlinear differential equations with a mea-sure of noncompactness in Banach spacesrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 70 no 7 pp 2593ndash26012009

[20] L Olszowy ldquoExistence ofmild solutions for the semilinear non-local problem in Banach spacesrdquo Nonlinear Analysis TheoryMethods amp Applications A vol 81 pp 211ndash223 2013

[21] D Bothe ldquoMultivalued perturbations of119898-accretive differentialinclusionsrdquo Israel Journal of Mathematics vol 108 pp 109ndash1381998

[22] H-S Ding J Liang G M NrsquoGuerekata and T-J XiaoldquoMild pseudo-almost periodic solutions of nonautonomoussemilinear evolution equationsrdquo Mathematical and ComputerModelling vol 45 no 5-6 pp 579ndash584 2007

[23] J Garcıa-Falset ldquoThe asymptotic behavior of the solutions of theCauchy problem generated by 120593-accretive operatorsrdquo Journal ofMathematical Analysis and Applications vol 310 no 2 pp 594ndash608 2005

[24] Z Fan and G Li ldquoAsymptotic behavior of the solutions of non-autonomous systems in Banach spacesrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 68 no 12 pp 3733ndash37412008

[25] J Banas ldquoOn existence theorems for differential equations inBanach spacesrdquo Bulletin of the Australian Mathematical Societyvol 32 no 1 pp 73ndash82 1985

[26] J Banas ldquoMeasures of noncompactness in the space of contin-uous tempered functionsrdquo Demonstratio Mathematica vol 14no 1 pp 127ndash133 1981

Journal of Function Spaces and Applications 11

[27] J Banas and K GoebelMeasures of Noncompactness in BanachSpaces vol 60 ofLectureNotes in Pure andAppliedMathematicsMarcel Dekker New York NY USA 1980

[28] J Banas and M Lecko ldquoSolvability of infinite systems ofdifferential equations in Banach sequence spacesrdquo Journal ofComputational andAppliedMathematics vol 137 no 2 pp 363ndash375 2001

[29] J Banas and K Sadarangani ldquoCompactness conditions inthe study of functional differential and integral equationsrdquoAbstract and Applied Analysis vol 2013 Article ID 819315 14pages 2013

[30] M Mursaleen and S A Mohiuddine ldquoApplications of mea-sures of noncompactness to the infinite system of differentialequations in ℓ

119901spacesrdquo Nonlinear Analysis Theory Methods amp

Applications A vol 75 no 4 pp 2111ndash2115 2012[31] L Olszowy ldquoSolvability of some functional integral equationrdquo

Dynamic Systems and Applications vol 18 no 3-4 pp 667ndash6762009

[32] L Olszowy ldquoOn some measures of noncompactness in theFrechet spaces of continuous functionsrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 72 pp 2119ndash2126 2010

[33] L Olszowy ldquoOn existence of solutions of a quadratic Urysohnintegral equation on an unbounded intervalrdquo CommentationesMathematicae vol 48 no 1 pp 103ndash112 2008

[34] L Olszowy ldquoExistence ofmild solutions for semilinear nonlocalCauchy problems in separable Banach spacesrdquo Journal of Anal-ysis and its Applications vol 32 no 2 pp 215ndash232 2013

[35] L Olszowy ldquoFixed point theorems in the Frechet space 119862(119877+)

and functional integral equations on an unbounded intervalrdquoApplied Mathematics and Computation vol 218 no 18 pp9066ndash9074 2012

[36] J Banas and D OrsquoRegan ldquoOn existence and local attractivity ofsolutions of a quadratic Volterra integral equation of fractionalorderrdquo Journal of Mathematical Analysis and Applications vol345 no 1 pp 573ndash582 2008

[37] L D Landau EM Lifshitz and L P Pitaevski Electrodynamicsof Continuous Media vol 8 Butterworth-Heinemann 2ndedition 1984

[38] L D Landau and E M Lifshitz Fluid Mechanics vol bButterworth-Heinemann 2nd edition 1987

[39] R Temam Infinite-Dimensional Dynamical Systems inMechan-ics and Physics vol 68 Springer New York NY USA 2ndedition 1997

[40] J K Hale Asymptotic Behavior of Dissipative Systems vol 25 ofMathematical Surveys and Monographs American Mathemati-cal Society Providence RI USA 1988

[41] R A Adams and J J F Fournier Sobolev Spaces vol 140Elsevier 2nd edition 2003

[42] V Mazrsquoya Sobolev Spaces with Applications to Elliptic PartialDifferential Equations Springer Heidelberg Germany 2ndedition 2011

[43] V Georgescu ldquoSome boundary value problems for differentialforms on compact Riemannian manifoldsrdquo Annali di Matemat-ica Pura ed Applicata vol 122 pp 159ndash198 1979

[44] D Henry Geometric Theory of Semilinear Parabolic Equationsvol 840 of Lecture Notes in Mathematics Springer New YorkNY USA 1981

[45] V Girault and P-A Raviart Finite Element Methods for Navier-Stokes Equations Springer New York NY USA 1986

[46] J Bardeen L N Cooper and J R Schrieffer ldquoTheory ofsuperconductivityrdquo Physical Review vol 108 pp 1175ndash12041957

[47] A A Abrikosov Fundamental of the Theory of Metals North-Holland Amstedram The Netherlands 1988

[48] M Tinkham Introduction To Superconductivity McGraw-HillNew York NY USA 2nd edition 1996

[49] P DeGennes Superconductivity inMetals and Alloys BenjaminNew York NY USA 1966

[50] H Brezis Functional Analysis Sobolev Spaces and PartialDifferential Equations Springer New York NY USA 2011

[51] E Zeidler Nonlinear Functional Analysis and its Applicationsvol 2 Springer New York NY USA 1990

[52] A Schmid ldquoA time dependent Ginzburg-Landau equation andits application to the problem of resistivity in the mixed staterdquoPhysik der KondensiertenMaterie vol 5 no 4 pp 302ndash317 1966

[53] C M Elliot and Q Tang Existence Theorems For an Evolution-ary Superconductivity Model University of Sussex BrightonUK 1992

[54] Q Tang ldquoOn an evolutionary system of Ginzburg-Landauequations with fixed total magnetic fluxrdquo Communications inPartial Differential Equations vol 20 no 1-2 pp 1ndash36 1995

[55] Q Du ldquoGlobal existence and uniqueness of solutions of thetime-dependent Ginzburg-Landau model for superconductiv-ityrdquo Applicable Analysis vol 53 no 1-2 pp 1ndash18 1994

[56] Z M Chen K-H Hoffmann and J Liang ldquoOn a nonstation-ary Ginzburg-Landau superconductivity modelrdquoMathematicalMethods in the Applied Sciences vol 16 no 12 pp 855ndash875 1993

[57] J Liang and Q Tang ldquoAsymptotic behaviour of the solutionsof an evolutionaryGinzburg-Landau superconductivitymodelrdquoJournal of Mathematical Analysis and Applications vol 195 no1 pp 92ndash107 1995

[58] L P Gorrsquokov and G M Eliasherberg ldquoGeneralization of theGinzburg-Landau equations for an nonstationary problemsin the case of alloys with paramagnetic impartiesrdquo ZhurnalEksperimentalrsquonoi i Teoreticheskoi Fiziki vol 54 pp 612ndash6261968 Soviet Physics vol 27 pp 328ndash334 1968

[59] J Fleckinger-Pelle and H G Kaper ldquoGauges for the Ginzburg-Landau equations of superconductivityrdquo Zeitschrift fur Ange-wandte Mathematik und Mechanik vol 76 no 2 pp 345ndash3481996

[60] P Takac ldquoOn the dynamical process generated by supercon-ductivity modelrdquo Zeitschrift fur Angewandte Mathematik undMechanik vol 76 no 52 pp 349ndash352 1995

[61] W D Gropp H G Kaper DM Levine M Palumbo and VMVinokur ldquoNumerical simulation of vortex dynamics in type-IIsuperconductorsrdquo Journal of Computational Physics vol 123 no2 pp 254ndash266 1996

[62] D W Braun G W Crabtree H G Kaper et al ldquoStructure of amoving vortex latticerdquo Physical Review Letters vol 76 no 5 pp831ndash834 1996

[63] G W Crabtree G K Leaf H G Kaper et al ldquoTime-dependent Ginzburg-Landau simulations of vortex guidance bytwin boundariesrdquo Physica C vol 263 no 1ndash4 pp 401ndash408 1996

[64] G W Carabtree G K Leaf H G Kaper D W Braun V MVinokur andA E Koshelev ldquoDynamics vortex phases in super-conductors with correlated disorderrdquo Preprint ANLMCS-P590-0496 Mathematics and Computer Science DivisionAragonne National Laboratory 1996

[65] Th Gallay ldquoA center-stable manifold theorem for differentialequations in Banach spacesrdquo Communications in MathematicalPhysics vol 152 no 2 pp 249ndash268 1993

[66] J-M Ghidaglia and B Heron ldquoDimension of the attractorsassociated to the Ginzburg-Landau partial differential equa-tionrdquo Physica D vol 28 no 3 pp 282ndash304 1987

12 Journal of Function Spaces and Applications

[67] P Mattila Geometry of Sets and Measures in Euclidean Spacesvol 44 Cambridge University Press Cambridge UK 1995

[68] L HormanderThe Analysis of Linear Partial Differential Oper-ators vol 1 Springer Berlin Germany 2nd edition 1990

[69] H Amann Linear and Quasilinear Parabolic ProblemBikhauser Basel Switzerland 1995

[70] H-P Heinz ldquoOn the behaviour ofmeasures of noncompactnesswith respect to differentiation and integration of vector-valuedfunctionsrdquo Nonlinear Analysis Theory Methods amp ApplicationsA vol 7 no 12 pp 1351ndash1371 1983

[71] R P Agarwal M Meehan and D OrsquoRegan Fixed Point Theoryand Applications vol 141 of Cambridge Tracts in MathematicsCambridge University Press Cambridge UK 2001

[72] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer New York NY USA1983

[73] J Fleckinger-Pelle H G Kaper and P Takac ldquoDynamics of theGinzburg-Landau equations of superconductivityrdquo NonlinearAnalysis Theory Methods amp Applications A vol 32 no 5 pp647ndash665 1998

[74] H G Kaper and P Takac ldquoAn equivalence relation for theGinzburg-Landau equations of superconductivityrdquo Journal ofApplied Mathematics and Physics vol 48 no 4 pp 665ndash6751997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces and Applications 3

4119900 if 119880

119899infin

119899=1is a sequence of closed sets from M

119862such

that 119880119899+1

sub 119880119899(119899 = 1 2 ) and if lim

119899rarrinfin120583119879(119880119899) =

0 for each 119879 ge 0 then the intersection set 119880infin

=

⋂infin

119899=1119880119899is nonempty

Remark 1 Observe that in contrast to the definition of theconcept of a measure of noncompactness given in [27] ourmapping 120583

119879may take the valueinfin Moreover a single map-

ping 120583119879is not the measure of noncompactness in 119862(R

+X)

but the whole family 120583119879119879ge0

can be called family of measuresof noncompactness

Remark 2 Let us notice that the intersection set 119880infin

de-scribed in axiom 4

119900 is a member of the kernel of the familyof measure of noncompactness 120583

119879119879ge0

and therefore 119880infin

is compact in 119862(R+X) In fact the inequality 120583

119879(119880infin) le

120583119879(119880119899) for 119899 = 1 2 and 119879 ge 0 implies that 120583

119879(119880infin) = 0

Hence 119880infinisin ker120583

119879 This property of the set 119880

infinwill be

very important in our further investigations

Definition 3 A set 119890minusA119905119905ge0

of bounded linear operators onX is called 119862

0-semigroup if

(i) 119890minusA0 = 119868 119890minusA(119905+119904) = 119890minusA119905119890minusA119904 for 119905 119904 ge 0(ii) for all 119909 isin X the function R

+ni 119905 997891rarr 119890

minusA119905119909 is

continuous in R+

Further denote

119873(119879) = sup 10038171003817100381710038171003817119890minusA11990510038171003817100381710038171003817

119905 le 119879 for 119879 ge 0 (14)

Definition 4 A function 119909 isin 119862(R+X) is said to be a mild

solution of the nonlocal initial-value problem (6)-(7) if forevery 119905 isin R

+

119906 (119905) = 119890minusA1199051199060+ int

119905

0

119890minusA(119905minus119904)

F (119904 119906 (119904)) 119889119904 (15)

To prove the existence results in this paper we need thefollowing lemmas

Lemma 5 (see [21]) If 119884 is a bounded subset of Banach spaceX then for each 120576 gt 0 there is a sequence 119910

119896infin

119896=1sub 119884 such that

120594 (119884) le 2120594 (119910119896infin

119896=1) + 120576 (16)

We call a set119880 sub 1198711([0 119879]X)uniformly integrable if there

exists 120578 isin 1198711([0 119879]R+) such that 119906(119904) le 120578(119904) for 119906 isin 119880 and

ae 119904 isin [0 119879]

Lemma 6 (see [70]) If 119906119896infin

119896=1sub 1198711([0 119879]X) is uniformly

integrable then 120594(119906119896(119905)infin

119896=1) is measurable and

120594(int

119905

0

119906119896(119904) 119889119904

infin

119896=1

)le2int

119905

0

120594 (119906119896(119904)infin

119896=1) 119889119904 119891119900119903 119905isin[0 119879]

(17)

Lemma 7 (see [34]) Assume that a set 119880 sub 119862(R+X) is

bounded Then

(i) sup119905isin[0119879]

120594(119880(119905)) le 120594(119880([0 119879])) le 120596119879

0(119880) + sup

119905isin[0119879]

120594(119880(119905))(ii) sup

119905isin[0119879]120594(119880(119905)) le 120594

119862(119880) le 120596

119879

0(119880) + sup

119905isin[0119879]

120594(119880(119905))

for 119879 ge 0 where 119880([0 119879]) = 119906(119905) 119905 isin [0 119879] 119906 isin 119880

Lemma 8 (Gronwallrsquos inequality [71]) Assume that the func-tion 120601

1 R+rarr R

+is measurable and the function 120601

2

R+rarr R

+is locally integrable Moreover we assume that

int119905

01206011(119904)1206012(119904)119889119904 lt infin for 119905 ge 0 and a measurable function

120595 R+rarr R+satisfies the following inequality

120595 (119905) le 1206011(119905) + int

119905

0

1206012(119904) 120595 (119904) 119889119904 119891119900119903 119905 ge 0 (18)

Then

120595 (119905) le 1206011(119905) + int

119905

0

1206011(119904) 1206012(119904) exp(int

119905

119904

1206012(120591) 119889120591) 119889119904

119891119900119903 119905 ge 0

(19)

Next we consider the operator 119865 119862(R+X) rarr

119862(R+X) defined by the following formula

(119865119906) (119905) = int

119905

0

119890minusA(119905minus119904)

F (119904 119906 (119904)) 119889119904 119905 isin R+

119906 isin 119862 (R+X)

(20)

Lemma 9 (see [20]) Assume that conditions (1198671) (1198672) and

(1198674) are satisfied (see Section 3) a set 119880 sub 119862(R

+X) is

bounded 119879 gt 0 a function ℎ [0 119879] rarr R+is measurable

and

2119873 (119879) 120594 (F (119904 119880 (119904))) le ℎ (119904) (21)

for ae 119904 isin [0 119879] Then

120596119879

0(119865119880) le int

119879

0

ℎ (119904) 119889119904 (22)

3 Main Result

In this section we give an existence result for the semilinearequation of evolution (6)-(7) and we describe the asymptoticbehaviour of those solutions

First we will assume that the functions involved in (6)satisfy the following conditions

(1198671) A is a linear operator acting from 119863(A) sub X and A

generates 1198620-semigroup 119890minusA(119905minus119904)

119905ge0

(1198672) the mapping F R

+times X rarr X satisfies the

Caratheodory condition that isF(sdot 119906) is measurablefor 119906 isin X andF(119905 sdot) is continuous for ae 119905 isin R

+

(1198673) 119886 R

+rarr R

+and 119887 R

+rarr R

+are locally

integrable functions such that

F (119905 119906) le 119886 (119905) + 119887 (119905) 119906 (23)

for any 119906 isin X and ae 119905 isin R+

4 Journal of Function Spaces and Applications

(1198674) There exists a locally integrable function 119871 R

+rarr

R+such that for any bounded119863 sub X

120594 (F (119905 119863)) le 119871 (119905) 120594 (119863) (24)

for ae 119905 isin R+

Theorem 10 Under assumptions (1198671)ndash(1198674) (6) with initial

condition (7) has for every 1199060isin X at least one mild solution

119906 = 119906(119905) which satisfied the following estimate

119906 (119905) le 119872 (119905)

+ 119873 (119905) int

119905

0

119887 (119904)119872 (119904) exp(int119905

119904

119887 (120591)119873 (120591) 119889120591) 119889119904

(25)

where

119872(119905) = 119873 (119905) (100381710038171003817100381711990601003817100381710038171003817 + int

119905

0

119886 (119904) 119889119904) (26)

Proof Consider the operator 119866 119862(R+X) rarr 119862(R

+X)

defined by the following formula

119866119906 (119905) = 119890minusA1199051199060+ int

119905

0

119890minusA(119905minus119904)

F (119904 119906 (119904)) 119889119904 (27)

Now let us observe that for any continuous function 119906

R+rarr X in view of condition (119867

3) we get the following

estimate

119866119906 (119905) le 119873 (119905)100381710038171003817100381711990601003817100381710038171003817 + 119873 (119905) int

119905

0

(119886 (119904) + 119887 (119904) 119906 (119904)) 119889119904

(28)

which yields

119866119906 (119905) le 119872 (119905) + 119873 (119905) int

119905

0

119887 (119904) 119906 (119904) 119889119904 (29)

Next consider the following integral equation

P (119905) = 119872 (119905) + 119873 (119905) int

119905

0

119887 (119904)P (119904) 119889119904 (30)

Solving this equation by standard methods we get

P (119905)=119872 (119905)+119873 (119905)int

119905

0

119887 (119904)119872 (119904) exp(int119905

119904

119887 (120591)119873 (120591) 119889120591) 119889119904

(31)

The function P(119905) is continuous nonnegative and nonde-creasing Observe that the following implication is true

119906 (119905) le P (119905) 997904rArr 119866119906 (119905) le P (119905) for 119905 ge 0 (32)

Indeed linking (29) and (32) we have

119866119906 (119905) le 119872 (119905) + 119873 (119905) int

119905

0

119887 (119904) 119906 (119904) 119889119904

le 119872 (119905) + 119873 (119905) int

119905

0

119887 (119904)P (119904) 119889119904 = P (119905)

(33)

In the space 119862(R+X) let us consider the following set

Γ = 119906 isin 119862 (R+X) 119906 (119905) le P (119905) for 119905 ge 0 (34)

Obviously the set Γ is convex and closed Moreover in viewof (32) we have that 119866 is a self-mapping of Γ

Using the criterion of convergence (lowast) in 119862(R+X) and

standard techniques (see [31ndash33 35]) we can show that theoperator 119866 is continuous on 119862(R

+X)

Now we consider the sequence of sets Γ119899 defined by

induction as follows

Γ0= Γ Γ

119899+1= Conv (119866Γ

119899) for 119899 = 0 1 (35)

This sequence is decreasing that is Γ119899sup Γ119899+1

for 119899 =

0 1 2 Further let us fix 119879 gt 0 and for 119905 isin [0 119879] let us put

119892119899(119905) = sup 120594 (Γ

119899(119904)) 119904 isin [0 119905] (36)

119908119899= 120596119879

0(Γ119899) (37)

The sequences 119908119899 and 119892

119899(119905) are nonincreasing for all 119905 isin

[0 119879] so they have limits

119892infin(119905) = lim

119899rarrinfin119892119899(119905) 119905 isin [0 119879]

119908infin= lim119899rarrinfin

119908119899

(38)

Moreover each function 119892119899is nondecreasing therefore 119892

119899

and 119906infin

are measurable on [0 119879] for 119899 = 1 2 Now we apply the family of measures of noncompactness

120583119879119879ge0

defined in 119862(R+X) by formula (13) In view of the

above notation we have

120583119879(Γ119899) = 119908119899+ 119892119899(119879) (39)

We show that

119908infin= 119892infin(119879) = 0 (40)

To fix a number 119905 isin [0 119879] 119899 isin N and take an arbitrarynumber 119904 isin [0 119905]

We know from Lemma 5 that for any 120576 gt 0 there is asequence 119910

119896infin

119896=1sub 119866Γ119899(119904) such that

120594 (119866Γ119899(119904)) le 2120594 (119910

119896infin

119896=1) + 120576 (41)

This implies that there is a sequence 119906119896infin

119896=1sub Γ119899 such that

119910119896= (119866119906

119896) (119904) for 119896 = 1 2 (42)

Journal of Function Spaces and Applications 5

Hence in view of Lemma 6 (1198674) and (36) we obtain

120594 (119866Γ119899(119904)) le 2120594 (119910

119896infin

119896=1) + 120576

= 2120594 ((119866119906119896) (119904)infin

119896=1) + 120576 le 2120594 (119890

minusA1199041199060)

+ 2120594(int

119904

0

119890minusA(119904minus120591)

F (120591 119906119896(120591)) 119889120591

infin

119896=1

)

+ 120576 le 4int

119904

0

120594 (119890minusA(119904minus120591)

F (120591 119906119896(120591)infin

119896=1)) 119889120591 + 120576

le 4119873 (119879)int

119904

0

119871 (120591) 120594 (119906119896(120591)infin

119896=1) 119889120591 + 120576

le 4119873 (119879)int

119905

0

119871 (120591) 119892119899(120591) 119889120591 + 120576

(43)

Since 120576 gt 0 is arbitrary it follows from the above inequalitiesthat

120594 (119866Γ119899(119904)) le 4119873 (119879)int

119905

0

119871 (120591) 119892119899(120591) 119889120591 119904 isin [0 119905] (44)

Using (36) we have

119892119899+1

(119905) = sup 120594 (119866Γ119899(119904)) 119904 isin [0 119905]

le 4119873 (119879)int

119905

0

119871 (120591) 119892119899(120591) 119889120591 119905 isin [0 119879]

(45)

Letting 119899 rarr infin we derive the following inequality

119892infin(119905) le 4119873 (119879)int

119905

0

119871 (120591) 119892infin(120591) 119889120591 119905 isin [0 119879] (46)

This inequality together with Gronwallrsquos Lemma 8 impliesthat

119892infin(119905) = 0 119905 isin [0 119879] (47)

Next let us notice that in view of (1198674) and (36) we have

120594 (F (119904 Γ119899(119904))) le 119871 (119904) 120594 (Γ

119899(119904)) le 119871 (119904) 119892

119899(119904) (48)

for ae 119904 isin [0 119879] and the function

[0 119879] ni 119904 997891997888rarr 2119873 (119879) 119871 (119904) 119892119899(119904) (49)

is measurable Then in virtue of equality 1205961198790(119866Γ119899) = 120596119879

0(119865Γ119899)

and Lemma 9 we get

120596119879

0(119866Γ119899) le 2119873 (119879)int

119879

0

119871 (119904) 119892119899(119904) 119889119904 (50)

Hence we derive

119908119899+1

= 120596119879

0(Γ119899+1) = 120596119879

0(119866Γ119899)

le 2119873 (119879)int

119879

0

119871 (119904) 119892119899(119904) 119889119904

(51)

Letting 119899 rarr infin we get

119908infinle 2119873 (119879)int

119879

0

119871 (119904) 119892infin(119904) 119889119904 (52)

Keeping in mind (47) we deduce that

119908infin= 0 (53)

This together with (39) and (47) yields

lim119899rarrinfin

120583119879(Γ119899) = lim119899rarrinfin

(119908119899+ 119892119899(119879)) = 119908

infin+ 119892infin= 0 (54)

Finally using Remark 2 for the measure 120583119879 we deduce

that the set Γinfin= ⋂infin

119899=0Γ119899is nonempty convex and compact

Then by the Schauder-Tychonoff theorem we conclude thatoperator 119865 Γ

infinrarr Γinfin

has at least one fixed-point 119906 = 119906(119905)Obviously the function 119906 = 119906(119905) is a solution of problem (6)-(7) and in view of the definition of the set Γ

infin the estimate

119906(119905) le P(119905) holds to be trueThis completes the proof

4 Local and Global Attractivity

Following the concepts introduced in [36] we introduce firsta few definitions of various kinds of the concept of attractivityof mild solution of (6)

Definition 11 The mild solution 119906 = 119906(119905) of (6) with initialcondition (7) is said to be globally attractive if for each mildsolution V = V(119905) of (6) with initial condition V(0) = V

0we

have that

lim119905rarrinfin

(119906 (119905) minus V (119905)) = 120579 (55)

In other words we may say that solutions of (6) are globallyattractive if for arbitrary solutions 119906(119905) and V(119905) of thisequation condition (55) are satisfied

Definition 12 We say that mild solution 119906 = 119906(119905) of (6) withinitial condition (7) is locally attractive if there exists a ball119861(119906(0) 119903) in the space X such that for arbitrary solution V(119905)of (1) with initial-value V(0) isin 119861(119906(0) 119903) condition (55) doeshold

In the case when the limit (55) is uniform with respect toall solutions V(119905) that is when for each 120576 gt 0 there exist119879 gt 0such that

119906 (119905) minus V (119905) le 120576 (56)

for all V(119905) being solutions of (6) with initial-value V(0) isin119861(119906(0) 119903) and for 119905 ge 119879 we will say that solution 119906 = 119906(119905)

is uniformly locally attractive on R+

Now we formulate themain result of this sectionWe willconsider (6) under the following conditions

(1198671015840

1) A is the infinitesimal generator of an exponentiallystable119862

0-semigroup 119890minusA119905

119905ge0 that is there exist119872 gt

0 120588 gt 0 such that 119890minusA119905 le 119872119890minus120588119905 for all 119905 ge 0

6 Journal of Function Spaces and Applications

(1198675) there exist locally integrable functions119898 R

+rarr R+

such that

F (119905 119906) minusF (119905 V) le 119898 (119905) 119906 minus V (57)

for 119905 ge 0 and 119906 V isin X Moreover we assume that

lim119905rarrinfin

(int

119905

0

119898(119904) 119889119904 minus120588

119872119905) = minusinfin (58)

Remark 13 The property (11986710158401) is generally satisfied in diffu-

sion problem A necessary and sufficient condition for (11986710158401)

is presented in [72]

The main result of this section is shown in the giventheorem below

Theorem 14 Under assumptions (11986710158401) and (119867

2)ndash(1198675) prob-

lem (6)-(7) has amild solution 119906 = 119906(119905) for each 1199060isin X which

is globally attractive and locally uniformly attractive

Proof Existence of a solution 119906 = 119906(119905) is a consequence ofTheorem 10 Let us fix 119903 gt 0 and V

0isin 119861(119906(0) 119903)

Let V = V(119905) denote a mild solution of (6) with the initialcondition V(0) = V

0 Using (1198671015840

1) and (119867

3) we get

V (119905) le10038171003817100381710038171003817119890minusA119905V0

10038171003817100381710038171003817+ int

119905

0

10038171003817100381710038171003817119890minusA(119905minus119904)10038171003817100381710038171003817

sdot F (119904 V (119904)) 119889119904

le 119872(100381710038171003817100381711990601003817100381710038171003817 + 119903) +119872int

119905

0

119890120588119904119886 (119904) 119889119904

+ int

119905

0

119872119890120588119904119887 (119904) V (119904) 119889119904

(59)

Now let us put

P1(119905) = 119872 (

1003817100381710038171003817V01003817100381710038171003817 + 119903) +119872int

119905

0

119890120588119904119886 (119904) 119889119904

P2(119905) = 119872119890

120588119904119887 (119905)

(60)

Taking into account Lemma 5 we obtain

V (119905) le P1(119905) + int

119905

0

P2(119904)P1(119904) exp(int

119904

0

P2(120591) 119889120591) 119889119904

(61)

Further let 119878 be the set of all mild solutions V(119905) of (6) withthe initial-value V

0= V(0) isin 119861(119906(0) 119903)

Put

119892 (119905) = sup 119906 (119905) minus V (119905) V isin 119878 (62)

The estimate (61) implies that the function119892(119905) is well definedApplying (119867

5) and (1198671015840

1) we get

119906 (119905) minus V (119905) le10038171003817100381710038171003817119890minusA11990510038171003817100381710038171003817

sdot10038171003817100381710038171199060 minus V

0

1003817100381710038171003817

+ int

119905

0

10038171003817100381710038171003817119890minusA(119905minus119904)10038171003817100381710038171003817

119898 (119904) 119906 (119904) minus V (119904) 119889119904

le 119872119890minus120588119905119903 + int

119905

0

119872119890minus119892(119905minus119904)

119898(119904) 119892 (119904) 119889119904

(63)

or equivalently

119890120588119905119892 (119905) le 119872119903 + int

119905

0

119872119898(119904) 119890120588119904119892 (119904) 119889119904 (64)

Using again Lemma 5 for the above estimate (where ℎ(119905) =119890120588119905119892(119905)) we obtain

119890120588119905119892 (119905) le 119872119903

+ int

119905

0

1198722119903119898 (119904) exp(int

119905

119904

119872119898(120591) 119889120591) 119889119904

(65)

Elementary calculations lead to the following equality

int

119905

0

1198722119903119898 (119904) exp(int

119905

119904

119872119898(120591) 119889120591) 119889119904

= 119872119903(exp(int119905

0

119872119898(119904) 119889119904) minus 1)

(66)

Hence

119892 (119905) le 119872119903 exp (minus120588119905) + 119872119903 exp(int119905

0

119872119898(119904) 119889119904 minus 120588119905)

minus119872119903 exp (minus120588119905) (67)

Applying assumption (1198675) we derive

lim119905rarrinfin

119892 (119905) = 0 (68)

and this proves that 119906(119905) is locally attractive Finally thisequality together with definition of the function 119892(119905) impliesthat 119906(119905) is globally attractive The proof is complete

Remark 15 In the case when 119898(119905) equiv 119898 is constant the fol-lowing condition

lim119905rarrinfin

(int

119905

0

119898(119904) 119889119904 minus120588

119872119905) = minusinfin (69)

means that 119898 lt 120588119872 Observe that this condition cannotbe weakened This observation is illustrated by the followingexampled

Example 16 Let X = R 119898(119905) equiv 119898 F(119905 119906) = sin 119905 + 119898119906119872 = 1 and 119890minusA119905 = 119890minus120588119905 Then the equation

119906 (119905) = 119890minusA1199051199060+ int

119905

0

119890minusA(119905minus119904)

F (119904 119906 (119904)) 119889119904 (70)

(for any fixed 1199060isin R) has the solution 119906(119905) expressed by the

following formula

119906 (119905) =(120588 minus 119898) sin 119905 minus cos 119905

(120588 minus 119898)2

+ 1+ (1199060+

1

(120588 minus 119898)2

+ 1) 119890(119898minus120588)119905

(71)

Notice that for 119898 ge 120588 the solution 119906(119905) is neither globallyattractive nor locally uniformly attractive because for each

Journal of Function Spaces and Applications 7

other solution V(119905) with initial condition V(0) = V0 obviously

described by similar formula as 119906(119905) we would have acontradiction

|119906 (119905) minus V (119905)| =

10038161003816100381610038161199060 minus V0

1003816100381610038161003816 119890(119898minus120588)119905

997888rarr infin

or10038161003816100381610038161199060 minus V

0

1003816100381610038161003816

(72)

5 Asymptotic Behaviour

In this section we will give a theorem describing asymptoticbehaviour of mild solutions of (6) with condition (7) Thistheorem generalizes the result included in [72Theorem 44]First we formulate the assumptions

(1198671015840

3) This condition is almost identical with (119867

3) and the

only difference is that we assume the functions 119886 and119887 are locally essentially bounded on R

+

(1198671015840

5) There exists 119906

infinisin X such that there exists the limit

lim119905rarrinfin

F(119905 119906infin) and

lim119905rarrinfin

F (119905 119906infin) = A

minus1119906infin (73)

Moreover there exists a number119898 lt 120588119872 such that1003817100381710038171003817F (119905 119906) minusF (119905 119906

infin)1003817100381710038171003817 le 119898

1003817100381710038171003817119906 minus 119906infin1003817100381710038171003817 (74)

for 119905 ge 0 and 119906 isin X

Remark 17 The condition (11986710158405) in conjunction with (1198671015840

1)

ensures the existence ofAminus1 (see [72]) Clearly (11986710158403) implies

(1198673)

Theorem 18 Under assumptions (11986710158401) (1198672) (11986710158403) (1198674) and

(11986710158405) (6) with condition (7) has a mild solution 119906 = 119906(119905) for

each 1199060isin X such that lim

119905rarrinfin119906(119905) = 119906

infin

Proof The existence of a mild solution 119906(119905) is guaranteed byTheorem 10 Let us put

119892 (119905) =1003817100381710038171003817119906 (119905) minus 119906infin

1003817100381710038171003817

Finfin= lim119905rarrinfin

F (119905 119906infin)

(75)

We show that

lim119905rarrinfin

119892 (119905) = 0 (76)

Recall that if assumption (11986710158401) is fulfilled then for each 119911 isin X

we have (see [72])

int

infin

0

119890minusA119904

119911 119889119904 = minusAminus1119911 (77)

Using the above fact and (11986710158405) we get

119906infin= (minusA

minus1(minusAminus1119906infin)) = minusA

minus1Finfin

= int

infin

0

119890minusA119904

Finfin119889119904

(78)

Linking the above equality with (11986710158401) we obtain

119892 (119905)

=

10038171003817100381710038171003817100381710038171003817119890minusA1199051199060+ int

119905

0

119890minusA(119905minus119904)

F (119904 119906 (119904)) 119889119904 minus int

infin

0

119890minusA119904

Finfin119889119904

10038171003817100381710038171003817100381710038171003817

le 119872119890minus120588119905 10038171003817100381710038171199060

1003817100381710038171003817 +

10038171003817100381710038171003817100381710038171003817int

infin

119905

119890minusA119904

Finfin119889119904

10038171003817100381710038171003817100381710038171003817

+

10038171003817100381710038171003817100381710038171003817int

119905

0

119890minusA(119905minus119904)

F (119904 119906 (119904)) 119889119904 minus int

119905

0

119890minusA119904

Finfin119889119904

10038171003817100381710038171003817100381710038171003817

le 119872119890minus120588119905 10038171003817100381710038171199060

1003817100381710038171003817 +119872

119890minus1205881199051205881003817100381710038171003817Finfin

1003817100381710038171003817

+

10038171003817100381710038171003817100381710038171003817int

119905

0

119890minusA(119905minus119904)

F (119904 119906 (119904)) 119889119904 minus int

119905

0

119890minusA(119905minus119904)

Finfin119889119904

10038171003817100381710038171003817100381710038171003817

le 119872119890minus120588119905(100381710038171003817100381711990601003817100381710038171003817 +

1003817100381710038171003817Finfin1003817100381710038171003817

120588)

+ int

119905

0

119872119890minus120588(119905minus119904) 1003817100381710038171003817F (119904 119906 (119904)) minusF

infin

1003817100381710038171003817 119889119904

le 119872119890minus120588119905(100381710038171003817100381711990601003817100381710038171003817 +

1003817100381710038171003817Finfin1003817100381710038171003817

120588)

+119872119890minus120588119905int

119905

0

119890120588119904 1003817100381710038171003817F (119904 119906

infin) minusF

infin

1003817100381710038171003817 119889119904

+119872119890minus120588119905int

119905

0

119890120588119904 1003817100381710038171003817F (119904 119906 (119904)) minusF (119904 119906

infin)1003817100381710038171003817 119889119904

(79)

Next putting

119888 (119905) = 119872119890minus120588(119905)

(100381710038171003817100381711990601003817100381710038171003817 +

1003817100381710038171003817Finfin1003817100381710038171003817

120588)

+119872119890minus120588119905int

119905

0

119890120588119904 1003817100381710038171003817F (119904 119906

infin) minusF

infin

1003817100381710038171003817 119889119904

(80)

and applying (11986710158405) we derive the following inequality

119892 (119905) le 119888 (119905) + 119898119872119890minus120588119905int

119905

0

119890120588119904119892 (119904) 119889119904 (81)

Hence

119890120588119905119892 (119905) le 119890

120588(119905)119888 (119905) + 119898119872int

119905

0

119890120588119904119892 (119904) 119889119904 (82)

The above inequality in conjunction with Lemma 5 gives

119890120588119905119892 (119905) le 119890

120588119905119888 (119905)

+ int

119905

0

119890120588119904119888 (119904)119898119872 sdot exp (119898119872(119905 minus 119904)) 119889119904

= 119890120588119905119888 (119905)

+ 119898119872119890119898119872119905

int

119905

0

119888 (119904) exp ((120588 minus 119898119872) 119904) 119889119904

(83)

8 Journal of Function Spaces and Applications

Hence

119892 (119905) le 119888 (119905) +119898119872int

119905

0119888 (119904) 119890(119892minus119898119872)119904

119889119904

exp ((120588 minus 119898119872) 119905) (84)

Before proving (76) we first show that

lim119905rarrinfin

119888 (119905) = 0 (85)

To prove this equality it is sufficient to show that the lastcomponent in the formula expressing 119888(119905) that is 119872119890minus120588119905

int119905

0119890120588119904F(119904 119906

infin) minusF

infin119889119904 tends to zero as 119905 rarr infin

To this end let us fix 120576 gt 0 Assumption (11986710158405) implies that

there exists 1199050such that

sup 1003817100381710038171003817F (119904 119906infin) minusF

infin

1003817100381710038171003817 119904 ge 1199050 le120576120588

2119872 (86)

This inequality together with (11986710158403) implies

ess sup 1003817100381710038171003817F (119904 119906infin) minusF

infin

1003817100381710038171003817 119904 isin R lt infin (87)

Now fix 1199051isin R such that 119905

1ge 119905 and

ess sup 1003817100381710038171003817F (119904 119906infin) minusF

infin

1003817100381710038171003817 119904 isin R

le120576120588 exp (120588119905

1)

2119872 (exp (1205881199050) minus 1)

(88)

Then using (86) and (88) we conclude that for 119905 ge 1199051we have

119872119890minus120588119905int

119905

0

119890120588119904 1003817100381710038171003817F (119904 119906

infin) minusF

infin

1003817100381710038171003817 119889119904

le 119872119890minus120588119905int

1199050

0

119890120588119904 1003817100381710038171003817F (119904 119906

infin) minusF

infin

1003817100381710038171003817 119889119904

+119872119890minus120588119905int

119905

1199050

119890120588119904 1003817100381710038171003817F (119904 119906

infin) minusF

infin

1003817100381710038171003817 119889119904 le120576

2+120576

2= 120576

(89)

This fact proves (85)Further using (1198671015840

5) and (85) and employing de

LrsquoHospitalrsquos rule for the fraction on the right-hand sideof inequality (84) we obtain that condition (76) is satisfiedThis fact completes the proof

6 An Application to the Ginzburg-LandauEquations of Superconductivity

In this section we formulate the gauged time-dependentGinzburg-Landau (TDGL) equations as an abstract evolutionequation in a Hilbert space Moreover we show applicationsof the above theorems to TDGL equations

We assume that Ω is a bounded domain in R119899 withboundary 120597Ω of class11986211That isΩ is an open and connectedset whose boundary 120597Ω is a compact (119899 minus 1)-manifolddescribed by Lipschitz continuous differentiable charts Weconsider two- and three-dimensional problems (119899 = 2 and119899 = 3 resp) Assume that the vector potential A takes its

values in R119899 The vector H will represent the (externally)applied magnetic field which is a function of space andtime similarly to A it takes its values in R119899 The function120574 is defined and satisfies Lipschitz condition on 120597Ω and120574(119909) ge 0 for 119909 isin 120597Ω The parameters in the TDGLequations are 120578 a (dimensionless) friction coefficient and 120581the (dimensionless) Ginzburg-Landau parameter

The order parameter should be thought of as the wavefunction of the center-of-massmotion of the ldquosuperelectronsrdquo(Cooper pairs) whose density is 119899

119904= |120595|

2 and whose fluxis J119904 The vector potential A determines the electromagnetic

field E = minus120597119905A minus nabla120601 is the electric field and B = nabla times A

is the magnetic induction where J the total current is thesum of a ldquonormalrdquo current J

119899= E the supercurrent J

119904 and

the transport current J119905= nabla times H The normal current obeys

Ohmrsquos law J119899= 120590119899E the ldquonormal conductivityrdquo coefficient 120590

119899

is equal to one in the adopted system of units The differenceM = BminusH is known as themagnetizationThe trivial solution(120595 = 0 B = H E = 0) represents the normal state where allsuperconducting properties have been lost

Nowwe accept the following notion all Banach spaces arereal the (real) dual of a Banach space119883 is denoted by1198831015840Thesymbol 119871119901(Ω) for 1 le 119901 le infin denotes the usual Lebesguespace with norm sdot

119871119901 (sdot sdot) is the inner product in 1198712(Ω)

1198821198982(Ω) for nonnegative integer 119898 is the usual Sobolev

space with norm sdot 1198821198982 1198821198982(Ω) is a Hilbert space for the

inner product (sdot sdot)1198982

given by (119906 V)1198982

= sum|120572|le119898

(120597120572119906 120597120572V)

for 119906 V isin 1198821198982(Ω) Fractional Sobolev space1198821199042(Ω) with afractional 119904 is defined by interpolation ([40 Chap VII] and[41 49 50]) 119862]

(Ω) for ] ge 0 ] = 119898 + 120582 with 0 le 120582 lt 1 isthe space of 119898 times continuous differentiable functions onΩ those 119898th-order derivatives satisfy the Holder conditionwith exponent 120582 if ] is a proper fraction the norm sdot

119862] is

defined in the usual wayThe definitions extend to the space of vector-valued

functions in the standard way with the caveat that the innerproduct in [1198712(Ω)]119899 is defined by (119906 V) = int

Ω119906 sdot V where the

symbol (119906 V) indicates the scalar product in R119899 Complex-valued functions are interpreted as vector-valued functionswith two real components

Functions that vary in space and time like the orderparameter and the vector potential are considered as map-pings from the time domain which is a subinterval of [0infin)into spaces of complex- or vector-valued functions definedin Ω Let 119883 = (119883 sdot

119883) be a Banach space of functions

defined inΩ Then the functions are defined in Ω Then thefunctions of space and time defined on Ω times (0 119879) for 119879 gt 0may be considered as elements of 119871119901(0 119879119883) for 1 le 119901 le infinor1198822119898(0 119879119883) for nonnegative119898 or119862]

(0 119879119883) for ] ge 0] = 119898 + 120582 with 0 le 120582 lt 1 Detailed definitions can be foundfor example in [43]

Obviously function spaces of ordered pairs (120595A) where120595 Ω rarr R2 and A Ω rarr R119899 (119899 = 2 3) play an importantrole in the study of the gauged TDGL equationsWe thereforeadapt the following special notationH = [119883(Ω)]

2times[119883(Ω)]

119899for any Banach space for the order parameter 120595 and thevector potential A respectively A suitable framework forthe functional analysis of the gauged TDGL equations is the

Journal of Function Spaces and Applications 9

Cartesian product W1+1205722 = [1198821+1205722

(Ω)]2times [1198821+1205722

(Ω)]119899

where 120572 isin (12 1) This space is continuously imbedded inW12 capLinfin

Assume H = 0 and H isin [1198712(Ω)]119899 Let AH be a minimizer

of the convex quadratic form 119869120596equiv J120596[119860]

119869120596 [A] = int

Ω

[120596(nabla sdot A)2 + |nabla times A minusH|2] 119889119909 (90)

on the domain

D (119869120596) = A isin [119882

12(Ω)]119899

n sdot A = 0 on 120597Ω (91)

We now introduce the reduced vector potential A1015840

A1015840 = A minus AH (92)

In terms of 120595 and A1015840 the gauged TDGL equations have thefollowing form

120597120595

120597119905minus

1

1205781205812Δ120595 = 120593 in Ω times (0infin)

120597A1015840

120597119905+ nabla times nabla times A1015840 minus 120596nabla (nabla sdot A1015840) = F in Ω times (0infin)

n sdot nabla + 120574120595 = 0 n sdot A1015840 = 0

n times (nabla sdot A1015840) = 0 on 120597Ω times (0infin)

(93)

Here 120593 and F are nonlinear functions of 120595 and A1015840

120593 equiv 120593 (119905 120595A1015840) = 1

120578[minus2119894

120581(nabla120595) sdot (A1015840 + AH)

minus119894

120581(1 minus 120578120581

2120596)120595 (nabla sdot A1015840)

minus12059510038161003816100381610038161003816A1015840 + AH

10038161003816100381610038161003816

2

+ (1 minus100381610038161003816100381612059510038161003816100381610038162

) 120595]

(94)

F equiv F (119905 120595A1015840) = 1

2119894119896(120595lowastnabla120595 minus 120595nabla120595

lowast)

minus100381610038161003816100381612059510038161003816100381610038162

(A1015840 + AH) minus120597AH120597119905

(95)

The equations are supplemented by initial data which is inthe followimg form

120595 = 1205950 A = A

0on Ω times 0 (96)

where 1205950and A

0are given and by (92) we have

120595 = 1205950 A1015840 = A

0minus AH (0) on Ω times 0 (97)

In the next part we connect the evolution of the solution(120595A1015840) of the system of (93) with the initial data (120595

0A0minus

AH(0)) with the dynamics of a vector 119906 in a Hilbert space

L2= [119871119901(Ω)]2

times [119871119901(Ω)]119899

(98)

The following analysis is restricted to the case 120596 gt 0 andthe case 120596 = 0 (see [73])

Let vector 119906 [0infin) rarr L2 represent the pair (120595A1015840)

119906 = (120595A1015840) equiv (120595A minus AH) (99)

and letA be the linear self-adjoint operator inL2 associatedwith the quadratic form 119876

120596equiv 119876120596[119906] given by the following

formula

119876120596 [119906] = int

Ω

[1

12057812058121003816100381610038161003816nabla120595

10038161003816100381610038162

+ 120596(nabla sdot A1015840)2

+10038161003816100381610038161003816nabla times A101584010038161003816100381610038161003816

2

] 119889119909

+ int120597Ω

120574

1205781205812100381610038161003816100381612059510038161003816100381610038162

119889120590 (119909)

(100)

on the domain

D (119876120596) = D (A

12)

= 119906 = (120595A1015840) isinW12 n sdot A1015840 = 0 on 120597Ω

(101)

The quadratic form 119876120596is nonnegative Furthermore since

120596 gt 0119876120596[120595A1015840]+119888 120595

1198712 is coercive onW12 for any constant

119888 gt 0 Hence A is positively definite in L2 ([44 Chap 1]equation (545)) If it does not lead to confusion we use thesame symbol A for the restriction A

120595and AA of A to the

respective linear subspace [1198712(Ω)]2 = [1198712(Ω)]2 times 0 (for 120595)and [1198712(Ω)]119899 equiv 0 times [1198712(Ω)]119899 (for A) ofL2

Now consider the initial-value problems (6) and (7) inL2 whereF(119905 119906) = (120593 F) 120593 and F is given by (94) and (95)and 119906

0= (1205950A0minus AH(0))

With 120582 isin (12 1) and 1199060isin W1+1205822 we say that 119906 are a

solution of (6) and (7) on the interval [0 119879] for some 119879 gt 0if 119906 [0 119879] rarr W1+1205822 is continuous and

119906 (119905) = 119890minusA(119905)

+ int

119905

0

119890minusA(119905minus119904)

F (119904 119906 (119904)) 119889119904 for 119905 isin [0 119879]

(102)

in L2 A mild solution of the initial-value problems (6) and(7) defines a weak solution (120595A1015840) of the boundary valueproblem (93) which in turn defines a weak solution (120595A)of the gauged TDGL equations provided AH is sufficientlyregular

Namely let us assume that X = L2 and 1199060isin L2 and

119906 R+times R119899 rarr R119899 for 119899 isin N is an unknown function 119906 =

119906(119905 119909) In order to applyTheorems 10 and 14 we are not goingto consider 119906 as a function of 119905 and 119909 together but rather asa mapping 119906 of variable 119905 into the spaceX =L2 of functions119909 that is 119906 R

+rarr L2 119906(119905)(119909) = 119906(119905 119909) 119909 isin R119899 119905 isin R

+

Remark 19 Since A is the linear self-adjoint operator inL2

associated with the quadratic form (100) thenA generates asemigroup 119890minusA119905

119905ge0(see [74])

Below we formulate the principal theorem of this paperThis theorem is a simple consequence of Theorem 10

10 Journal of Function Spaces and Applications

Theorem 20 Under assumptions (1198671)ndash(1198674) problem (6)-(7)

is equivalent to the system of (1)ndash(5) (TDGL equations) andhas for each 119906

0isin L2 at least one mild solution in sense of

Definition 4 119906 = 119906(119905) Moreover each solution 119906(119905) satisfiesestimate (25)

We observe that in the case when 119890minusA119905119905ge0

is the semi-group of contractions and 119886(119905) equiv 119886 119887(119905) equiv 119887 for 119905 isin

R+are constant after simple calculations based on estimate

(25) we get that the solution 119906 = 119906(119905) has the asymptoticcharacterization

119906 (119905) = 119874 (1199053) as 119905 997888rarr infin (103)

For further purposes let us formulate the followingassumption

(A) A is the infinitesimal generator of an exponentiallystable 119862

0-semigroup 119890minusA119905

119905ge0

The next result of this chapter is shown in the giventheorem below

Theorem21 Under assumptions (A) and (1198672)ndash(1198674) problem

(6)-(7) which is equivalent to the systems equations (1)ndash(5)(TDGL equations) has a mild solution 119906 = 119906(119905) for each1199060isin L2 which is globally attractive and locally uniformly

attractive

References

[1] V L Ginzburg and L D Landau ldquoOn the theory of supercon-ductivityrdquo Zhurnal Eksperimentalrsquonoi i Teoreticheskoi Fiziki vol20 pp 1064ndash1082 1950 English translation in L D LandauMen of Physics D ter Haar ed Vol I Pergamon Press OxfordUK pp 138ndash167 1965

[2] S Aizicovici and H Lee ldquoNonlinear nonlocal Cauchy problemsin Banach spacesrdquo Applied Mathematics Letters vol 18 no 4pp 401ndash407 2005

[3] S Aizicovici and M McKibben ldquoExistence results for a classof abstract nonlocal Cauchy problemsrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 39 no 5 pp 649ndash6682000

[4] MMcKibbenDiscovering Evolution Equation with ApplicationCRC Press Taylor amp Francis Group 2011

[5] S Aizicovici and V Staicu ldquoMultivalued evolution equationswith nonlocal initial conditions in Banach spacesrdquo NonlinearDifferential Equations and Applications vol 14 no 3-4 pp 361ndash376 2007

[6] R P Agarwal and D OrsquoRegan Infinite Interval Problems for Dif-ferential Difference and Integral Equations Kluwer AcademicPublishers Dordrecht The Netherlands 2001

[7] Z Fan ldquoExistence of nondensely defined evolution equationswith nonlocal conditionsrdquo Nonlinear Analysis Theory Methodsamp Applications A vol 70 no 11 pp 3829ndash3836 2009

[8] Z Fan ldquoImpulsive problems for semilinear differential equa-tions with nonlocal conditionsrdquo Nonlinear Analysis TheoryMethods amp Applications A vol 72 no 2 pp 1104ndash1109 2010

[9] Z Fan and G Li ldquoExistence results for semilinear differentialequations with nonlocal and impulsive conditionsrdquo Journal ofFunctional Analysis vol 258 no 5 pp 1709ndash1727 2010

[10] J Garcıa-Falset ldquoExistence results and asymptotic behavior fornonlocal abstract Cauchy problemsrdquo Journal of MathematicalAnalysis and Applications vol 338 no 1 pp 639ndash652 2008

[11] J Liang J H Liu and T-J Xiao ldquoNonlocal impulsive problemsfor nonlinear differential equations in Banach spacesrdquo Mathe-matical and Computer Modelling vol 49 no 3-4 pp 798ndash8042009

[12] L Zhu and G Li ldquoOn a nonlocal problem for semilinear differ-ential equations with upper semicontinuous nonlinearities ingeneral Banach spacesrdquo Journal of Mathematical Analysis andApplications vol 341 no 1 pp 660ndash675 2008

[13] Q Liu and R Yuan ldquoExistence of mild solutions for semilinearevolution equations with non-local initial conditionsrdquo Nonlin-ear AnalysisTheoryMethodsampApplications A vol 71 no 9 pp4177ndash4184 2009

[14] C Xie W Zhong and Z Fan ldquoExistence results for nonlinearnonlocal problems in Banach spacesrdquo Applied MathematicsLetters vol 22 no 7 pp 998ndash1002 2009

[15] T-J Xiao and J Liang ldquoExistence of classical solutions tononautonomous nonlocal parabolic problemsrdquoNonlinear Anal-ysisTheoryMethods and Applications vol 63 no 5-7 pp e225ndashe232 2005

[16] X Xue ldquo119871119901 theory for semilinear nonlocal problems with mea-sure of noncompactness in separable Banach spacesrdquo Journal ofFixed Point Theory and Applications vol 5 no 1 pp 129ndash1442009

[17] X Xue ldquoNonlinear differential equations with nonlocal condi-tions in Banach spacesrdquo Nonlinear Analysis Theory Methods ampApplications A vol 63 no 4 pp 575ndash586 2005

[18] X Xue ldquoExistence of solutions for semilinear nonlocal Cauchyproblems in Banach spacesrdquo Electronic Journal of DifferentialEquations vol 64 p 7 2005

[19] X Xue ldquoNonlocal nonlinear differential equations with a mea-sure of noncompactness in Banach spacesrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 70 no 7 pp 2593ndash26012009

[20] L Olszowy ldquoExistence ofmild solutions for the semilinear non-local problem in Banach spacesrdquo Nonlinear Analysis TheoryMethods amp Applications A vol 81 pp 211ndash223 2013

[21] D Bothe ldquoMultivalued perturbations of119898-accretive differentialinclusionsrdquo Israel Journal of Mathematics vol 108 pp 109ndash1381998

[22] H-S Ding J Liang G M NrsquoGuerekata and T-J XiaoldquoMild pseudo-almost periodic solutions of nonautonomoussemilinear evolution equationsrdquo Mathematical and ComputerModelling vol 45 no 5-6 pp 579ndash584 2007

[23] J Garcıa-Falset ldquoThe asymptotic behavior of the solutions of theCauchy problem generated by 120593-accretive operatorsrdquo Journal ofMathematical Analysis and Applications vol 310 no 2 pp 594ndash608 2005

[24] Z Fan and G Li ldquoAsymptotic behavior of the solutions of non-autonomous systems in Banach spacesrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 68 no 12 pp 3733ndash37412008

[25] J Banas ldquoOn existence theorems for differential equations inBanach spacesrdquo Bulletin of the Australian Mathematical Societyvol 32 no 1 pp 73ndash82 1985

[26] J Banas ldquoMeasures of noncompactness in the space of contin-uous tempered functionsrdquo Demonstratio Mathematica vol 14no 1 pp 127ndash133 1981

Journal of Function Spaces and Applications 11

[27] J Banas and K GoebelMeasures of Noncompactness in BanachSpaces vol 60 ofLectureNotes in Pure andAppliedMathematicsMarcel Dekker New York NY USA 1980

[28] J Banas and M Lecko ldquoSolvability of infinite systems ofdifferential equations in Banach sequence spacesrdquo Journal ofComputational andAppliedMathematics vol 137 no 2 pp 363ndash375 2001

[29] J Banas and K Sadarangani ldquoCompactness conditions inthe study of functional differential and integral equationsrdquoAbstract and Applied Analysis vol 2013 Article ID 819315 14pages 2013

[30] M Mursaleen and S A Mohiuddine ldquoApplications of mea-sures of noncompactness to the infinite system of differentialequations in ℓ

119901spacesrdquo Nonlinear Analysis Theory Methods amp

Applications A vol 75 no 4 pp 2111ndash2115 2012[31] L Olszowy ldquoSolvability of some functional integral equationrdquo

Dynamic Systems and Applications vol 18 no 3-4 pp 667ndash6762009

[32] L Olszowy ldquoOn some measures of noncompactness in theFrechet spaces of continuous functionsrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 72 pp 2119ndash2126 2010

[33] L Olszowy ldquoOn existence of solutions of a quadratic Urysohnintegral equation on an unbounded intervalrdquo CommentationesMathematicae vol 48 no 1 pp 103ndash112 2008

[34] L Olszowy ldquoExistence ofmild solutions for semilinear nonlocalCauchy problems in separable Banach spacesrdquo Journal of Anal-ysis and its Applications vol 32 no 2 pp 215ndash232 2013

[35] L Olszowy ldquoFixed point theorems in the Frechet space 119862(119877+)

and functional integral equations on an unbounded intervalrdquoApplied Mathematics and Computation vol 218 no 18 pp9066ndash9074 2012

[36] J Banas and D OrsquoRegan ldquoOn existence and local attractivity ofsolutions of a quadratic Volterra integral equation of fractionalorderrdquo Journal of Mathematical Analysis and Applications vol345 no 1 pp 573ndash582 2008

[37] L D Landau EM Lifshitz and L P Pitaevski Electrodynamicsof Continuous Media vol 8 Butterworth-Heinemann 2ndedition 1984

[38] L D Landau and E M Lifshitz Fluid Mechanics vol bButterworth-Heinemann 2nd edition 1987

[39] R Temam Infinite-Dimensional Dynamical Systems inMechan-ics and Physics vol 68 Springer New York NY USA 2ndedition 1997

[40] J K Hale Asymptotic Behavior of Dissipative Systems vol 25 ofMathematical Surveys and Monographs American Mathemati-cal Society Providence RI USA 1988

[41] R A Adams and J J F Fournier Sobolev Spaces vol 140Elsevier 2nd edition 2003

[42] V Mazrsquoya Sobolev Spaces with Applications to Elliptic PartialDifferential Equations Springer Heidelberg Germany 2ndedition 2011

[43] V Georgescu ldquoSome boundary value problems for differentialforms on compact Riemannian manifoldsrdquo Annali di Matemat-ica Pura ed Applicata vol 122 pp 159ndash198 1979

[44] D Henry Geometric Theory of Semilinear Parabolic Equationsvol 840 of Lecture Notes in Mathematics Springer New YorkNY USA 1981

[45] V Girault and P-A Raviart Finite Element Methods for Navier-Stokes Equations Springer New York NY USA 1986

[46] J Bardeen L N Cooper and J R Schrieffer ldquoTheory ofsuperconductivityrdquo Physical Review vol 108 pp 1175ndash12041957

[47] A A Abrikosov Fundamental of the Theory of Metals North-Holland Amstedram The Netherlands 1988

[48] M Tinkham Introduction To Superconductivity McGraw-HillNew York NY USA 2nd edition 1996

[49] P DeGennes Superconductivity inMetals and Alloys BenjaminNew York NY USA 1966

[50] H Brezis Functional Analysis Sobolev Spaces and PartialDifferential Equations Springer New York NY USA 2011

[51] E Zeidler Nonlinear Functional Analysis and its Applicationsvol 2 Springer New York NY USA 1990

[52] A Schmid ldquoA time dependent Ginzburg-Landau equation andits application to the problem of resistivity in the mixed staterdquoPhysik der KondensiertenMaterie vol 5 no 4 pp 302ndash317 1966

[53] C M Elliot and Q Tang Existence Theorems For an Evolution-ary Superconductivity Model University of Sussex BrightonUK 1992

[54] Q Tang ldquoOn an evolutionary system of Ginzburg-Landauequations with fixed total magnetic fluxrdquo Communications inPartial Differential Equations vol 20 no 1-2 pp 1ndash36 1995

[55] Q Du ldquoGlobal existence and uniqueness of solutions of thetime-dependent Ginzburg-Landau model for superconductiv-ityrdquo Applicable Analysis vol 53 no 1-2 pp 1ndash18 1994

[56] Z M Chen K-H Hoffmann and J Liang ldquoOn a nonstation-ary Ginzburg-Landau superconductivity modelrdquoMathematicalMethods in the Applied Sciences vol 16 no 12 pp 855ndash875 1993

[57] J Liang and Q Tang ldquoAsymptotic behaviour of the solutionsof an evolutionaryGinzburg-Landau superconductivitymodelrdquoJournal of Mathematical Analysis and Applications vol 195 no1 pp 92ndash107 1995

[58] L P Gorrsquokov and G M Eliasherberg ldquoGeneralization of theGinzburg-Landau equations for an nonstationary problemsin the case of alloys with paramagnetic impartiesrdquo ZhurnalEksperimentalrsquonoi i Teoreticheskoi Fiziki vol 54 pp 612ndash6261968 Soviet Physics vol 27 pp 328ndash334 1968

[59] J Fleckinger-Pelle and H G Kaper ldquoGauges for the Ginzburg-Landau equations of superconductivityrdquo Zeitschrift fur Ange-wandte Mathematik und Mechanik vol 76 no 2 pp 345ndash3481996

[60] P Takac ldquoOn the dynamical process generated by supercon-ductivity modelrdquo Zeitschrift fur Angewandte Mathematik undMechanik vol 76 no 52 pp 349ndash352 1995

[61] W D Gropp H G Kaper DM Levine M Palumbo and VMVinokur ldquoNumerical simulation of vortex dynamics in type-IIsuperconductorsrdquo Journal of Computational Physics vol 123 no2 pp 254ndash266 1996

[62] D W Braun G W Crabtree H G Kaper et al ldquoStructure of amoving vortex latticerdquo Physical Review Letters vol 76 no 5 pp831ndash834 1996

[63] G W Crabtree G K Leaf H G Kaper et al ldquoTime-dependent Ginzburg-Landau simulations of vortex guidance bytwin boundariesrdquo Physica C vol 263 no 1ndash4 pp 401ndash408 1996

[64] G W Carabtree G K Leaf H G Kaper D W Braun V MVinokur andA E Koshelev ldquoDynamics vortex phases in super-conductors with correlated disorderrdquo Preprint ANLMCS-P590-0496 Mathematics and Computer Science DivisionAragonne National Laboratory 1996

[65] Th Gallay ldquoA center-stable manifold theorem for differentialequations in Banach spacesrdquo Communications in MathematicalPhysics vol 152 no 2 pp 249ndash268 1993

[66] J-M Ghidaglia and B Heron ldquoDimension of the attractorsassociated to the Ginzburg-Landau partial differential equa-tionrdquo Physica D vol 28 no 3 pp 282ndash304 1987

12 Journal of Function Spaces and Applications

[67] P Mattila Geometry of Sets and Measures in Euclidean Spacesvol 44 Cambridge University Press Cambridge UK 1995

[68] L HormanderThe Analysis of Linear Partial Differential Oper-ators vol 1 Springer Berlin Germany 2nd edition 1990

[69] H Amann Linear and Quasilinear Parabolic ProblemBikhauser Basel Switzerland 1995

[70] H-P Heinz ldquoOn the behaviour ofmeasures of noncompactnesswith respect to differentiation and integration of vector-valuedfunctionsrdquo Nonlinear Analysis Theory Methods amp ApplicationsA vol 7 no 12 pp 1351ndash1371 1983

[71] R P Agarwal M Meehan and D OrsquoRegan Fixed Point Theoryand Applications vol 141 of Cambridge Tracts in MathematicsCambridge University Press Cambridge UK 2001

[72] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer New York NY USA1983

[73] J Fleckinger-Pelle H G Kaper and P Takac ldquoDynamics of theGinzburg-Landau equations of superconductivityrdquo NonlinearAnalysis Theory Methods amp Applications A vol 32 no 5 pp647ndash665 1998

[74] H G Kaper and P Takac ldquoAn equivalence relation for theGinzburg-Landau equations of superconductivityrdquo Journal ofApplied Mathematics and Physics vol 48 no 4 pp 665ndash6751997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Operations ResearchAdvances in

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Function Spaces and Applications

(1198674) There exists a locally integrable function 119871 R

+rarr

R+such that for any bounded119863 sub X

120594 (F (119905 119863)) le 119871 (119905) 120594 (119863) (24)

for ae 119905 isin R+

Theorem 10 Under assumptions (1198671)ndash(1198674) (6) with initial

condition (7) has for every 1199060isin X at least one mild solution

119906 = 119906(119905) which satisfied the following estimate

119906 (119905) le 119872 (119905)

+ 119873 (119905) int

119905

0

119887 (119904)119872 (119904) exp(int119905

119904

119887 (120591)119873 (120591) 119889120591) 119889119904

(25)

where

119872(119905) = 119873 (119905) (100381710038171003817100381711990601003817100381710038171003817 + int

119905

0

119886 (119904) 119889119904) (26)

Proof Consider the operator 119866 119862(R+X) rarr 119862(R

+X)

defined by the following formula

119866119906 (119905) = 119890minusA1199051199060+ int

119905

0

119890minusA(119905minus119904)

F (119904 119906 (119904)) 119889119904 (27)

Now let us observe that for any continuous function 119906

R+rarr X in view of condition (119867

3) we get the following

estimate

119866119906 (119905) le 119873 (119905)100381710038171003817100381711990601003817100381710038171003817 + 119873 (119905) int

119905

0

(119886 (119904) + 119887 (119904) 119906 (119904)) 119889119904

(28)

which yields

119866119906 (119905) le 119872 (119905) + 119873 (119905) int

119905

0

119887 (119904) 119906 (119904) 119889119904 (29)

Next consider the following integral equation

P (119905) = 119872 (119905) + 119873 (119905) int

119905

0

119887 (119904)P (119904) 119889119904 (30)

Solving this equation by standard methods we get

P (119905)=119872 (119905)+119873 (119905)int

119905

0

119887 (119904)119872 (119904) exp(int119905

119904

119887 (120591)119873 (120591) 119889120591) 119889119904

(31)

The function P(119905) is continuous nonnegative and nonde-creasing Observe that the following implication is true

119906 (119905) le P (119905) 997904rArr 119866119906 (119905) le P (119905) for 119905 ge 0 (32)

Indeed linking (29) and (32) we have

119866119906 (119905) le 119872 (119905) + 119873 (119905) int

119905

0

119887 (119904) 119906 (119904) 119889119904

le 119872 (119905) + 119873 (119905) int

119905

0

119887 (119904)P (119904) 119889119904 = P (119905)

(33)

In the space 119862(R+X) let us consider the following set

Γ = 119906 isin 119862 (R+X) 119906 (119905) le P (119905) for 119905 ge 0 (34)

Obviously the set Γ is convex and closed Moreover in viewof (32) we have that 119866 is a self-mapping of Γ

Using the criterion of convergence (lowast) in 119862(R+X) and

standard techniques (see [31ndash33 35]) we can show that theoperator 119866 is continuous on 119862(R

+X)

Now we consider the sequence of sets Γ119899 defined by

induction as follows

Γ0= Γ Γ

119899+1= Conv (119866Γ

119899) for 119899 = 0 1 (35)

This sequence is decreasing that is Γ119899sup Γ119899+1

for 119899 =

0 1 2 Further let us fix 119879 gt 0 and for 119905 isin [0 119879] let us put

119892119899(119905) = sup 120594 (Γ

119899(119904)) 119904 isin [0 119905] (36)

119908119899= 120596119879

0(Γ119899) (37)

The sequences 119908119899 and 119892

119899(119905) are nonincreasing for all 119905 isin

[0 119879] so they have limits

119892infin(119905) = lim

119899rarrinfin119892119899(119905) 119905 isin [0 119879]

119908infin= lim119899rarrinfin

119908119899

(38)

Moreover each function 119892119899is nondecreasing therefore 119892

119899

and 119906infin

are measurable on [0 119879] for 119899 = 1 2 Now we apply the family of measures of noncompactness

120583119879119879ge0

defined in 119862(R+X) by formula (13) In view of the

above notation we have

120583119879(Γ119899) = 119908119899+ 119892119899(119879) (39)

We show that

119908infin= 119892infin(119879) = 0 (40)

To fix a number 119905 isin [0 119879] 119899 isin N and take an arbitrarynumber 119904 isin [0 119905]

We know from Lemma 5 that for any 120576 gt 0 there is asequence 119910

119896infin

119896=1sub 119866Γ119899(119904) such that

120594 (119866Γ119899(119904)) le 2120594 (119910

119896infin

119896=1) + 120576 (41)

This implies that there is a sequence 119906119896infin

119896=1sub Γ119899 such that

119910119896= (119866119906

119896) (119904) for 119896 = 1 2 (42)

Journal of Function Spaces and Applications 5

Hence in view of Lemma 6 (1198674) and (36) we obtain

120594 (119866Γ119899(119904)) le 2120594 (119910

119896infin

119896=1) + 120576

= 2120594 ((119866119906119896) (119904)infin

119896=1) + 120576 le 2120594 (119890

minusA1199041199060)

+ 2120594(int

119904

0

119890minusA(119904minus120591)

F (120591 119906119896(120591)) 119889120591

infin

119896=1

)

+ 120576 le 4int

119904

0

120594 (119890minusA(119904minus120591)

F (120591 119906119896(120591)infin

119896=1)) 119889120591 + 120576

le 4119873 (119879)int

119904

0

119871 (120591) 120594 (119906119896(120591)infin

119896=1) 119889120591 + 120576

le 4119873 (119879)int

119905

0

119871 (120591) 119892119899(120591) 119889120591 + 120576

(43)

Since 120576 gt 0 is arbitrary it follows from the above inequalitiesthat

120594 (119866Γ119899(119904)) le 4119873 (119879)int

119905

0

119871 (120591) 119892119899(120591) 119889120591 119904 isin [0 119905] (44)

Using (36) we have

119892119899+1

(119905) = sup 120594 (119866Γ119899(119904)) 119904 isin [0 119905]

le 4119873 (119879)int

119905

0

119871 (120591) 119892119899(120591) 119889120591 119905 isin [0 119879]

(45)

Letting 119899 rarr infin we derive the following inequality

119892infin(119905) le 4119873 (119879)int

119905

0

119871 (120591) 119892infin(120591) 119889120591 119905 isin [0 119879] (46)

This inequality together with Gronwallrsquos Lemma 8 impliesthat

119892infin(119905) = 0 119905 isin [0 119879] (47)

Next let us notice that in view of (1198674) and (36) we have

120594 (F (119904 Γ119899(119904))) le 119871 (119904) 120594 (Γ

119899(119904)) le 119871 (119904) 119892

119899(119904) (48)

for ae 119904 isin [0 119879] and the function

[0 119879] ni 119904 997891997888rarr 2119873 (119879) 119871 (119904) 119892119899(119904) (49)

is measurable Then in virtue of equality 1205961198790(119866Γ119899) = 120596119879

0(119865Γ119899)

and Lemma 9 we get

120596119879

0(119866Γ119899) le 2119873 (119879)int

119879

0

119871 (119904) 119892119899(119904) 119889119904 (50)

Hence we derive

119908119899+1

= 120596119879

0(Γ119899+1) = 120596119879

0(119866Γ119899)

le 2119873 (119879)int

119879

0

119871 (119904) 119892119899(119904) 119889119904

(51)

Letting 119899 rarr infin we get

119908infinle 2119873 (119879)int

119879

0

119871 (119904) 119892infin(119904) 119889119904 (52)

Keeping in mind (47) we deduce that

119908infin= 0 (53)

This together with (39) and (47) yields

lim119899rarrinfin

120583119879(Γ119899) = lim119899rarrinfin

(119908119899+ 119892119899(119879)) = 119908

infin+ 119892infin= 0 (54)

Finally using Remark 2 for the measure 120583119879 we deduce

that the set Γinfin= ⋂infin

119899=0Γ119899is nonempty convex and compact

Then by the Schauder-Tychonoff theorem we conclude thatoperator 119865 Γ

infinrarr Γinfin

has at least one fixed-point 119906 = 119906(119905)Obviously the function 119906 = 119906(119905) is a solution of problem (6)-(7) and in view of the definition of the set Γ

infin the estimate

119906(119905) le P(119905) holds to be trueThis completes the proof

4 Local and Global Attractivity

Following the concepts introduced in [36] we introduce firsta few definitions of various kinds of the concept of attractivityof mild solution of (6)

Definition 11 The mild solution 119906 = 119906(119905) of (6) with initialcondition (7) is said to be globally attractive if for each mildsolution V = V(119905) of (6) with initial condition V(0) = V

0we

have that

lim119905rarrinfin

(119906 (119905) minus V (119905)) = 120579 (55)

In other words we may say that solutions of (6) are globallyattractive if for arbitrary solutions 119906(119905) and V(119905) of thisequation condition (55) are satisfied

Definition 12 We say that mild solution 119906 = 119906(119905) of (6) withinitial condition (7) is locally attractive if there exists a ball119861(119906(0) 119903) in the space X such that for arbitrary solution V(119905)of (1) with initial-value V(0) isin 119861(119906(0) 119903) condition (55) doeshold

In the case when the limit (55) is uniform with respect toall solutions V(119905) that is when for each 120576 gt 0 there exist119879 gt 0such that

119906 (119905) minus V (119905) le 120576 (56)

for all V(119905) being solutions of (6) with initial-value V(0) isin119861(119906(0) 119903) and for 119905 ge 119879 we will say that solution 119906 = 119906(119905)

is uniformly locally attractive on R+

Now we formulate themain result of this sectionWe willconsider (6) under the following conditions

(1198671015840

1) A is the infinitesimal generator of an exponentiallystable119862

0-semigroup 119890minusA119905

119905ge0 that is there exist119872 gt

0 120588 gt 0 such that 119890minusA119905 le 119872119890minus120588119905 for all 119905 ge 0

6 Journal of Function Spaces and Applications

(1198675) there exist locally integrable functions119898 R

+rarr R+

such that

F (119905 119906) minusF (119905 V) le 119898 (119905) 119906 minus V (57)

for 119905 ge 0 and 119906 V isin X Moreover we assume that

lim119905rarrinfin

(int

119905

0

119898(119904) 119889119904 minus120588

119872119905) = minusinfin (58)

Remark 13 The property (11986710158401) is generally satisfied in diffu-

sion problem A necessary and sufficient condition for (11986710158401)

is presented in [72]

The main result of this section is shown in the giventheorem below

Theorem 14 Under assumptions (11986710158401) and (119867

2)ndash(1198675) prob-

lem (6)-(7) has amild solution 119906 = 119906(119905) for each 1199060isin X which

is globally attractive and locally uniformly attractive

Proof Existence of a solution 119906 = 119906(119905) is a consequence ofTheorem 10 Let us fix 119903 gt 0 and V

0isin 119861(119906(0) 119903)

Let V = V(119905) denote a mild solution of (6) with the initialcondition V(0) = V

0 Using (1198671015840

1) and (119867

3) we get

V (119905) le10038171003817100381710038171003817119890minusA119905V0

10038171003817100381710038171003817+ int

119905

0

10038171003817100381710038171003817119890minusA(119905minus119904)10038171003817100381710038171003817

sdot F (119904 V (119904)) 119889119904

le 119872(100381710038171003817100381711990601003817100381710038171003817 + 119903) +119872int

119905

0

119890120588119904119886 (119904) 119889119904

+ int

119905

0

119872119890120588119904119887 (119904) V (119904) 119889119904

(59)

Now let us put

P1(119905) = 119872 (

1003817100381710038171003817V01003817100381710038171003817 + 119903) +119872int

119905

0

119890120588119904119886 (119904) 119889119904

P2(119905) = 119872119890

120588119904119887 (119905)

(60)

Taking into account Lemma 5 we obtain

V (119905) le P1(119905) + int

119905

0

P2(119904)P1(119904) exp(int

119904

0

P2(120591) 119889120591) 119889119904

(61)

Further let 119878 be the set of all mild solutions V(119905) of (6) withthe initial-value V

0= V(0) isin 119861(119906(0) 119903)

Put

119892 (119905) = sup 119906 (119905) minus V (119905) V isin 119878 (62)

The estimate (61) implies that the function119892(119905) is well definedApplying (119867

5) and (1198671015840

1) we get

119906 (119905) minus V (119905) le10038171003817100381710038171003817119890minusA11990510038171003817100381710038171003817

sdot10038171003817100381710038171199060 minus V

0

1003817100381710038171003817

+ int

119905

0

10038171003817100381710038171003817119890minusA(119905minus119904)10038171003817100381710038171003817

119898 (119904) 119906 (119904) minus V (119904) 119889119904

le 119872119890minus120588119905119903 + int

119905

0

119872119890minus119892(119905minus119904)

119898(119904) 119892 (119904) 119889119904

(63)

or equivalently

119890120588119905119892 (119905) le 119872119903 + int

119905

0

119872119898(119904) 119890120588119904119892 (119904) 119889119904 (64)

Using again Lemma 5 for the above estimate (where ℎ(119905) =119890120588119905119892(119905)) we obtain

119890120588119905119892 (119905) le 119872119903

+ int

119905

0

1198722119903119898 (119904) exp(int

119905

119904

119872119898(120591) 119889120591) 119889119904

(65)

Elementary calculations lead to the following equality

int

119905

0

1198722119903119898 (119904) exp(int

119905

119904

119872119898(120591) 119889120591) 119889119904

= 119872119903(exp(int119905

0

119872119898(119904) 119889119904) minus 1)

(66)

Hence

119892 (119905) le 119872119903 exp (minus120588119905) + 119872119903 exp(int119905

0

119872119898(119904) 119889119904 minus 120588119905)

minus119872119903 exp (minus120588119905) (67)

Applying assumption (1198675) we derive

lim119905rarrinfin

119892 (119905) = 0 (68)

and this proves that 119906(119905) is locally attractive Finally thisequality together with definition of the function 119892(119905) impliesthat 119906(119905) is globally attractive The proof is complete

Remark 15 In the case when 119898(119905) equiv 119898 is constant the fol-lowing condition

lim119905rarrinfin

(int

119905

0

119898(119904) 119889119904 minus120588

119872119905) = minusinfin (69)

means that 119898 lt 120588119872 Observe that this condition cannotbe weakened This observation is illustrated by the followingexampled

Example 16 Let X = R 119898(119905) equiv 119898 F(119905 119906) = sin 119905 + 119898119906119872 = 1 and 119890minusA119905 = 119890minus120588119905 Then the equation

119906 (119905) = 119890minusA1199051199060+ int

119905

0

119890minusA(119905minus119904)

F (119904 119906 (119904)) 119889119904 (70)

(for any fixed 1199060isin R) has the solution 119906(119905) expressed by the

following formula

119906 (119905) =(120588 minus 119898) sin 119905 minus cos 119905

(120588 minus 119898)2

+ 1+ (1199060+

1

(120588 minus 119898)2

+ 1) 119890(119898minus120588)119905

(71)

Notice that for 119898 ge 120588 the solution 119906(119905) is neither globallyattractive nor locally uniformly attractive because for each

Journal of Function Spaces and Applications 7

other solution V(119905) with initial condition V(0) = V0 obviously

described by similar formula as 119906(119905) we would have acontradiction

|119906 (119905) minus V (119905)| =

10038161003816100381610038161199060 minus V0

1003816100381610038161003816 119890(119898minus120588)119905

997888rarr infin

or10038161003816100381610038161199060 minus V

0

1003816100381610038161003816

(72)

5 Asymptotic Behaviour

In this section we will give a theorem describing asymptoticbehaviour of mild solutions of (6) with condition (7) Thistheorem generalizes the result included in [72Theorem 44]First we formulate the assumptions

(1198671015840

3) This condition is almost identical with (119867

3) and the

only difference is that we assume the functions 119886 and119887 are locally essentially bounded on R

+

(1198671015840

5) There exists 119906

infinisin X such that there exists the limit

lim119905rarrinfin

F(119905 119906infin) and

lim119905rarrinfin

F (119905 119906infin) = A

minus1119906infin (73)

Moreover there exists a number119898 lt 120588119872 such that1003817100381710038171003817F (119905 119906) minusF (119905 119906

infin)1003817100381710038171003817 le 119898

1003817100381710038171003817119906 minus 119906infin1003817100381710038171003817 (74)

for 119905 ge 0 and 119906 isin X

Remark 17 The condition (11986710158405) in conjunction with (1198671015840

1)

ensures the existence ofAminus1 (see [72]) Clearly (11986710158403) implies

(1198673)

Theorem 18 Under assumptions (11986710158401) (1198672) (11986710158403) (1198674) and

(11986710158405) (6) with condition (7) has a mild solution 119906 = 119906(119905) for

each 1199060isin X such that lim

119905rarrinfin119906(119905) = 119906

infin

Proof The existence of a mild solution 119906(119905) is guaranteed byTheorem 10 Let us put

119892 (119905) =1003817100381710038171003817119906 (119905) minus 119906infin

1003817100381710038171003817

Finfin= lim119905rarrinfin

F (119905 119906infin)

(75)

We show that

lim119905rarrinfin

119892 (119905) = 0 (76)

Recall that if assumption (11986710158401) is fulfilled then for each 119911 isin X

we have (see [72])

int

infin

0

119890minusA119904

119911 119889119904 = minusAminus1119911 (77)

Using the above fact and (11986710158405) we get

119906infin= (minusA

minus1(minusAminus1119906infin)) = minusA

minus1Finfin

= int

infin

0

119890minusA119904

Finfin119889119904

(78)

Linking the above equality with (11986710158401) we obtain

119892 (119905)

=

10038171003817100381710038171003817100381710038171003817119890minusA1199051199060+ int

119905

0

119890minusA(119905minus119904)

F (119904 119906 (119904)) 119889119904 minus int

infin

0

119890minusA119904

Finfin119889119904

10038171003817100381710038171003817100381710038171003817

le 119872119890minus120588119905 10038171003817100381710038171199060

1003817100381710038171003817 +

10038171003817100381710038171003817100381710038171003817int

infin

119905

119890minusA119904

Finfin119889119904

10038171003817100381710038171003817100381710038171003817

+

10038171003817100381710038171003817100381710038171003817int

119905

0

119890minusA(119905minus119904)

F (119904 119906 (119904)) 119889119904 minus int

119905

0

119890minusA119904

Finfin119889119904

10038171003817100381710038171003817100381710038171003817

le 119872119890minus120588119905 10038171003817100381710038171199060

1003817100381710038171003817 +119872

119890minus1205881199051205881003817100381710038171003817Finfin

1003817100381710038171003817

+

10038171003817100381710038171003817100381710038171003817int

119905

0

119890minusA(119905minus119904)

F (119904 119906 (119904)) 119889119904 minus int

119905

0

119890minusA(119905minus119904)

Finfin119889119904

10038171003817100381710038171003817100381710038171003817

le 119872119890minus120588119905(100381710038171003817100381711990601003817100381710038171003817 +

1003817100381710038171003817Finfin1003817100381710038171003817

120588)

+ int

119905

0

119872119890minus120588(119905minus119904) 1003817100381710038171003817F (119904 119906 (119904)) minusF

infin

1003817100381710038171003817 119889119904

le 119872119890minus120588119905(100381710038171003817100381711990601003817100381710038171003817 +

1003817100381710038171003817Finfin1003817100381710038171003817

120588)

+119872119890minus120588119905int

119905

0

119890120588119904 1003817100381710038171003817F (119904 119906

infin) minusF

infin

1003817100381710038171003817 119889119904

+119872119890minus120588119905int

119905

0

119890120588119904 1003817100381710038171003817F (119904 119906 (119904)) minusF (119904 119906

infin)1003817100381710038171003817 119889119904

(79)

Next putting

119888 (119905) = 119872119890minus120588(119905)

(100381710038171003817100381711990601003817100381710038171003817 +

1003817100381710038171003817Finfin1003817100381710038171003817

120588)

+119872119890minus120588119905int

119905

0

119890120588119904 1003817100381710038171003817F (119904 119906

infin) minusF

infin

1003817100381710038171003817 119889119904

(80)

and applying (11986710158405) we derive the following inequality

119892 (119905) le 119888 (119905) + 119898119872119890minus120588119905int

119905

0

119890120588119904119892 (119904) 119889119904 (81)

Hence

119890120588119905119892 (119905) le 119890

120588(119905)119888 (119905) + 119898119872int

119905

0

119890120588119904119892 (119904) 119889119904 (82)

The above inequality in conjunction with Lemma 5 gives

119890120588119905119892 (119905) le 119890

120588119905119888 (119905)

+ int

119905

0

119890120588119904119888 (119904)119898119872 sdot exp (119898119872(119905 minus 119904)) 119889119904

= 119890120588119905119888 (119905)

+ 119898119872119890119898119872119905

int

119905

0

119888 (119904) exp ((120588 minus 119898119872) 119904) 119889119904

(83)

8 Journal of Function Spaces and Applications

Hence

119892 (119905) le 119888 (119905) +119898119872int

119905

0119888 (119904) 119890(119892minus119898119872)119904

119889119904

exp ((120588 minus 119898119872) 119905) (84)

Before proving (76) we first show that

lim119905rarrinfin

119888 (119905) = 0 (85)

To prove this equality it is sufficient to show that the lastcomponent in the formula expressing 119888(119905) that is 119872119890minus120588119905

int119905

0119890120588119904F(119904 119906

infin) minusF

infin119889119904 tends to zero as 119905 rarr infin

To this end let us fix 120576 gt 0 Assumption (11986710158405) implies that

there exists 1199050such that

sup 1003817100381710038171003817F (119904 119906infin) minusF

infin

1003817100381710038171003817 119904 ge 1199050 le120576120588

2119872 (86)

This inequality together with (11986710158403) implies

ess sup 1003817100381710038171003817F (119904 119906infin) minusF

infin

1003817100381710038171003817 119904 isin R lt infin (87)

Now fix 1199051isin R such that 119905

1ge 119905 and

ess sup 1003817100381710038171003817F (119904 119906infin) minusF

infin

1003817100381710038171003817 119904 isin R

le120576120588 exp (120588119905

1)

2119872 (exp (1205881199050) minus 1)

(88)

Then using (86) and (88) we conclude that for 119905 ge 1199051we have

119872119890minus120588119905int

119905

0

119890120588119904 1003817100381710038171003817F (119904 119906

infin) minusF

infin

1003817100381710038171003817 119889119904

le 119872119890minus120588119905int

1199050

0

119890120588119904 1003817100381710038171003817F (119904 119906

infin) minusF

infin

1003817100381710038171003817 119889119904

+119872119890minus120588119905int

119905

1199050

119890120588119904 1003817100381710038171003817F (119904 119906

infin) minusF

infin

1003817100381710038171003817 119889119904 le120576

2+120576

2= 120576

(89)

This fact proves (85)Further using (1198671015840

5) and (85) and employing de

LrsquoHospitalrsquos rule for the fraction on the right-hand sideof inequality (84) we obtain that condition (76) is satisfiedThis fact completes the proof

6 An Application to the Ginzburg-LandauEquations of Superconductivity

In this section we formulate the gauged time-dependentGinzburg-Landau (TDGL) equations as an abstract evolutionequation in a Hilbert space Moreover we show applicationsof the above theorems to TDGL equations

We assume that Ω is a bounded domain in R119899 withboundary 120597Ω of class11986211That isΩ is an open and connectedset whose boundary 120597Ω is a compact (119899 minus 1)-manifolddescribed by Lipschitz continuous differentiable charts Weconsider two- and three-dimensional problems (119899 = 2 and119899 = 3 resp) Assume that the vector potential A takes its

values in R119899 The vector H will represent the (externally)applied magnetic field which is a function of space andtime similarly to A it takes its values in R119899 The function120574 is defined and satisfies Lipschitz condition on 120597Ω and120574(119909) ge 0 for 119909 isin 120597Ω The parameters in the TDGLequations are 120578 a (dimensionless) friction coefficient and 120581the (dimensionless) Ginzburg-Landau parameter

The order parameter should be thought of as the wavefunction of the center-of-massmotion of the ldquosuperelectronsrdquo(Cooper pairs) whose density is 119899

119904= |120595|

2 and whose fluxis J119904 The vector potential A determines the electromagnetic

field E = minus120597119905A minus nabla120601 is the electric field and B = nabla times A

is the magnetic induction where J the total current is thesum of a ldquonormalrdquo current J

119899= E the supercurrent J

119904 and

the transport current J119905= nabla times H The normal current obeys

Ohmrsquos law J119899= 120590119899E the ldquonormal conductivityrdquo coefficient 120590

119899

is equal to one in the adopted system of units The differenceM = BminusH is known as themagnetizationThe trivial solution(120595 = 0 B = H E = 0) represents the normal state where allsuperconducting properties have been lost

Nowwe accept the following notion all Banach spaces arereal the (real) dual of a Banach space119883 is denoted by1198831015840Thesymbol 119871119901(Ω) for 1 le 119901 le infin denotes the usual Lebesguespace with norm sdot

119871119901 (sdot sdot) is the inner product in 1198712(Ω)

1198821198982(Ω) for nonnegative integer 119898 is the usual Sobolev

space with norm sdot 1198821198982 1198821198982(Ω) is a Hilbert space for the

inner product (sdot sdot)1198982

given by (119906 V)1198982

= sum|120572|le119898

(120597120572119906 120597120572V)

for 119906 V isin 1198821198982(Ω) Fractional Sobolev space1198821199042(Ω) with afractional 119904 is defined by interpolation ([40 Chap VII] and[41 49 50]) 119862]

(Ω) for ] ge 0 ] = 119898 + 120582 with 0 le 120582 lt 1 isthe space of 119898 times continuous differentiable functions onΩ those 119898th-order derivatives satisfy the Holder conditionwith exponent 120582 if ] is a proper fraction the norm sdot

119862] is

defined in the usual wayThe definitions extend to the space of vector-valued

functions in the standard way with the caveat that the innerproduct in [1198712(Ω)]119899 is defined by (119906 V) = int

Ω119906 sdot V where the

symbol (119906 V) indicates the scalar product in R119899 Complex-valued functions are interpreted as vector-valued functionswith two real components

Functions that vary in space and time like the orderparameter and the vector potential are considered as map-pings from the time domain which is a subinterval of [0infin)into spaces of complex- or vector-valued functions definedin Ω Let 119883 = (119883 sdot

119883) be a Banach space of functions

defined inΩ Then the functions are defined in Ω Then thefunctions of space and time defined on Ω times (0 119879) for 119879 gt 0may be considered as elements of 119871119901(0 119879119883) for 1 le 119901 le infinor1198822119898(0 119879119883) for nonnegative119898 or119862]

(0 119879119883) for ] ge 0] = 119898 + 120582 with 0 le 120582 lt 1 Detailed definitions can be foundfor example in [43]

Obviously function spaces of ordered pairs (120595A) where120595 Ω rarr R2 and A Ω rarr R119899 (119899 = 2 3) play an importantrole in the study of the gauged TDGL equationsWe thereforeadapt the following special notationH = [119883(Ω)]

2times[119883(Ω)]

119899for any Banach space for the order parameter 120595 and thevector potential A respectively A suitable framework forthe functional analysis of the gauged TDGL equations is the

Journal of Function Spaces and Applications 9

Cartesian product W1+1205722 = [1198821+1205722

(Ω)]2times [1198821+1205722

(Ω)]119899

where 120572 isin (12 1) This space is continuously imbedded inW12 capLinfin

Assume H = 0 and H isin [1198712(Ω)]119899 Let AH be a minimizer

of the convex quadratic form 119869120596equiv J120596[119860]

119869120596 [A] = int

Ω

[120596(nabla sdot A)2 + |nabla times A minusH|2] 119889119909 (90)

on the domain

D (119869120596) = A isin [119882

12(Ω)]119899

n sdot A = 0 on 120597Ω (91)

We now introduce the reduced vector potential A1015840

A1015840 = A minus AH (92)

In terms of 120595 and A1015840 the gauged TDGL equations have thefollowing form

120597120595

120597119905minus

1

1205781205812Δ120595 = 120593 in Ω times (0infin)

120597A1015840

120597119905+ nabla times nabla times A1015840 minus 120596nabla (nabla sdot A1015840) = F in Ω times (0infin)

n sdot nabla + 120574120595 = 0 n sdot A1015840 = 0

n times (nabla sdot A1015840) = 0 on 120597Ω times (0infin)

(93)

Here 120593 and F are nonlinear functions of 120595 and A1015840

120593 equiv 120593 (119905 120595A1015840) = 1

120578[minus2119894

120581(nabla120595) sdot (A1015840 + AH)

minus119894

120581(1 minus 120578120581

2120596)120595 (nabla sdot A1015840)

minus12059510038161003816100381610038161003816A1015840 + AH

10038161003816100381610038161003816

2

+ (1 minus100381610038161003816100381612059510038161003816100381610038162

) 120595]

(94)

F equiv F (119905 120595A1015840) = 1

2119894119896(120595lowastnabla120595 minus 120595nabla120595

lowast)

minus100381610038161003816100381612059510038161003816100381610038162

(A1015840 + AH) minus120597AH120597119905

(95)

The equations are supplemented by initial data which is inthe followimg form

120595 = 1205950 A = A

0on Ω times 0 (96)

where 1205950and A

0are given and by (92) we have

120595 = 1205950 A1015840 = A

0minus AH (0) on Ω times 0 (97)

In the next part we connect the evolution of the solution(120595A1015840) of the system of (93) with the initial data (120595

0A0minus

AH(0)) with the dynamics of a vector 119906 in a Hilbert space

L2= [119871119901(Ω)]2

times [119871119901(Ω)]119899

(98)

The following analysis is restricted to the case 120596 gt 0 andthe case 120596 = 0 (see [73])

Let vector 119906 [0infin) rarr L2 represent the pair (120595A1015840)

119906 = (120595A1015840) equiv (120595A minus AH) (99)

and letA be the linear self-adjoint operator inL2 associatedwith the quadratic form 119876

120596equiv 119876120596[119906] given by the following

formula

119876120596 [119906] = int

Ω

[1

12057812058121003816100381610038161003816nabla120595

10038161003816100381610038162

+ 120596(nabla sdot A1015840)2

+10038161003816100381610038161003816nabla times A101584010038161003816100381610038161003816

2

] 119889119909

+ int120597Ω

120574

1205781205812100381610038161003816100381612059510038161003816100381610038162

119889120590 (119909)

(100)

on the domain

D (119876120596) = D (A

12)

= 119906 = (120595A1015840) isinW12 n sdot A1015840 = 0 on 120597Ω

(101)

The quadratic form 119876120596is nonnegative Furthermore since

120596 gt 0119876120596[120595A1015840]+119888 120595

1198712 is coercive onW12 for any constant

119888 gt 0 Hence A is positively definite in L2 ([44 Chap 1]equation (545)) If it does not lead to confusion we use thesame symbol A for the restriction A

120595and AA of A to the

respective linear subspace [1198712(Ω)]2 = [1198712(Ω)]2 times 0 (for 120595)and [1198712(Ω)]119899 equiv 0 times [1198712(Ω)]119899 (for A) ofL2

Now consider the initial-value problems (6) and (7) inL2 whereF(119905 119906) = (120593 F) 120593 and F is given by (94) and (95)and 119906

0= (1205950A0minus AH(0))

With 120582 isin (12 1) and 1199060isin W1+1205822 we say that 119906 are a

solution of (6) and (7) on the interval [0 119879] for some 119879 gt 0if 119906 [0 119879] rarr W1+1205822 is continuous and

119906 (119905) = 119890minusA(119905)

+ int

119905

0

119890minusA(119905minus119904)

F (119904 119906 (119904)) 119889119904 for 119905 isin [0 119879]

(102)

in L2 A mild solution of the initial-value problems (6) and(7) defines a weak solution (120595A1015840) of the boundary valueproblem (93) which in turn defines a weak solution (120595A)of the gauged TDGL equations provided AH is sufficientlyregular

Namely let us assume that X = L2 and 1199060isin L2 and

119906 R+times R119899 rarr R119899 for 119899 isin N is an unknown function 119906 =

119906(119905 119909) In order to applyTheorems 10 and 14 we are not goingto consider 119906 as a function of 119905 and 119909 together but rather asa mapping 119906 of variable 119905 into the spaceX =L2 of functions119909 that is 119906 R

+rarr L2 119906(119905)(119909) = 119906(119905 119909) 119909 isin R119899 119905 isin R

+

Remark 19 Since A is the linear self-adjoint operator inL2

associated with the quadratic form (100) thenA generates asemigroup 119890minusA119905

119905ge0(see [74])

Below we formulate the principal theorem of this paperThis theorem is a simple consequence of Theorem 10

10 Journal of Function Spaces and Applications

Theorem 20 Under assumptions (1198671)ndash(1198674) problem (6)-(7)

is equivalent to the system of (1)ndash(5) (TDGL equations) andhas for each 119906

0isin L2 at least one mild solution in sense of

Definition 4 119906 = 119906(119905) Moreover each solution 119906(119905) satisfiesestimate (25)

We observe that in the case when 119890minusA119905119905ge0

is the semi-group of contractions and 119886(119905) equiv 119886 119887(119905) equiv 119887 for 119905 isin

R+are constant after simple calculations based on estimate

(25) we get that the solution 119906 = 119906(119905) has the asymptoticcharacterization

119906 (119905) = 119874 (1199053) as 119905 997888rarr infin (103)

For further purposes let us formulate the followingassumption

(A) A is the infinitesimal generator of an exponentiallystable 119862

0-semigroup 119890minusA119905

119905ge0

The next result of this chapter is shown in the giventheorem below

Theorem21 Under assumptions (A) and (1198672)ndash(1198674) problem

(6)-(7) which is equivalent to the systems equations (1)ndash(5)(TDGL equations) has a mild solution 119906 = 119906(119905) for each1199060isin L2 which is globally attractive and locally uniformly

attractive

References

[1] V L Ginzburg and L D Landau ldquoOn the theory of supercon-ductivityrdquo Zhurnal Eksperimentalrsquonoi i Teoreticheskoi Fiziki vol20 pp 1064ndash1082 1950 English translation in L D LandauMen of Physics D ter Haar ed Vol I Pergamon Press OxfordUK pp 138ndash167 1965

[2] S Aizicovici and H Lee ldquoNonlinear nonlocal Cauchy problemsin Banach spacesrdquo Applied Mathematics Letters vol 18 no 4pp 401ndash407 2005

[3] S Aizicovici and M McKibben ldquoExistence results for a classof abstract nonlocal Cauchy problemsrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 39 no 5 pp 649ndash6682000

[4] MMcKibbenDiscovering Evolution Equation with ApplicationCRC Press Taylor amp Francis Group 2011

[5] S Aizicovici and V Staicu ldquoMultivalued evolution equationswith nonlocal initial conditions in Banach spacesrdquo NonlinearDifferential Equations and Applications vol 14 no 3-4 pp 361ndash376 2007

[6] R P Agarwal and D OrsquoRegan Infinite Interval Problems for Dif-ferential Difference and Integral Equations Kluwer AcademicPublishers Dordrecht The Netherlands 2001

[7] Z Fan ldquoExistence of nondensely defined evolution equationswith nonlocal conditionsrdquo Nonlinear Analysis Theory Methodsamp Applications A vol 70 no 11 pp 3829ndash3836 2009

[8] Z Fan ldquoImpulsive problems for semilinear differential equa-tions with nonlocal conditionsrdquo Nonlinear Analysis TheoryMethods amp Applications A vol 72 no 2 pp 1104ndash1109 2010

[9] Z Fan and G Li ldquoExistence results for semilinear differentialequations with nonlocal and impulsive conditionsrdquo Journal ofFunctional Analysis vol 258 no 5 pp 1709ndash1727 2010

[10] J Garcıa-Falset ldquoExistence results and asymptotic behavior fornonlocal abstract Cauchy problemsrdquo Journal of MathematicalAnalysis and Applications vol 338 no 1 pp 639ndash652 2008

[11] J Liang J H Liu and T-J Xiao ldquoNonlocal impulsive problemsfor nonlinear differential equations in Banach spacesrdquo Mathe-matical and Computer Modelling vol 49 no 3-4 pp 798ndash8042009

[12] L Zhu and G Li ldquoOn a nonlocal problem for semilinear differ-ential equations with upper semicontinuous nonlinearities ingeneral Banach spacesrdquo Journal of Mathematical Analysis andApplications vol 341 no 1 pp 660ndash675 2008

[13] Q Liu and R Yuan ldquoExistence of mild solutions for semilinearevolution equations with non-local initial conditionsrdquo Nonlin-ear AnalysisTheoryMethodsampApplications A vol 71 no 9 pp4177ndash4184 2009

[14] C Xie W Zhong and Z Fan ldquoExistence results for nonlinearnonlocal problems in Banach spacesrdquo Applied MathematicsLetters vol 22 no 7 pp 998ndash1002 2009

[15] T-J Xiao and J Liang ldquoExistence of classical solutions tononautonomous nonlocal parabolic problemsrdquoNonlinear Anal-ysisTheoryMethods and Applications vol 63 no 5-7 pp e225ndashe232 2005

[16] X Xue ldquo119871119901 theory for semilinear nonlocal problems with mea-sure of noncompactness in separable Banach spacesrdquo Journal ofFixed Point Theory and Applications vol 5 no 1 pp 129ndash1442009

[17] X Xue ldquoNonlinear differential equations with nonlocal condi-tions in Banach spacesrdquo Nonlinear Analysis Theory Methods ampApplications A vol 63 no 4 pp 575ndash586 2005

[18] X Xue ldquoExistence of solutions for semilinear nonlocal Cauchyproblems in Banach spacesrdquo Electronic Journal of DifferentialEquations vol 64 p 7 2005

[19] X Xue ldquoNonlocal nonlinear differential equations with a mea-sure of noncompactness in Banach spacesrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 70 no 7 pp 2593ndash26012009

[20] L Olszowy ldquoExistence ofmild solutions for the semilinear non-local problem in Banach spacesrdquo Nonlinear Analysis TheoryMethods amp Applications A vol 81 pp 211ndash223 2013

[21] D Bothe ldquoMultivalued perturbations of119898-accretive differentialinclusionsrdquo Israel Journal of Mathematics vol 108 pp 109ndash1381998

[22] H-S Ding J Liang G M NrsquoGuerekata and T-J XiaoldquoMild pseudo-almost periodic solutions of nonautonomoussemilinear evolution equationsrdquo Mathematical and ComputerModelling vol 45 no 5-6 pp 579ndash584 2007

[23] J Garcıa-Falset ldquoThe asymptotic behavior of the solutions of theCauchy problem generated by 120593-accretive operatorsrdquo Journal ofMathematical Analysis and Applications vol 310 no 2 pp 594ndash608 2005

[24] Z Fan and G Li ldquoAsymptotic behavior of the solutions of non-autonomous systems in Banach spacesrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 68 no 12 pp 3733ndash37412008

[25] J Banas ldquoOn existence theorems for differential equations inBanach spacesrdquo Bulletin of the Australian Mathematical Societyvol 32 no 1 pp 73ndash82 1985

[26] J Banas ldquoMeasures of noncompactness in the space of contin-uous tempered functionsrdquo Demonstratio Mathematica vol 14no 1 pp 127ndash133 1981

Journal of Function Spaces and Applications 11

[27] J Banas and K GoebelMeasures of Noncompactness in BanachSpaces vol 60 ofLectureNotes in Pure andAppliedMathematicsMarcel Dekker New York NY USA 1980

[28] J Banas and M Lecko ldquoSolvability of infinite systems ofdifferential equations in Banach sequence spacesrdquo Journal ofComputational andAppliedMathematics vol 137 no 2 pp 363ndash375 2001

[29] J Banas and K Sadarangani ldquoCompactness conditions inthe study of functional differential and integral equationsrdquoAbstract and Applied Analysis vol 2013 Article ID 819315 14pages 2013

[30] M Mursaleen and S A Mohiuddine ldquoApplications of mea-sures of noncompactness to the infinite system of differentialequations in ℓ

119901spacesrdquo Nonlinear Analysis Theory Methods amp

Applications A vol 75 no 4 pp 2111ndash2115 2012[31] L Olszowy ldquoSolvability of some functional integral equationrdquo

Dynamic Systems and Applications vol 18 no 3-4 pp 667ndash6762009

[32] L Olszowy ldquoOn some measures of noncompactness in theFrechet spaces of continuous functionsrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 72 pp 2119ndash2126 2010

[33] L Olszowy ldquoOn existence of solutions of a quadratic Urysohnintegral equation on an unbounded intervalrdquo CommentationesMathematicae vol 48 no 1 pp 103ndash112 2008

[34] L Olszowy ldquoExistence ofmild solutions for semilinear nonlocalCauchy problems in separable Banach spacesrdquo Journal of Anal-ysis and its Applications vol 32 no 2 pp 215ndash232 2013

[35] L Olszowy ldquoFixed point theorems in the Frechet space 119862(119877+)

and functional integral equations on an unbounded intervalrdquoApplied Mathematics and Computation vol 218 no 18 pp9066ndash9074 2012

[36] J Banas and D OrsquoRegan ldquoOn existence and local attractivity ofsolutions of a quadratic Volterra integral equation of fractionalorderrdquo Journal of Mathematical Analysis and Applications vol345 no 1 pp 573ndash582 2008

[37] L D Landau EM Lifshitz and L P Pitaevski Electrodynamicsof Continuous Media vol 8 Butterworth-Heinemann 2ndedition 1984

[38] L D Landau and E M Lifshitz Fluid Mechanics vol bButterworth-Heinemann 2nd edition 1987

[39] R Temam Infinite-Dimensional Dynamical Systems inMechan-ics and Physics vol 68 Springer New York NY USA 2ndedition 1997

[40] J K Hale Asymptotic Behavior of Dissipative Systems vol 25 ofMathematical Surveys and Monographs American Mathemati-cal Society Providence RI USA 1988

[41] R A Adams and J J F Fournier Sobolev Spaces vol 140Elsevier 2nd edition 2003

[42] V Mazrsquoya Sobolev Spaces with Applications to Elliptic PartialDifferential Equations Springer Heidelberg Germany 2ndedition 2011

[43] V Georgescu ldquoSome boundary value problems for differentialforms on compact Riemannian manifoldsrdquo Annali di Matemat-ica Pura ed Applicata vol 122 pp 159ndash198 1979

[44] D Henry Geometric Theory of Semilinear Parabolic Equationsvol 840 of Lecture Notes in Mathematics Springer New YorkNY USA 1981

[45] V Girault and P-A Raviart Finite Element Methods for Navier-Stokes Equations Springer New York NY USA 1986

[46] J Bardeen L N Cooper and J R Schrieffer ldquoTheory ofsuperconductivityrdquo Physical Review vol 108 pp 1175ndash12041957

[47] A A Abrikosov Fundamental of the Theory of Metals North-Holland Amstedram The Netherlands 1988

[48] M Tinkham Introduction To Superconductivity McGraw-HillNew York NY USA 2nd edition 1996

[49] P DeGennes Superconductivity inMetals and Alloys BenjaminNew York NY USA 1966

[50] H Brezis Functional Analysis Sobolev Spaces and PartialDifferential Equations Springer New York NY USA 2011

[51] E Zeidler Nonlinear Functional Analysis and its Applicationsvol 2 Springer New York NY USA 1990

[52] A Schmid ldquoA time dependent Ginzburg-Landau equation andits application to the problem of resistivity in the mixed staterdquoPhysik der KondensiertenMaterie vol 5 no 4 pp 302ndash317 1966

[53] C M Elliot and Q Tang Existence Theorems For an Evolution-ary Superconductivity Model University of Sussex BrightonUK 1992

[54] Q Tang ldquoOn an evolutionary system of Ginzburg-Landauequations with fixed total magnetic fluxrdquo Communications inPartial Differential Equations vol 20 no 1-2 pp 1ndash36 1995

[55] Q Du ldquoGlobal existence and uniqueness of solutions of thetime-dependent Ginzburg-Landau model for superconductiv-ityrdquo Applicable Analysis vol 53 no 1-2 pp 1ndash18 1994

[56] Z M Chen K-H Hoffmann and J Liang ldquoOn a nonstation-ary Ginzburg-Landau superconductivity modelrdquoMathematicalMethods in the Applied Sciences vol 16 no 12 pp 855ndash875 1993

[57] J Liang and Q Tang ldquoAsymptotic behaviour of the solutionsof an evolutionaryGinzburg-Landau superconductivitymodelrdquoJournal of Mathematical Analysis and Applications vol 195 no1 pp 92ndash107 1995

[58] L P Gorrsquokov and G M Eliasherberg ldquoGeneralization of theGinzburg-Landau equations for an nonstationary problemsin the case of alloys with paramagnetic impartiesrdquo ZhurnalEksperimentalrsquonoi i Teoreticheskoi Fiziki vol 54 pp 612ndash6261968 Soviet Physics vol 27 pp 328ndash334 1968

[59] J Fleckinger-Pelle and H G Kaper ldquoGauges for the Ginzburg-Landau equations of superconductivityrdquo Zeitschrift fur Ange-wandte Mathematik und Mechanik vol 76 no 2 pp 345ndash3481996

[60] P Takac ldquoOn the dynamical process generated by supercon-ductivity modelrdquo Zeitschrift fur Angewandte Mathematik undMechanik vol 76 no 52 pp 349ndash352 1995

[61] W D Gropp H G Kaper DM Levine M Palumbo and VMVinokur ldquoNumerical simulation of vortex dynamics in type-IIsuperconductorsrdquo Journal of Computational Physics vol 123 no2 pp 254ndash266 1996

[62] D W Braun G W Crabtree H G Kaper et al ldquoStructure of amoving vortex latticerdquo Physical Review Letters vol 76 no 5 pp831ndash834 1996

[63] G W Crabtree G K Leaf H G Kaper et al ldquoTime-dependent Ginzburg-Landau simulations of vortex guidance bytwin boundariesrdquo Physica C vol 263 no 1ndash4 pp 401ndash408 1996

[64] G W Carabtree G K Leaf H G Kaper D W Braun V MVinokur andA E Koshelev ldquoDynamics vortex phases in super-conductors with correlated disorderrdquo Preprint ANLMCS-P590-0496 Mathematics and Computer Science DivisionAragonne National Laboratory 1996

[65] Th Gallay ldquoA center-stable manifold theorem for differentialequations in Banach spacesrdquo Communications in MathematicalPhysics vol 152 no 2 pp 249ndash268 1993

[66] J-M Ghidaglia and B Heron ldquoDimension of the attractorsassociated to the Ginzburg-Landau partial differential equa-tionrdquo Physica D vol 28 no 3 pp 282ndash304 1987

12 Journal of Function Spaces and Applications

[67] P Mattila Geometry of Sets and Measures in Euclidean Spacesvol 44 Cambridge University Press Cambridge UK 1995

[68] L HormanderThe Analysis of Linear Partial Differential Oper-ators vol 1 Springer Berlin Germany 2nd edition 1990

[69] H Amann Linear and Quasilinear Parabolic ProblemBikhauser Basel Switzerland 1995

[70] H-P Heinz ldquoOn the behaviour ofmeasures of noncompactnesswith respect to differentiation and integration of vector-valuedfunctionsrdquo Nonlinear Analysis Theory Methods amp ApplicationsA vol 7 no 12 pp 1351ndash1371 1983

[71] R P Agarwal M Meehan and D OrsquoRegan Fixed Point Theoryand Applications vol 141 of Cambridge Tracts in MathematicsCambridge University Press Cambridge UK 2001

[72] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer New York NY USA1983

[73] J Fleckinger-Pelle H G Kaper and P Takac ldquoDynamics of theGinzburg-Landau equations of superconductivityrdquo NonlinearAnalysis Theory Methods amp Applications A vol 32 no 5 pp647ndash665 1998

[74] H G Kaper and P Takac ldquoAn equivalence relation for theGinzburg-Landau equations of superconductivityrdquo Journal ofApplied Mathematics and Physics vol 48 no 4 pp 665ndash6751997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces and Applications 5

Hence in view of Lemma 6 (1198674) and (36) we obtain

120594 (119866Γ119899(119904)) le 2120594 (119910

119896infin

119896=1) + 120576

= 2120594 ((119866119906119896) (119904)infin

119896=1) + 120576 le 2120594 (119890

minusA1199041199060)

+ 2120594(int

119904

0

119890minusA(119904minus120591)

F (120591 119906119896(120591)) 119889120591

infin

119896=1

)

+ 120576 le 4int

119904

0

120594 (119890minusA(119904minus120591)

F (120591 119906119896(120591)infin

119896=1)) 119889120591 + 120576

le 4119873 (119879)int

119904

0

119871 (120591) 120594 (119906119896(120591)infin

119896=1) 119889120591 + 120576

le 4119873 (119879)int

119905

0

119871 (120591) 119892119899(120591) 119889120591 + 120576

(43)

Since 120576 gt 0 is arbitrary it follows from the above inequalitiesthat

120594 (119866Γ119899(119904)) le 4119873 (119879)int

119905

0

119871 (120591) 119892119899(120591) 119889120591 119904 isin [0 119905] (44)

Using (36) we have

119892119899+1

(119905) = sup 120594 (119866Γ119899(119904)) 119904 isin [0 119905]

le 4119873 (119879)int

119905

0

119871 (120591) 119892119899(120591) 119889120591 119905 isin [0 119879]

(45)

Letting 119899 rarr infin we derive the following inequality

119892infin(119905) le 4119873 (119879)int

119905

0

119871 (120591) 119892infin(120591) 119889120591 119905 isin [0 119879] (46)

This inequality together with Gronwallrsquos Lemma 8 impliesthat

119892infin(119905) = 0 119905 isin [0 119879] (47)

Next let us notice that in view of (1198674) and (36) we have

120594 (F (119904 Γ119899(119904))) le 119871 (119904) 120594 (Γ

119899(119904)) le 119871 (119904) 119892

119899(119904) (48)

for ae 119904 isin [0 119879] and the function

[0 119879] ni 119904 997891997888rarr 2119873 (119879) 119871 (119904) 119892119899(119904) (49)

is measurable Then in virtue of equality 1205961198790(119866Γ119899) = 120596119879

0(119865Γ119899)

and Lemma 9 we get

120596119879

0(119866Γ119899) le 2119873 (119879)int

119879

0

119871 (119904) 119892119899(119904) 119889119904 (50)

Hence we derive

119908119899+1

= 120596119879

0(Γ119899+1) = 120596119879

0(119866Γ119899)

le 2119873 (119879)int

119879

0

119871 (119904) 119892119899(119904) 119889119904

(51)

Letting 119899 rarr infin we get

119908infinle 2119873 (119879)int

119879

0

119871 (119904) 119892infin(119904) 119889119904 (52)

Keeping in mind (47) we deduce that

119908infin= 0 (53)

This together with (39) and (47) yields

lim119899rarrinfin

120583119879(Γ119899) = lim119899rarrinfin

(119908119899+ 119892119899(119879)) = 119908

infin+ 119892infin= 0 (54)

Finally using Remark 2 for the measure 120583119879 we deduce

that the set Γinfin= ⋂infin

119899=0Γ119899is nonempty convex and compact

Then by the Schauder-Tychonoff theorem we conclude thatoperator 119865 Γ

infinrarr Γinfin

has at least one fixed-point 119906 = 119906(119905)Obviously the function 119906 = 119906(119905) is a solution of problem (6)-(7) and in view of the definition of the set Γ

infin the estimate

119906(119905) le P(119905) holds to be trueThis completes the proof

4 Local and Global Attractivity

Following the concepts introduced in [36] we introduce firsta few definitions of various kinds of the concept of attractivityof mild solution of (6)

Definition 11 The mild solution 119906 = 119906(119905) of (6) with initialcondition (7) is said to be globally attractive if for each mildsolution V = V(119905) of (6) with initial condition V(0) = V

0we

have that

lim119905rarrinfin

(119906 (119905) minus V (119905)) = 120579 (55)

In other words we may say that solutions of (6) are globallyattractive if for arbitrary solutions 119906(119905) and V(119905) of thisequation condition (55) are satisfied

Definition 12 We say that mild solution 119906 = 119906(119905) of (6) withinitial condition (7) is locally attractive if there exists a ball119861(119906(0) 119903) in the space X such that for arbitrary solution V(119905)of (1) with initial-value V(0) isin 119861(119906(0) 119903) condition (55) doeshold

In the case when the limit (55) is uniform with respect toall solutions V(119905) that is when for each 120576 gt 0 there exist119879 gt 0such that

119906 (119905) minus V (119905) le 120576 (56)

for all V(119905) being solutions of (6) with initial-value V(0) isin119861(119906(0) 119903) and for 119905 ge 119879 we will say that solution 119906 = 119906(119905)

is uniformly locally attractive on R+

Now we formulate themain result of this sectionWe willconsider (6) under the following conditions

(1198671015840

1) A is the infinitesimal generator of an exponentiallystable119862

0-semigroup 119890minusA119905

119905ge0 that is there exist119872 gt

0 120588 gt 0 such that 119890minusA119905 le 119872119890minus120588119905 for all 119905 ge 0

6 Journal of Function Spaces and Applications

(1198675) there exist locally integrable functions119898 R

+rarr R+

such that

F (119905 119906) minusF (119905 V) le 119898 (119905) 119906 minus V (57)

for 119905 ge 0 and 119906 V isin X Moreover we assume that

lim119905rarrinfin

(int

119905

0

119898(119904) 119889119904 minus120588

119872119905) = minusinfin (58)

Remark 13 The property (11986710158401) is generally satisfied in diffu-

sion problem A necessary and sufficient condition for (11986710158401)

is presented in [72]

The main result of this section is shown in the giventheorem below

Theorem 14 Under assumptions (11986710158401) and (119867

2)ndash(1198675) prob-

lem (6)-(7) has amild solution 119906 = 119906(119905) for each 1199060isin X which

is globally attractive and locally uniformly attractive

Proof Existence of a solution 119906 = 119906(119905) is a consequence ofTheorem 10 Let us fix 119903 gt 0 and V

0isin 119861(119906(0) 119903)

Let V = V(119905) denote a mild solution of (6) with the initialcondition V(0) = V

0 Using (1198671015840

1) and (119867

3) we get

V (119905) le10038171003817100381710038171003817119890minusA119905V0

10038171003817100381710038171003817+ int

119905

0

10038171003817100381710038171003817119890minusA(119905minus119904)10038171003817100381710038171003817

sdot F (119904 V (119904)) 119889119904

le 119872(100381710038171003817100381711990601003817100381710038171003817 + 119903) +119872int

119905

0

119890120588119904119886 (119904) 119889119904

+ int

119905

0

119872119890120588119904119887 (119904) V (119904) 119889119904

(59)

Now let us put

P1(119905) = 119872 (

1003817100381710038171003817V01003817100381710038171003817 + 119903) +119872int

119905

0

119890120588119904119886 (119904) 119889119904

P2(119905) = 119872119890

120588119904119887 (119905)

(60)

Taking into account Lemma 5 we obtain

V (119905) le P1(119905) + int

119905

0

P2(119904)P1(119904) exp(int

119904

0

P2(120591) 119889120591) 119889119904

(61)

Further let 119878 be the set of all mild solutions V(119905) of (6) withthe initial-value V

0= V(0) isin 119861(119906(0) 119903)

Put

119892 (119905) = sup 119906 (119905) minus V (119905) V isin 119878 (62)

The estimate (61) implies that the function119892(119905) is well definedApplying (119867

5) and (1198671015840

1) we get

119906 (119905) minus V (119905) le10038171003817100381710038171003817119890minusA11990510038171003817100381710038171003817

sdot10038171003817100381710038171199060 minus V

0

1003817100381710038171003817

+ int

119905

0

10038171003817100381710038171003817119890minusA(119905minus119904)10038171003817100381710038171003817

119898 (119904) 119906 (119904) minus V (119904) 119889119904

le 119872119890minus120588119905119903 + int

119905

0

119872119890minus119892(119905minus119904)

119898(119904) 119892 (119904) 119889119904

(63)

or equivalently

119890120588119905119892 (119905) le 119872119903 + int

119905

0

119872119898(119904) 119890120588119904119892 (119904) 119889119904 (64)

Using again Lemma 5 for the above estimate (where ℎ(119905) =119890120588119905119892(119905)) we obtain

119890120588119905119892 (119905) le 119872119903

+ int

119905

0

1198722119903119898 (119904) exp(int

119905

119904

119872119898(120591) 119889120591) 119889119904

(65)

Elementary calculations lead to the following equality

int

119905

0

1198722119903119898 (119904) exp(int

119905

119904

119872119898(120591) 119889120591) 119889119904

= 119872119903(exp(int119905

0

119872119898(119904) 119889119904) minus 1)

(66)

Hence

119892 (119905) le 119872119903 exp (minus120588119905) + 119872119903 exp(int119905

0

119872119898(119904) 119889119904 minus 120588119905)

minus119872119903 exp (minus120588119905) (67)

Applying assumption (1198675) we derive

lim119905rarrinfin

119892 (119905) = 0 (68)

and this proves that 119906(119905) is locally attractive Finally thisequality together with definition of the function 119892(119905) impliesthat 119906(119905) is globally attractive The proof is complete

Remark 15 In the case when 119898(119905) equiv 119898 is constant the fol-lowing condition

lim119905rarrinfin

(int

119905

0

119898(119904) 119889119904 minus120588

119872119905) = minusinfin (69)

means that 119898 lt 120588119872 Observe that this condition cannotbe weakened This observation is illustrated by the followingexampled

Example 16 Let X = R 119898(119905) equiv 119898 F(119905 119906) = sin 119905 + 119898119906119872 = 1 and 119890minusA119905 = 119890minus120588119905 Then the equation

119906 (119905) = 119890minusA1199051199060+ int

119905

0

119890minusA(119905minus119904)

F (119904 119906 (119904)) 119889119904 (70)

(for any fixed 1199060isin R) has the solution 119906(119905) expressed by the

following formula

119906 (119905) =(120588 minus 119898) sin 119905 minus cos 119905

(120588 minus 119898)2

+ 1+ (1199060+

1

(120588 minus 119898)2

+ 1) 119890(119898minus120588)119905

(71)

Notice that for 119898 ge 120588 the solution 119906(119905) is neither globallyattractive nor locally uniformly attractive because for each

Journal of Function Spaces and Applications 7

other solution V(119905) with initial condition V(0) = V0 obviously

described by similar formula as 119906(119905) we would have acontradiction

|119906 (119905) minus V (119905)| =

10038161003816100381610038161199060 minus V0

1003816100381610038161003816 119890(119898minus120588)119905

997888rarr infin

or10038161003816100381610038161199060 minus V

0

1003816100381610038161003816

(72)

5 Asymptotic Behaviour

In this section we will give a theorem describing asymptoticbehaviour of mild solutions of (6) with condition (7) Thistheorem generalizes the result included in [72Theorem 44]First we formulate the assumptions

(1198671015840

3) This condition is almost identical with (119867

3) and the

only difference is that we assume the functions 119886 and119887 are locally essentially bounded on R

+

(1198671015840

5) There exists 119906

infinisin X such that there exists the limit

lim119905rarrinfin

F(119905 119906infin) and

lim119905rarrinfin

F (119905 119906infin) = A

minus1119906infin (73)

Moreover there exists a number119898 lt 120588119872 such that1003817100381710038171003817F (119905 119906) minusF (119905 119906

infin)1003817100381710038171003817 le 119898

1003817100381710038171003817119906 minus 119906infin1003817100381710038171003817 (74)

for 119905 ge 0 and 119906 isin X

Remark 17 The condition (11986710158405) in conjunction with (1198671015840

1)

ensures the existence ofAminus1 (see [72]) Clearly (11986710158403) implies

(1198673)

Theorem 18 Under assumptions (11986710158401) (1198672) (11986710158403) (1198674) and

(11986710158405) (6) with condition (7) has a mild solution 119906 = 119906(119905) for

each 1199060isin X such that lim

119905rarrinfin119906(119905) = 119906

infin

Proof The existence of a mild solution 119906(119905) is guaranteed byTheorem 10 Let us put

119892 (119905) =1003817100381710038171003817119906 (119905) minus 119906infin

1003817100381710038171003817

Finfin= lim119905rarrinfin

F (119905 119906infin)

(75)

We show that

lim119905rarrinfin

119892 (119905) = 0 (76)

Recall that if assumption (11986710158401) is fulfilled then for each 119911 isin X

we have (see [72])

int

infin

0

119890minusA119904

119911 119889119904 = minusAminus1119911 (77)

Using the above fact and (11986710158405) we get

119906infin= (minusA

minus1(minusAminus1119906infin)) = minusA

minus1Finfin

= int

infin

0

119890minusA119904

Finfin119889119904

(78)

Linking the above equality with (11986710158401) we obtain

119892 (119905)

=

10038171003817100381710038171003817100381710038171003817119890minusA1199051199060+ int

119905

0

119890minusA(119905minus119904)

F (119904 119906 (119904)) 119889119904 minus int

infin

0

119890minusA119904

Finfin119889119904

10038171003817100381710038171003817100381710038171003817

le 119872119890minus120588119905 10038171003817100381710038171199060

1003817100381710038171003817 +

10038171003817100381710038171003817100381710038171003817int

infin

119905

119890minusA119904

Finfin119889119904

10038171003817100381710038171003817100381710038171003817

+

10038171003817100381710038171003817100381710038171003817int

119905

0

119890minusA(119905minus119904)

F (119904 119906 (119904)) 119889119904 minus int

119905

0

119890minusA119904

Finfin119889119904

10038171003817100381710038171003817100381710038171003817

le 119872119890minus120588119905 10038171003817100381710038171199060

1003817100381710038171003817 +119872

119890minus1205881199051205881003817100381710038171003817Finfin

1003817100381710038171003817

+

10038171003817100381710038171003817100381710038171003817int

119905

0

119890minusA(119905minus119904)

F (119904 119906 (119904)) 119889119904 minus int

119905

0

119890minusA(119905minus119904)

Finfin119889119904

10038171003817100381710038171003817100381710038171003817

le 119872119890minus120588119905(100381710038171003817100381711990601003817100381710038171003817 +

1003817100381710038171003817Finfin1003817100381710038171003817

120588)

+ int

119905

0

119872119890minus120588(119905minus119904) 1003817100381710038171003817F (119904 119906 (119904)) minusF

infin

1003817100381710038171003817 119889119904

le 119872119890minus120588119905(100381710038171003817100381711990601003817100381710038171003817 +

1003817100381710038171003817Finfin1003817100381710038171003817

120588)

+119872119890minus120588119905int

119905

0

119890120588119904 1003817100381710038171003817F (119904 119906

infin) minusF

infin

1003817100381710038171003817 119889119904

+119872119890minus120588119905int

119905

0

119890120588119904 1003817100381710038171003817F (119904 119906 (119904)) minusF (119904 119906

infin)1003817100381710038171003817 119889119904

(79)

Next putting

119888 (119905) = 119872119890minus120588(119905)

(100381710038171003817100381711990601003817100381710038171003817 +

1003817100381710038171003817Finfin1003817100381710038171003817

120588)

+119872119890minus120588119905int

119905

0

119890120588119904 1003817100381710038171003817F (119904 119906

infin) minusF

infin

1003817100381710038171003817 119889119904

(80)

and applying (11986710158405) we derive the following inequality

119892 (119905) le 119888 (119905) + 119898119872119890minus120588119905int

119905

0

119890120588119904119892 (119904) 119889119904 (81)

Hence

119890120588119905119892 (119905) le 119890

120588(119905)119888 (119905) + 119898119872int

119905

0

119890120588119904119892 (119904) 119889119904 (82)

The above inequality in conjunction with Lemma 5 gives

119890120588119905119892 (119905) le 119890

120588119905119888 (119905)

+ int

119905

0

119890120588119904119888 (119904)119898119872 sdot exp (119898119872(119905 minus 119904)) 119889119904

= 119890120588119905119888 (119905)

+ 119898119872119890119898119872119905

int

119905

0

119888 (119904) exp ((120588 minus 119898119872) 119904) 119889119904

(83)

8 Journal of Function Spaces and Applications

Hence

119892 (119905) le 119888 (119905) +119898119872int

119905

0119888 (119904) 119890(119892minus119898119872)119904

119889119904

exp ((120588 minus 119898119872) 119905) (84)

Before proving (76) we first show that

lim119905rarrinfin

119888 (119905) = 0 (85)

To prove this equality it is sufficient to show that the lastcomponent in the formula expressing 119888(119905) that is 119872119890minus120588119905

int119905

0119890120588119904F(119904 119906

infin) minusF

infin119889119904 tends to zero as 119905 rarr infin

To this end let us fix 120576 gt 0 Assumption (11986710158405) implies that

there exists 1199050such that

sup 1003817100381710038171003817F (119904 119906infin) minusF

infin

1003817100381710038171003817 119904 ge 1199050 le120576120588

2119872 (86)

This inequality together with (11986710158403) implies

ess sup 1003817100381710038171003817F (119904 119906infin) minusF

infin

1003817100381710038171003817 119904 isin R lt infin (87)

Now fix 1199051isin R such that 119905

1ge 119905 and

ess sup 1003817100381710038171003817F (119904 119906infin) minusF

infin

1003817100381710038171003817 119904 isin R

le120576120588 exp (120588119905

1)

2119872 (exp (1205881199050) minus 1)

(88)

Then using (86) and (88) we conclude that for 119905 ge 1199051we have

119872119890minus120588119905int

119905

0

119890120588119904 1003817100381710038171003817F (119904 119906

infin) minusF

infin

1003817100381710038171003817 119889119904

le 119872119890minus120588119905int

1199050

0

119890120588119904 1003817100381710038171003817F (119904 119906

infin) minusF

infin

1003817100381710038171003817 119889119904

+119872119890minus120588119905int

119905

1199050

119890120588119904 1003817100381710038171003817F (119904 119906

infin) minusF

infin

1003817100381710038171003817 119889119904 le120576

2+120576

2= 120576

(89)

This fact proves (85)Further using (1198671015840

5) and (85) and employing de

LrsquoHospitalrsquos rule for the fraction on the right-hand sideof inequality (84) we obtain that condition (76) is satisfiedThis fact completes the proof

6 An Application to the Ginzburg-LandauEquations of Superconductivity

In this section we formulate the gauged time-dependentGinzburg-Landau (TDGL) equations as an abstract evolutionequation in a Hilbert space Moreover we show applicationsof the above theorems to TDGL equations

We assume that Ω is a bounded domain in R119899 withboundary 120597Ω of class11986211That isΩ is an open and connectedset whose boundary 120597Ω is a compact (119899 minus 1)-manifolddescribed by Lipschitz continuous differentiable charts Weconsider two- and three-dimensional problems (119899 = 2 and119899 = 3 resp) Assume that the vector potential A takes its

values in R119899 The vector H will represent the (externally)applied magnetic field which is a function of space andtime similarly to A it takes its values in R119899 The function120574 is defined and satisfies Lipschitz condition on 120597Ω and120574(119909) ge 0 for 119909 isin 120597Ω The parameters in the TDGLequations are 120578 a (dimensionless) friction coefficient and 120581the (dimensionless) Ginzburg-Landau parameter

The order parameter should be thought of as the wavefunction of the center-of-massmotion of the ldquosuperelectronsrdquo(Cooper pairs) whose density is 119899

119904= |120595|

2 and whose fluxis J119904 The vector potential A determines the electromagnetic

field E = minus120597119905A minus nabla120601 is the electric field and B = nabla times A

is the magnetic induction where J the total current is thesum of a ldquonormalrdquo current J

119899= E the supercurrent J

119904 and

the transport current J119905= nabla times H The normal current obeys

Ohmrsquos law J119899= 120590119899E the ldquonormal conductivityrdquo coefficient 120590

119899

is equal to one in the adopted system of units The differenceM = BminusH is known as themagnetizationThe trivial solution(120595 = 0 B = H E = 0) represents the normal state where allsuperconducting properties have been lost

Nowwe accept the following notion all Banach spaces arereal the (real) dual of a Banach space119883 is denoted by1198831015840Thesymbol 119871119901(Ω) for 1 le 119901 le infin denotes the usual Lebesguespace with norm sdot

119871119901 (sdot sdot) is the inner product in 1198712(Ω)

1198821198982(Ω) for nonnegative integer 119898 is the usual Sobolev

space with norm sdot 1198821198982 1198821198982(Ω) is a Hilbert space for the

inner product (sdot sdot)1198982

given by (119906 V)1198982

= sum|120572|le119898

(120597120572119906 120597120572V)

for 119906 V isin 1198821198982(Ω) Fractional Sobolev space1198821199042(Ω) with afractional 119904 is defined by interpolation ([40 Chap VII] and[41 49 50]) 119862]

(Ω) for ] ge 0 ] = 119898 + 120582 with 0 le 120582 lt 1 isthe space of 119898 times continuous differentiable functions onΩ those 119898th-order derivatives satisfy the Holder conditionwith exponent 120582 if ] is a proper fraction the norm sdot

119862] is

defined in the usual wayThe definitions extend to the space of vector-valued

functions in the standard way with the caveat that the innerproduct in [1198712(Ω)]119899 is defined by (119906 V) = int

Ω119906 sdot V where the

symbol (119906 V) indicates the scalar product in R119899 Complex-valued functions are interpreted as vector-valued functionswith two real components

Functions that vary in space and time like the orderparameter and the vector potential are considered as map-pings from the time domain which is a subinterval of [0infin)into spaces of complex- or vector-valued functions definedin Ω Let 119883 = (119883 sdot

119883) be a Banach space of functions

defined inΩ Then the functions are defined in Ω Then thefunctions of space and time defined on Ω times (0 119879) for 119879 gt 0may be considered as elements of 119871119901(0 119879119883) for 1 le 119901 le infinor1198822119898(0 119879119883) for nonnegative119898 or119862]

(0 119879119883) for ] ge 0] = 119898 + 120582 with 0 le 120582 lt 1 Detailed definitions can be foundfor example in [43]

Obviously function spaces of ordered pairs (120595A) where120595 Ω rarr R2 and A Ω rarr R119899 (119899 = 2 3) play an importantrole in the study of the gauged TDGL equationsWe thereforeadapt the following special notationH = [119883(Ω)]

2times[119883(Ω)]

119899for any Banach space for the order parameter 120595 and thevector potential A respectively A suitable framework forthe functional analysis of the gauged TDGL equations is the

Journal of Function Spaces and Applications 9

Cartesian product W1+1205722 = [1198821+1205722

(Ω)]2times [1198821+1205722

(Ω)]119899

where 120572 isin (12 1) This space is continuously imbedded inW12 capLinfin

Assume H = 0 and H isin [1198712(Ω)]119899 Let AH be a minimizer

of the convex quadratic form 119869120596equiv J120596[119860]

119869120596 [A] = int

Ω

[120596(nabla sdot A)2 + |nabla times A minusH|2] 119889119909 (90)

on the domain

D (119869120596) = A isin [119882

12(Ω)]119899

n sdot A = 0 on 120597Ω (91)

We now introduce the reduced vector potential A1015840

A1015840 = A minus AH (92)

In terms of 120595 and A1015840 the gauged TDGL equations have thefollowing form

120597120595

120597119905minus

1

1205781205812Δ120595 = 120593 in Ω times (0infin)

120597A1015840

120597119905+ nabla times nabla times A1015840 minus 120596nabla (nabla sdot A1015840) = F in Ω times (0infin)

n sdot nabla + 120574120595 = 0 n sdot A1015840 = 0

n times (nabla sdot A1015840) = 0 on 120597Ω times (0infin)

(93)

Here 120593 and F are nonlinear functions of 120595 and A1015840

120593 equiv 120593 (119905 120595A1015840) = 1

120578[minus2119894

120581(nabla120595) sdot (A1015840 + AH)

minus119894

120581(1 minus 120578120581

2120596)120595 (nabla sdot A1015840)

minus12059510038161003816100381610038161003816A1015840 + AH

10038161003816100381610038161003816

2

+ (1 minus100381610038161003816100381612059510038161003816100381610038162

) 120595]

(94)

F equiv F (119905 120595A1015840) = 1

2119894119896(120595lowastnabla120595 minus 120595nabla120595

lowast)

minus100381610038161003816100381612059510038161003816100381610038162

(A1015840 + AH) minus120597AH120597119905

(95)

The equations are supplemented by initial data which is inthe followimg form

120595 = 1205950 A = A

0on Ω times 0 (96)

where 1205950and A

0are given and by (92) we have

120595 = 1205950 A1015840 = A

0minus AH (0) on Ω times 0 (97)

In the next part we connect the evolution of the solution(120595A1015840) of the system of (93) with the initial data (120595

0A0minus

AH(0)) with the dynamics of a vector 119906 in a Hilbert space

L2= [119871119901(Ω)]2

times [119871119901(Ω)]119899

(98)

The following analysis is restricted to the case 120596 gt 0 andthe case 120596 = 0 (see [73])

Let vector 119906 [0infin) rarr L2 represent the pair (120595A1015840)

119906 = (120595A1015840) equiv (120595A minus AH) (99)

and letA be the linear self-adjoint operator inL2 associatedwith the quadratic form 119876

120596equiv 119876120596[119906] given by the following

formula

119876120596 [119906] = int

Ω

[1

12057812058121003816100381610038161003816nabla120595

10038161003816100381610038162

+ 120596(nabla sdot A1015840)2

+10038161003816100381610038161003816nabla times A101584010038161003816100381610038161003816

2

] 119889119909

+ int120597Ω

120574

1205781205812100381610038161003816100381612059510038161003816100381610038162

119889120590 (119909)

(100)

on the domain

D (119876120596) = D (A

12)

= 119906 = (120595A1015840) isinW12 n sdot A1015840 = 0 on 120597Ω

(101)

The quadratic form 119876120596is nonnegative Furthermore since

120596 gt 0119876120596[120595A1015840]+119888 120595

1198712 is coercive onW12 for any constant

119888 gt 0 Hence A is positively definite in L2 ([44 Chap 1]equation (545)) If it does not lead to confusion we use thesame symbol A for the restriction A

120595and AA of A to the

respective linear subspace [1198712(Ω)]2 = [1198712(Ω)]2 times 0 (for 120595)and [1198712(Ω)]119899 equiv 0 times [1198712(Ω)]119899 (for A) ofL2

Now consider the initial-value problems (6) and (7) inL2 whereF(119905 119906) = (120593 F) 120593 and F is given by (94) and (95)and 119906

0= (1205950A0minus AH(0))

With 120582 isin (12 1) and 1199060isin W1+1205822 we say that 119906 are a

solution of (6) and (7) on the interval [0 119879] for some 119879 gt 0if 119906 [0 119879] rarr W1+1205822 is continuous and

119906 (119905) = 119890minusA(119905)

+ int

119905

0

119890minusA(119905minus119904)

F (119904 119906 (119904)) 119889119904 for 119905 isin [0 119879]

(102)

in L2 A mild solution of the initial-value problems (6) and(7) defines a weak solution (120595A1015840) of the boundary valueproblem (93) which in turn defines a weak solution (120595A)of the gauged TDGL equations provided AH is sufficientlyregular

Namely let us assume that X = L2 and 1199060isin L2 and

119906 R+times R119899 rarr R119899 for 119899 isin N is an unknown function 119906 =

119906(119905 119909) In order to applyTheorems 10 and 14 we are not goingto consider 119906 as a function of 119905 and 119909 together but rather asa mapping 119906 of variable 119905 into the spaceX =L2 of functions119909 that is 119906 R

+rarr L2 119906(119905)(119909) = 119906(119905 119909) 119909 isin R119899 119905 isin R

+

Remark 19 Since A is the linear self-adjoint operator inL2

associated with the quadratic form (100) thenA generates asemigroup 119890minusA119905

119905ge0(see [74])

Below we formulate the principal theorem of this paperThis theorem is a simple consequence of Theorem 10

10 Journal of Function Spaces and Applications

Theorem 20 Under assumptions (1198671)ndash(1198674) problem (6)-(7)

is equivalent to the system of (1)ndash(5) (TDGL equations) andhas for each 119906

0isin L2 at least one mild solution in sense of

Definition 4 119906 = 119906(119905) Moreover each solution 119906(119905) satisfiesestimate (25)

We observe that in the case when 119890minusA119905119905ge0

is the semi-group of contractions and 119886(119905) equiv 119886 119887(119905) equiv 119887 for 119905 isin

R+are constant after simple calculations based on estimate

(25) we get that the solution 119906 = 119906(119905) has the asymptoticcharacterization

119906 (119905) = 119874 (1199053) as 119905 997888rarr infin (103)

For further purposes let us formulate the followingassumption

(A) A is the infinitesimal generator of an exponentiallystable 119862

0-semigroup 119890minusA119905

119905ge0

The next result of this chapter is shown in the giventheorem below

Theorem21 Under assumptions (A) and (1198672)ndash(1198674) problem

(6)-(7) which is equivalent to the systems equations (1)ndash(5)(TDGL equations) has a mild solution 119906 = 119906(119905) for each1199060isin L2 which is globally attractive and locally uniformly

attractive

References

[1] V L Ginzburg and L D Landau ldquoOn the theory of supercon-ductivityrdquo Zhurnal Eksperimentalrsquonoi i Teoreticheskoi Fiziki vol20 pp 1064ndash1082 1950 English translation in L D LandauMen of Physics D ter Haar ed Vol I Pergamon Press OxfordUK pp 138ndash167 1965

[2] S Aizicovici and H Lee ldquoNonlinear nonlocal Cauchy problemsin Banach spacesrdquo Applied Mathematics Letters vol 18 no 4pp 401ndash407 2005

[3] S Aizicovici and M McKibben ldquoExistence results for a classof abstract nonlocal Cauchy problemsrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 39 no 5 pp 649ndash6682000

[4] MMcKibbenDiscovering Evolution Equation with ApplicationCRC Press Taylor amp Francis Group 2011

[5] S Aizicovici and V Staicu ldquoMultivalued evolution equationswith nonlocal initial conditions in Banach spacesrdquo NonlinearDifferential Equations and Applications vol 14 no 3-4 pp 361ndash376 2007

[6] R P Agarwal and D OrsquoRegan Infinite Interval Problems for Dif-ferential Difference and Integral Equations Kluwer AcademicPublishers Dordrecht The Netherlands 2001

[7] Z Fan ldquoExistence of nondensely defined evolution equationswith nonlocal conditionsrdquo Nonlinear Analysis Theory Methodsamp Applications A vol 70 no 11 pp 3829ndash3836 2009

[8] Z Fan ldquoImpulsive problems for semilinear differential equa-tions with nonlocal conditionsrdquo Nonlinear Analysis TheoryMethods amp Applications A vol 72 no 2 pp 1104ndash1109 2010

[9] Z Fan and G Li ldquoExistence results for semilinear differentialequations with nonlocal and impulsive conditionsrdquo Journal ofFunctional Analysis vol 258 no 5 pp 1709ndash1727 2010

[10] J Garcıa-Falset ldquoExistence results and asymptotic behavior fornonlocal abstract Cauchy problemsrdquo Journal of MathematicalAnalysis and Applications vol 338 no 1 pp 639ndash652 2008

[11] J Liang J H Liu and T-J Xiao ldquoNonlocal impulsive problemsfor nonlinear differential equations in Banach spacesrdquo Mathe-matical and Computer Modelling vol 49 no 3-4 pp 798ndash8042009

[12] L Zhu and G Li ldquoOn a nonlocal problem for semilinear differ-ential equations with upper semicontinuous nonlinearities ingeneral Banach spacesrdquo Journal of Mathematical Analysis andApplications vol 341 no 1 pp 660ndash675 2008

[13] Q Liu and R Yuan ldquoExistence of mild solutions for semilinearevolution equations with non-local initial conditionsrdquo Nonlin-ear AnalysisTheoryMethodsampApplications A vol 71 no 9 pp4177ndash4184 2009

[14] C Xie W Zhong and Z Fan ldquoExistence results for nonlinearnonlocal problems in Banach spacesrdquo Applied MathematicsLetters vol 22 no 7 pp 998ndash1002 2009

[15] T-J Xiao and J Liang ldquoExistence of classical solutions tononautonomous nonlocal parabolic problemsrdquoNonlinear Anal-ysisTheoryMethods and Applications vol 63 no 5-7 pp e225ndashe232 2005

[16] X Xue ldquo119871119901 theory for semilinear nonlocal problems with mea-sure of noncompactness in separable Banach spacesrdquo Journal ofFixed Point Theory and Applications vol 5 no 1 pp 129ndash1442009

[17] X Xue ldquoNonlinear differential equations with nonlocal condi-tions in Banach spacesrdquo Nonlinear Analysis Theory Methods ampApplications A vol 63 no 4 pp 575ndash586 2005

[18] X Xue ldquoExistence of solutions for semilinear nonlocal Cauchyproblems in Banach spacesrdquo Electronic Journal of DifferentialEquations vol 64 p 7 2005

[19] X Xue ldquoNonlocal nonlinear differential equations with a mea-sure of noncompactness in Banach spacesrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 70 no 7 pp 2593ndash26012009

[20] L Olszowy ldquoExistence ofmild solutions for the semilinear non-local problem in Banach spacesrdquo Nonlinear Analysis TheoryMethods amp Applications A vol 81 pp 211ndash223 2013

[21] D Bothe ldquoMultivalued perturbations of119898-accretive differentialinclusionsrdquo Israel Journal of Mathematics vol 108 pp 109ndash1381998

[22] H-S Ding J Liang G M NrsquoGuerekata and T-J XiaoldquoMild pseudo-almost periodic solutions of nonautonomoussemilinear evolution equationsrdquo Mathematical and ComputerModelling vol 45 no 5-6 pp 579ndash584 2007

[23] J Garcıa-Falset ldquoThe asymptotic behavior of the solutions of theCauchy problem generated by 120593-accretive operatorsrdquo Journal ofMathematical Analysis and Applications vol 310 no 2 pp 594ndash608 2005

[24] Z Fan and G Li ldquoAsymptotic behavior of the solutions of non-autonomous systems in Banach spacesrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 68 no 12 pp 3733ndash37412008

[25] J Banas ldquoOn existence theorems for differential equations inBanach spacesrdquo Bulletin of the Australian Mathematical Societyvol 32 no 1 pp 73ndash82 1985

[26] J Banas ldquoMeasures of noncompactness in the space of contin-uous tempered functionsrdquo Demonstratio Mathematica vol 14no 1 pp 127ndash133 1981

Journal of Function Spaces and Applications 11

[27] J Banas and K GoebelMeasures of Noncompactness in BanachSpaces vol 60 ofLectureNotes in Pure andAppliedMathematicsMarcel Dekker New York NY USA 1980

[28] J Banas and M Lecko ldquoSolvability of infinite systems ofdifferential equations in Banach sequence spacesrdquo Journal ofComputational andAppliedMathematics vol 137 no 2 pp 363ndash375 2001

[29] J Banas and K Sadarangani ldquoCompactness conditions inthe study of functional differential and integral equationsrdquoAbstract and Applied Analysis vol 2013 Article ID 819315 14pages 2013

[30] M Mursaleen and S A Mohiuddine ldquoApplications of mea-sures of noncompactness to the infinite system of differentialequations in ℓ

119901spacesrdquo Nonlinear Analysis Theory Methods amp

Applications A vol 75 no 4 pp 2111ndash2115 2012[31] L Olszowy ldquoSolvability of some functional integral equationrdquo

Dynamic Systems and Applications vol 18 no 3-4 pp 667ndash6762009

[32] L Olszowy ldquoOn some measures of noncompactness in theFrechet spaces of continuous functionsrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 72 pp 2119ndash2126 2010

[33] L Olszowy ldquoOn existence of solutions of a quadratic Urysohnintegral equation on an unbounded intervalrdquo CommentationesMathematicae vol 48 no 1 pp 103ndash112 2008

[34] L Olszowy ldquoExistence ofmild solutions for semilinear nonlocalCauchy problems in separable Banach spacesrdquo Journal of Anal-ysis and its Applications vol 32 no 2 pp 215ndash232 2013

[35] L Olszowy ldquoFixed point theorems in the Frechet space 119862(119877+)

and functional integral equations on an unbounded intervalrdquoApplied Mathematics and Computation vol 218 no 18 pp9066ndash9074 2012

[36] J Banas and D OrsquoRegan ldquoOn existence and local attractivity ofsolutions of a quadratic Volterra integral equation of fractionalorderrdquo Journal of Mathematical Analysis and Applications vol345 no 1 pp 573ndash582 2008

[37] L D Landau EM Lifshitz and L P Pitaevski Electrodynamicsof Continuous Media vol 8 Butterworth-Heinemann 2ndedition 1984

[38] L D Landau and E M Lifshitz Fluid Mechanics vol bButterworth-Heinemann 2nd edition 1987

[39] R Temam Infinite-Dimensional Dynamical Systems inMechan-ics and Physics vol 68 Springer New York NY USA 2ndedition 1997

[40] J K Hale Asymptotic Behavior of Dissipative Systems vol 25 ofMathematical Surveys and Monographs American Mathemati-cal Society Providence RI USA 1988

[41] R A Adams and J J F Fournier Sobolev Spaces vol 140Elsevier 2nd edition 2003

[42] V Mazrsquoya Sobolev Spaces with Applications to Elliptic PartialDifferential Equations Springer Heidelberg Germany 2ndedition 2011

[43] V Georgescu ldquoSome boundary value problems for differentialforms on compact Riemannian manifoldsrdquo Annali di Matemat-ica Pura ed Applicata vol 122 pp 159ndash198 1979

[44] D Henry Geometric Theory of Semilinear Parabolic Equationsvol 840 of Lecture Notes in Mathematics Springer New YorkNY USA 1981

[45] V Girault and P-A Raviart Finite Element Methods for Navier-Stokes Equations Springer New York NY USA 1986

[46] J Bardeen L N Cooper and J R Schrieffer ldquoTheory ofsuperconductivityrdquo Physical Review vol 108 pp 1175ndash12041957

[47] A A Abrikosov Fundamental of the Theory of Metals North-Holland Amstedram The Netherlands 1988

[48] M Tinkham Introduction To Superconductivity McGraw-HillNew York NY USA 2nd edition 1996

[49] P DeGennes Superconductivity inMetals and Alloys BenjaminNew York NY USA 1966

[50] H Brezis Functional Analysis Sobolev Spaces and PartialDifferential Equations Springer New York NY USA 2011

[51] E Zeidler Nonlinear Functional Analysis and its Applicationsvol 2 Springer New York NY USA 1990

[52] A Schmid ldquoA time dependent Ginzburg-Landau equation andits application to the problem of resistivity in the mixed staterdquoPhysik der KondensiertenMaterie vol 5 no 4 pp 302ndash317 1966

[53] C M Elliot and Q Tang Existence Theorems For an Evolution-ary Superconductivity Model University of Sussex BrightonUK 1992

[54] Q Tang ldquoOn an evolutionary system of Ginzburg-Landauequations with fixed total magnetic fluxrdquo Communications inPartial Differential Equations vol 20 no 1-2 pp 1ndash36 1995

[55] Q Du ldquoGlobal existence and uniqueness of solutions of thetime-dependent Ginzburg-Landau model for superconductiv-ityrdquo Applicable Analysis vol 53 no 1-2 pp 1ndash18 1994

[56] Z M Chen K-H Hoffmann and J Liang ldquoOn a nonstation-ary Ginzburg-Landau superconductivity modelrdquoMathematicalMethods in the Applied Sciences vol 16 no 12 pp 855ndash875 1993

[57] J Liang and Q Tang ldquoAsymptotic behaviour of the solutionsof an evolutionaryGinzburg-Landau superconductivitymodelrdquoJournal of Mathematical Analysis and Applications vol 195 no1 pp 92ndash107 1995

[58] L P Gorrsquokov and G M Eliasherberg ldquoGeneralization of theGinzburg-Landau equations for an nonstationary problemsin the case of alloys with paramagnetic impartiesrdquo ZhurnalEksperimentalrsquonoi i Teoreticheskoi Fiziki vol 54 pp 612ndash6261968 Soviet Physics vol 27 pp 328ndash334 1968

[59] J Fleckinger-Pelle and H G Kaper ldquoGauges for the Ginzburg-Landau equations of superconductivityrdquo Zeitschrift fur Ange-wandte Mathematik und Mechanik vol 76 no 2 pp 345ndash3481996

[60] P Takac ldquoOn the dynamical process generated by supercon-ductivity modelrdquo Zeitschrift fur Angewandte Mathematik undMechanik vol 76 no 52 pp 349ndash352 1995

[61] W D Gropp H G Kaper DM Levine M Palumbo and VMVinokur ldquoNumerical simulation of vortex dynamics in type-IIsuperconductorsrdquo Journal of Computational Physics vol 123 no2 pp 254ndash266 1996

[62] D W Braun G W Crabtree H G Kaper et al ldquoStructure of amoving vortex latticerdquo Physical Review Letters vol 76 no 5 pp831ndash834 1996

[63] G W Crabtree G K Leaf H G Kaper et al ldquoTime-dependent Ginzburg-Landau simulations of vortex guidance bytwin boundariesrdquo Physica C vol 263 no 1ndash4 pp 401ndash408 1996

[64] G W Carabtree G K Leaf H G Kaper D W Braun V MVinokur andA E Koshelev ldquoDynamics vortex phases in super-conductors with correlated disorderrdquo Preprint ANLMCS-P590-0496 Mathematics and Computer Science DivisionAragonne National Laboratory 1996

[65] Th Gallay ldquoA center-stable manifold theorem for differentialequations in Banach spacesrdquo Communications in MathematicalPhysics vol 152 no 2 pp 249ndash268 1993

[66] J-M Ghidaglia and B Heron ldquoDimension of the attractorsassociated to the Ginzburg-Landau partial differential equa-tionrdquo Physica D vol 28 no 3 pp 282ndash304 1987

12 Journal of Function Spaces and Applications

[67] P Mattila Geometry of Sets and Measures in Euclidean Spacesvol 44 Cambridge University Press Cambridge UK 1995

[68] L HormanderThe Analysis of Linear Partial Differential Oper-ators vol 1 Springer Berlin Germany 2nd edition 1990

[69] H Amann Linear and Quasilinear Parabolic ProblemBikhauser Basel Switzerland 1995

[70] H-P Heinz ldquoOn the behaviour ofmeasures of noncompactnesswith respect to differentiation and integration of vector-valuedfunctionsrdquo Nonlinear Analysis Theory Methods amp ApplicationsA vol 7 no 12 pp 1351ndash1371 1983

[71] R P Agarwal M Meehan and D OrsquoRegan Fixed Point Theoryand Applications vol 141 of Cambridge Tracts in MathematicsCambridge University Press Cambridge UK 2001

[72] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer New York NY USA1983

[73] J Fleckinger-Pelle H G Kaper and P Takac ldquoDynamics of theGinzburg-Landau equations of superconductivityrdquo NonlinearAnalysis Theory Methods amp Applications A vol 32 no 5 pp647ndash665 1998

[74] H G Kaper and P Takac ldquoAn equivalence relation for theGinzburg-Landau equations of superconductivityrdquo Journal ofApplied Mathematics and Physics vol 48 no 4 pp 665ndash6751997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Function Spaces and Applications

(1198675) there exist locally integrable functions119898 R

+rarr R+

such that

F (119905 119906) minusF (119905 V) le 119898 (119905) 119906 minus V (57)

for 119905 ge 0 and 119906 V isin X Moreover we assume that

lim119905rarrinfin

(int

119905

0

119898(119904) 119889119904 minus120588

119872119905) = minusinfin (58)

Remark 13 The property (11986710158401) is generally satisfied in diffu-

sion problem A necessary and sufficient condition for (11986710158401)

is presented in [72]

The main result of this section is shown in the giventheorem below

Theorem 14 Under assumptions (11986710158401) and (119867

2)ndash(1198675) prob-

lem (6)-(7) has amild solution 119906 = 119906(119905) for each 1199060isin X which

is globally attractive and locally uniformly attractive

Proof Existence of a solution 119906 = 119906(119905) is a consequence ofTheorem 10 Let us fix 119903 gt 0 and V

0isin 119861(119906(0) 119903)

Let V = V(119905) denote a mild solution of (6) with the initialcondition V(0) = V

0 Using (1198671015840

1) and (119867

3) we get

V (119905) le10038171003817100381710038171003817119890minusA119905V0

10038171003817100381710038171003817+ int

119905

0

10038171003817100381710038171003817119890minusA(119905minus119904)10038171003817100381710038171003817

sdot F (119904 V (119904)) 119889119904

le 119872(100381710038171003817100381711990601003817100381710038171003817 + 119903) +119872int

119905

0

119890120588119904119886 (119904) 119889119904

+ int

119905

0

119872119890120588119904119887 (119904) V (119904) 119889119904

(59)

Now let us put

P1(119905) = 119872 (

1003817100381710038171003817V01003817100381710038171003817 + 119903) +119872int

119905

0

119890120588119904119886 (119904) 119889119904

P2(119905) = 119872119890

120588119904119887 (119905)

(60)

Taking into account Lemma 5 we obtain

V (119905) le P1(119905) + int

119905

0

P2(119904)P1(119904) exp(int

119904

0

P2(120591) 119889120591) 119889119904

(61)

Further let 119878 be the set of all mild solutions V(119905) of (6) withthe initial-value V

0= V(0) isin 119861(119906(0) 119903)

Put

119892 (119905) = sup 119906 (119905) minus V (119905) V isin 119878 (62)

The estimate (61) implies that the function119892(119905) is well definedApplying (119867

5) and (1198671015840

1) we get

119906 (119905) minus V (119905) le10038171003817100381710038171003817119890minusA11990510038171003817100381710038171003817

sdot10038171003817100381710038171199060 minus V

0

1003817100381710038171003817

+ int

119905

0

10038171003817100381710038171003817119890minusA(119905minus119904)10038171003817100381710038171003817

119898 (119904) 119906 (119904) minus V (119904) 119889119904

le 119872119890minus120588119905119903 + int

119905

0

119872119890minus119892(119905minus119904)

119898(119904) 119892 (119904) 119889119904

(63)

or equivalently

119890120588119905119892 (119905) le 119872119903 + int

119905

0

119872119898(119904) 119890120588119904119892 (119904) 119889119904 (64)

Using again Lemma 5 for the above estimate (where ℎ(119905) =119890120588119905119892(119905)) we obtain

119890120588119905119892 (119905) le 119872119903

+ int

119905

0

1198722119903119898 (119904) exp(int

119905

119904

119872119898(120591) 119889120591) 119889119904

(65)

Elementary calculations lead to the following equality

int

119905

0

1198722119903119898 (119904) exp(int

119905

119904

119872119898(120591) 119889120591) 119889119904

= 119872119903(exp(int119905

0

119872119898(119904) 119889119904) minus 1)

(66)

Hence

119892 (119905) le 119872119903 exp (minus120588119905) + 119872119903 exp(int119905

0

119872119898(119904) 119889119904 minus 120588119905)

minus119872119903 exp (minus120588119905) (67)

Applying assumption (1198675) we derive

lim119905rarrinfin

119892 (119905) = 0 (68)

and this proves that 119906(119905) is locally attractive Finally thisequality together with definition of the function 119892(119905) impliesthat 119906(119905) is globally attractive The proof is complete

Remark 15 In the case when 119898(119905) equiv 119898 is constant the fol-lowing condition

lim119905rarrinfin

(int

119905

0

119898(119904) 119889119904 minus120588

119872119905) = minusinfin (69)

means that 119898 lt 120588119872 Observe that this condition cannotbe weakened This observation is illustrated by the followingexampled

Example 16 Let X = R 119898(119905) equiv 119898 F(119905 119906) = sin 119905 + 119898119906119872 = 1 and 119890minusA119905 = 119890minus120588119905 Then the equation

119906 (119905) = 119890minusA1199051199060+ int

119905

0

119890minusA(119905minus119904)

F (119904 119906 (119904)) 119889119904 (70)

(for any fixed 1199060isin R) has the solution 119906(119905) expressed by the

following formula

119906 (119905) =(120588 minus 119898) sin 119905 minus cos 119905

(120588 minus 119898)2

+ 1+ (1199060+

1

(120588 minus 119898)2

+ 1) 119890(119898minus120588)119905

(71)

Notice that for 119898 ge 120588 the solution 119906(119905) is neither globallyattractive nor locally uniformly attractive because for each

Journal of Function Spaces and Applications 7

other solution V(119905) with initial condition V(0) = V0 obviously

described by similar formula as 119906(119905) we would have acontradiction

|119906 (119905) minus V (119905)| =

10038161003816100381610038161199060 minus V0

1003816100381610038161003816 119890(119898minus120588)119905

997888rarr infin

or10038161003816100381610038161199060 minus V

0

1003816100381610038161003816

(72)

5 Asymptotic Behaviour

In this section we will give a theorem describing asymptoticbehaviour of mild solutions of (6) with condition (7) Thistheorem generalizes the result included in [72Theorem 44]First we formulate the assumptions

(1198671015840

3) This condition is almost identical with (119867

3) and the

only difference is that we assume the functions 119886 and119887 are locally essentially bounded on R

+

(1198671015840

5) There exists 119906

infinisin X such that there exists the limit

lim119905rarrinfin

F(119905 119906infin) and

lim119905rarrinfin

F (119905 119906infin) = A

minus1119906infin (73)

Moreover there exists a number119898 lt 120588119872 such that1003817100381710038171003817F (119905 119906) minusF (119905 119906

infin)1003817100381710038171003817 le 119898

1003817100381710038171003817119906 minus 119906infin1003817100381710038171003817 (74)

for 119905 ge 0 and 119906 isin X

Remark 17 The condition (11986710158405) in conjunction with (1198671015840

1)

ensures the existence ofAminus1 (see [72]) Clearly (11986710158403) implies

(1198673)

Theorem 18 Under assumptions (11986710158401) (1198672) (11986710158403) (1198674) and

(11986710158405) (6) with condition (7) has a mild solution 119906 = 119906(119905) for

each 1199060isin X such that lim

119905rarrinfin119906(119905) = 119906

infin

Proof The existence of a mild solution 119906(119905) is guaranteed byTheorem 10 Let us put

119892 (119905) =1003817100381710038171003817119906 (119905) minus 119906infin

1003817100381710038171003817

Finfin= lim119905rarrinfin

F (119905 119906infin)

(75)

We show that

lim119905rarrinfin

119892 (119905) = 0 (76)

Recall that if assumption (11986710158401) is fulfilled then for each 119911 isin X

we have (see [72])

int

infin

0

119890minusA119904

119911 119889119904 = minusAminus1119911 (77)

Using the above fact and (11986710158405) we get

119906infin= (minusA

minus1(minusAminus1119906infin)) = minusA

minus1Finfin

= int

infin

0

119890minusA119904

Finfin119889119904

(78)

Linking the above equality with (11986710158401) we obtain

119892 (119905)

=

10038171003817100381710038171003817100381710038171003817119890minusA1199051199060+ int

119905

0

119890minusA(119905minus119904)

F (119904 119906 (119904)) 119889119904 minus int

infin

0

119890minusA119904

Finfin119889119904

10038171003817100381710038171003817100381710038171003817

le 119872119890minus120588119905 10038171003817100381710038171199060

1003817100381710038171003817 +

10038171003817100381710038171003817100381710038171003817int

infin

119905

119890minusA119904

Finfin119889119904

10038171003817100381710038171003817100381710038171003817

+

10038171003817100381710038171003817100381710038171003817int

119905

0

119890minusA(119905minus119904)

F (119904 119906 (119904)) 119889119904 minus int

119905

0

119890minusA119904

Finfin119889119904

10038171003817100381710038171003817100381710038171003817

le 119872119890minus120588119905 10038171003817100381710038171199060

1003817100381710038171003817 +119872

119890minus1205881199051205881003817100381710038171003817Finfin

1003817100381710038171003817

+

10038171003817100381710038171003817100381710038171003817int

119905

0

119890minusA(119905minus119904)

F (119904 119906 (119904)) 119889119904 minus int

119905

0

119890minusA(119905minus119904)

Finfin119889119904

10038171003817100381710038171003817100381710038171003817

le 119872119890minus120588119905(100381710038171003817100381711990601003817100381710038171003817 +

1003817100381710038171003817Finfin1003817100381710038171003817

120588)

+ int

119905

0

119872119890minus120588(119905minus119904) 1003817100381710038171003817F (119904 119906 (119904)) minusF

infin

1003817100381710038171003817 119889119904

le 119872119890minus120588119905(100381710038171003817100381711990601003817100381710038171003817 +

1003817100381710038171003817Finfin1003817100381710038171003817

120588)

+119872119890minus120588119905int

119905

0

119890120588119904 1003817100381710038171003817F (119904 119906

infin) minusF

infin

1003817100381710038171003817 119889119904

+119872119890minus120588119905int

119905

0

119890120588119904 1003817100381710038171003817F (119904 119906 (119904)) minusF (119904 119906

infin)1003817100381710038171003817 119889119904

(79)

Next putting

119888 (119905) = 119872119890minus120588(119905)

(100381710038171003817100381711990601003817100381710038171003817 +

1003817100381710038171003817Finfin1003817100381710038171003817

120588)

+119872119890minus120588119905int

119905

0

119890120588119904 1003817100381710038171003817F (119904 119906

infin) minusF

infin

1003817100381710038171003817 119889119904

(80)

and applying (11986710158405) we derive the following inequality

119892 (119905) le 119888 (119905) + 119898119872119890minus120588119905int

119905

0

119890120588119904119892 (119904) 119889119904 (81)

Hence

119890120588119905119892 (119905) le 119890

120588(119905)119888 (119905) + 119898119872int

119905

0

119890120588119904119892 (119904) 119889119904 (82)

The above inequality in conjunction with Lemma 5 gives

119890120588119905119892 (119905) le 119890

120588119905119888 (119905)

+ int

119905

0

119890120588119904119888 (119904)119898119872 sdot exp (119898119872(119905 minus 119904)) 119889119904

= 119890120588119905119888 (119905)

+ 119898119872119890119898119872119905

int

119905

0

119888 (119904) exp ((120588 minus 119898119872) 119904) 119889119904

(83)

8 Journal of Function Spaces and Applications

Hence

119892 (119905) le 119888 (119905) +119898119872int

119905

0119888 (119904) 119890(119892minus119898119872)119904

119889119904

exp ((120588 minus 119898119872) 119905) (84)

Before proving (76) we first show that

lim119905rarrinfin

119888 (119905) = 0 (85)

To prove this equality it is sufficient to show that the lastcomponent in the formula expressing 119888(119905) that is 119872119890minus120588119905

int119905

0119890120588119904F(119904 119906

infin) minusF

infin119889119904 tends to zero as 119905 rarr infin

To this end let us fix 120576 gt 0 Assumption (11986710158405) implies that

there exists 1199050such that

sup 1003817100381710038171003817F (119904 119906infin) minusF

infin

1003817100381710038171003817 119904 ge 1199050 le120576120588

2119872 (86)

This inequality together with (11986710158403) implies

ess sup 1003817100381710038171003817F (119904 119906infin) minusF

infin

1003817100381710038171003817 119904 isin R lt infin (87)

Now fix 1199051isin R such that 119905

1ge 119905 and

ess sup 1003817100381710038171003817F (119904 119906infin) minusF

infin

1003817100381710038171003817 119904 isin R

le120576120588 exp (120588119905

1)

2119872 (exp (1205881199050) minus 1)

(88)

Then using (86) and (88) we conclude that for 119905 ge 1199051we have

119872119890minus120588119905int

119905

0

119890120588119904 1003817100381710038171003817F (119904 119906

infin) minusF

infin

1003817100381710038171003817 119889119904

le 119872119890minus120588119905int

1199050

0

119890120588119904 1003817100381710038171003817F (119904 119906

infin) minusF

infin

1003817100381710038171003817 119889119904

+119872119890minus120588119905int

119905

1199050

119890120588119904 1003817100381710038171003817F (119904 119906

infin) minusF

infin

1003817100381710038171003817 119889119904 le120576

2+120576

2= 120576

(89)

This fact proves (85)Further using (1198671015840

5) and (85) and employing de

LrsquoHospitalrsquos rule for the fraction on the right-hand sideof inequality (84) we obtain that condition (76) is satisfiedThis fact completes the proof

6 An Application to the Ginzburg-LandauEquations of Superconductivity

In this section we formulate the gauged time-dependentGinzburg-Landau (TDGL) equations as an abstract evolutionequation in a Hilbert space Moreover we show applicationsof the above theorems to TDGL equations

We assume that Ω is a bounded domain in R119899 withboundary 120597Ω of class11986211That isΩ is an open and connectedset whose boundary 120597Ω is a compact (119899 minus 1)-manifolddescribed by Lipschitz continuous differentiable charts Weconsider two- and three-dimensional problems (119899 = 2 and119899 = 3 resp) Assume that the vector potential A takes its

values in R119899 The vector H will represent the (externally)applied magnetic field which is a function of space andtime similarly to A it takes its values in R119899 The function120574 is defined and satisfies Lipschitz condition on 120597Ω and120574(119909) ge 0 for 119909 isin 120597Ω The parameters in the TDGLequations are 120578 a (dimensionless) friction coefficient and 120581the (dimensionless) Ginzburg-Landau parameter

The order parameter should be thought of as the wavefunction of the center-of-massmotion of the ldquosuperelectronsrdquo(Cooper pairs) whose density is 119899

119904= |120595|

2 and whose fluxis J119904 The vector potential A determines the electromagnetic

field E = minus120597119905A minus nabla120601 is the electric field and B = nabla times A

is the magnetic induction where J the total current is thesum of a ldquonormalrdquo current J

119899= E the supercurrent J

119904 and

the transport current J119905= nabla times H The normal current obeys

Ohmrsquos law J119899= 120590119899E the ldquonormal conductivityrdquo coefficient 120590

119899

is equal to one in the adopted system of units The differenceM = BminusH is known as themagnetizationThe trivial solution(120595 = 0 B = H E = 0) represents the normal state where allsuperconducting properties have been lost

Nowwe accept the following notion all Banach spaces arereal the (real) dual of a Banach space119883 is denoted by1198831015840Thesymbol 119871119901(Ω) for 1 le 119901 le infin denotes the usual Lebesguespace with norm sdot

119871119901 (sdot sdot) is the inner product in 1198712(Ω)

1198821198982(Ω) for nonnegative integer 119898 is the usual Sobolev

space with norm sdot 1198821198982 1198821198982(Ω) is a Hilbert space for the

inner product (sdot sdot)1198982

given by (119906 V)1198982

= sum|120572|le119898

(120597120572119906 120597120572V)

for 119906 V isin 1198821198982(Ω) Fractional Sobolev space1198821199042(Ω) with afractional 119904 is defined by interpolation ([40 Chap VII] and[41 49 50]) 119862]

(Ω) for ] ge 0 ] = 119898 + 120582 with 0 le 120582 lt 1 isthe space of 119898 times continuous differentiable functions onΩ those 119898th-order derivatives satisfy the Holder conditionwith exponent 120582 if ] is a proper fraction the norm sdot

119862] is

defined in the usual wayThe definitions extend to the space of vector-valued

functions in the standard way with the caveat that the innerproduct in [1198712(Ω)]119899 is defined by (119906 V) = int

Ω119906 sdot V where the

symbol (119906 V) indicates the scalar product in R119899 Complex-valued functions are interpreted as vector-valued functionswith two real components

Functions that vary in space and time like the orderparameter and the vector potential are considered as map-pings from the time domain which is a subinterval of [0infin)into spaces of complex- or vector-valued functions definedin Ω Let 119883 = (119883 sdot

119883) be a Banach space of functions

defined inΩ Then the functions are defined in Ω Then thefunctions of space and time defined on Ω times (0 119879) for 119879 gt 0may be considered as elements of 119871119901(0 119879119883) for 1 le 119901 le infinor1198822119898(0 119879119883) for nonnegative119898 or119862]

(0 119879119883) for ] ge 0] = 119898 + 120582 with 0 le 120582 lt 1 Detailed definitions can be foundfor example in [43]

Obviously function spaces of ordered pairs (120595A) where120595 Ω rarr R2 and A Ω rarr R119899 (119899 = 2 3) play an importantrole in the study of the gauged TDGL equationsWe thereforeadapt the following special notationH = [119883(Ω)]

2times[119883(Ω)]

119899for any Banach space for the order parameter 120595 and thevector potential A respectively A suitable framework forthe functional analysis of the gauged TDGL equations is the

Journal of Function Spaces and Applications 9

Cartesian product W1+1205722 = [1198821+1205722

(Ω)]2times [1198821+1205722

(Ω)]119899

where 120572 isin (12 1) This space is continuously imbedded inW12 capLinfin

Assume H = 0 and H isin [1198712(Ω)]119899 Let AH be a minimizer

of the convex quadratic form 119869120596equiv J120596[119860]

119869120596 [A] = int

Ω

[120596(nabla sdot A)2 + |nabla times A minusH|2] 119889119909 (90)

on the domain

D (119869120596) = A isin [119882

12(Ω)]119899

n sdot A = 0 on 120597Ω (91)

We now introduce the reduced vector potential A1015840

A1015840 = A minus AH (92)

In terms of 120595 and A1015840 the gauged TDGL equations have thefollowing form

120597120595

120597119905minus

1

1205781205812Δ120595 = 120593 in Ω times (0infin)

120597A1015840

120597119905+ nabla times nabla times A1015840 minus 120596nabla (nabla sdot A1015840) = F in Ω times (0infin)

n sdot nabla + 120574120595 = 0 n sdot A1015840 = 0

n times (nabla sdot A1015840) = 0 on 120597Ω times (0infin)

(93)

Here 120593 and F are nonlinear functions of 120595 and A1015840

120593 equiv 120593 (119905 120595A1015840) = 1

120578[minus2119894

120581(nabla120595) sdot (A1015840 + AH)

minus119894

120581(1 minus 120578120581

2120596)120595 (nabla sdot A1015840)

minus12059510038161003816100381610038161003816A1015840 + AH

10038161003816100381610038161003816

2

+ (1 minus100381610038161003816100381612059510038161003816100381610038162

) 120595]

(94)

F equiv F (119905 120595A1015840) = 1

2119894119896(120595lowastnabla120595 minus 120595nabla120595

lowast)

minus100381610038161003816100381612059510038161003816100381610038162

(A1015840 + AH) minus120597AH120597119905

(95)

The equations are supplemented by initial data which is inthe followimg form

120595 = 1205950 A = A

0on Ω times 0 (96)

where 1205950and A

0are given and by (92) we have

120595 = 1205950 A1015840 = A

0minus AH (0) on Ω times 0 (97)

In the next part we connect the evolution of the solution(120595A1015840) of the system of (93) with the initial data (120595

0A0minus

AH(0)) with the dynamics of a vector 119906 in a Hilbert space

L2= [119871119901(Ω)]2

times [119871119901(Ω)]119899

(98)

The following analysis is restricted to the case 120596 gt 0 andthe case 120596 = 0 (see [73])

Let vector 119906 [0infin) rarr L2 represent the pair (120595A1015840)

119906 = (120595A1015840) equiv (120595A minus AH) (99)

and letA be the linear self-adjoint operator inL2 associatedwith the quadratic form 119876

120596equiv 119876120596[119906] given by the following

formula

119876120596 [119906] = int

Ω

[1

12057812058121003816100381610038161003816nabla120595

10038161003816100381610038162

+ 120596(nabla sdot A1015840)2

+10038161003816100381610038161003816nabla times A101584010038161003816100381610038161003816

2

] 119889119909

+ int120597Ω

120574

1205781205812100381610038161003816100381612059510038161003816100381610038162

119889120590 (119909)

(100)

on the domain

D (119876120596) = D (A

12)

= 119906 = (120595A1015840) isinW12 n sdot A1015840 = 0 on 120597Ω

(101)

The quadratic form 119876120596is nonnegative Furthermore since

120596 gt 0119876120596[120595A1015840]+119888 120595

1198712 is coercive onW12 for any constant

119888 gt 0 Hence A is positively definite in L2 ([44 Chap 1]equation (545)) If it does not lead to confusion we use thesame symbol A for the restriction A

120595and AA of A to the

respective linear subspace [1198712(Ω)]2 = [1198712(Ω)]2 times 0 (for 120595)and [1198712(Ω)]119899 equiv 0 times [1198712(Ω)]119899 (for A) ofL2

Now consider the initial-value problems (6) and (7) inL2 whereF(119905 119906) = (120593 F) 120593 and F is given by (94) and (95)and 119906

0= (1205950A0minus AH(0))

With 120582 isin (12 1) and 1199060isin W1+1205822 we say that 119906 are a

solution of (6) and (7) on the interval [0 119879] for some 119879 gt 0if 119906 [0 119879] rarr W1+1205822 is continuous and

119906 (119905) = 119890minusA(119905)

+ int

119905

0

119890minusA(119905minus119904)

F (119904 119906 (119904)) 119889119904 for 119905 isin [0 119879]

(102)

in L2 A mild solution of the initial-value problems (6) and(7) defines a weak solution (120595A1015840) of the boundary valueproblem (93) which in turn defines a weak solution (120595A)of the gauged TDGL equations provided AH is sufficientlyregular

Namely let us assume that X = L2 and 1199060isin L2 and

119906 R+times R119899 rarr R119899 for 119899 isin N is an unknown function 119906 =

119906(119905 119909) In order to applyTheorems 10 and 14 we are not goingto consider 119906 as a function of 119905 and 119909 together but rather asa mapping 119906 of variable 119905 into the spaceX =L2 of functions119909 that is 119906 R

+rarr L2 119906(119905)(119909) = 119906(119905 119909) 119909 isin R119899 119905 isin R

+

Remark 19 Since A is the linear self-adjoint operator inL2

associated with the quadratic form (100) thenA generates asemigroup 119890minusA119905

119905ge0(see [74])

Below we formulate the principal theorem of this paperThis theorem is a simple consequence of Theorem 10

10 Journal of Function Spaces and Applications

Theorem 20 Under assumptions (1198671)ndash(1198674) problem (6)-(7)

is equivalent to the system of (1)ndash(5) (TDGL equations) andhas for each 119906

0isin L2 at least one mild solution in sense of

Definition 4 119906 = 119906(119905) Moreover each solution 119906(119905) satisfiesestimate (25)

We observe that in the case when 119890minusA119905119905ge0

is the semi-group of contractions and 119886(119905) equiv 119886 119887(119905) equiv 119887 for 119905 isin

R+are constant after simple calculations based on estimate

(25) we get that the solution 119906 = 119906(119905) has the asymptoticcharacterization

119906 (119905) = 119874 (1199053) as 119905 997888rarr infin (103)

For further purposes let us formulate the followingassumption

(A) A is the infinitesimal generator of an exponentiallystable 119862

0-semigroup 119890minusA119905

119905ge0

The next result of this chapter is shown in the giventheorem below

Theorem21 Under assumptions (A) and (1198672)ndash(1198674) problem

(6)-(7) which is equivalent to the systems equations (1)ndash(5)(TDGL equations) has a mild solution 119906 = 119906(119905) for each1199060isin L2 which is globally attractive and locally uniformly

attractive

References

[1] V L Ginzburg and L D Landau ldquoOn the theory of supercon-ductivityrdquo Zhurnal Eksperimentalrsquonoi i Teoreticheskoi Fiziki vol20 pp 1064ndash1082 1950 English translation in L D LandauMen of Physics D ter Haar ed Vol I Pergamon Press OxfordUK pp 138ndash167 1965

[2] S Aizicovici and H Lee ldquoNonlinear nonlocal Cauchy problemsin Banach spacesrdquo Applied Mathematics Letters vol 18 no 4pp 401ndash407 2005

[3] S Aizicovici and M McKibben ldquoExistence results for a classof abstract nonlocal Cauchy problemsrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 39 no 5 pp 649ndash6682000

[4] MMcKibbenDiscovering Evolution Equation with ApplicationCRC Press Taylor amp Francis Group 2011

[5] S Aizicovici and V Staicu ldquoMultivalued evolution equationswith nonlocal initial conditions in Banach spacesrdquo NonlinearDifferential Equations and Applications vol 14 no 3-4 pp 361ndash376 2007

[6] R P Agarwal and D OrsquoRegan Infinite Interval Problems for Dif-ferential Difference and Integral Equations Kluwer AcademicPublishers Dordrecht The Netherlands 2001

[7] Z Fan ldquoExistence of nondensely defined evolution equationswith nonlocal conditionsrdquo Nonlinear Analysis Theory Methodsamp Applications A vol 70 no 11 pp 3829ndash3836 2009

[8] Z Fan ldquoImpulsive problems for semilinear differential equa-tions with nonlocal conditionsrdquo Nonlinear Analysis TheoryMethods amp Applications A vol 72 no 2 pp 1104ndash1109 2010

[9] Z Fan and G Li ldquoExistence results for semilinear differentialequations with nonlocal and impulsive conditionsrdquo Journal ofFunctional Analysis vol 258 no 5 pp 1709ndash1727 2010

[10] J Garcıa-Falset ldquoExistence results and asymptotic behavior fornonlocal abstract Cauchy problemsrdquo Journal of MathematicalAnalysis and Applications vol 338 no 1 pp 639ndash652 2008

[11] J Liang J H Liu and T-J Xiao ldquoNonlocal impulsive problemsfor nonlinear differential equations in Banach spacesrdquo Mathe-matical and Computer Modelling vol 49 no 3-4 pp 798ndash8042009

[12] L Zhu and G Li ldquoOn a nonlocal problem for semilinear differ-ential equations with upper semicontinuous nonlinearities ingeneral Banach spacesrdquo Journal of Mathematical Analysis andApplications vol 341 no 1 pp 660ndash675 2008

[13] Q Liu and R Yuan ldquoExistence of mild solutions for semilinearevolution equations with non-local initial conditionsrdquo Nonlin-ear AnalysisTheoryMethodsampApplications A vol 71 no 9 pp4177ndash4184 2009

[14] C Xie W Zhong and Z Fan ldquoExistence results for nonlinearnonlocal problems in Banach spacesrdquo Applied MathematicsLetters vol 22 no 7 pp 998ndash1002 2009

[15] T-J Xiao and J Liang ldquoExistence of classical solutions tononautonomous nonlocal parabolic problemsrdquoNonlinear Anal-ysisTheoryMethods and Applications vol 63 no 5-7 pp e225ndashe232 2005

[16] X Xue ldquo119871119901 theory for semilinear nonlocal problems with mea-sure of noncompactness in separable Banach spacesrdquo Journal ofFixed Point Theory and Applications vol 5 no 1 pp 129ndash1442009

[17] X Xue ldquoNonlinear differential equations with nonlocal condi-tions in Banach spacesrdquo Nonlinear Analysis Theory Methods ampApplications A vol 63 no 4 pp 575ndash586 2005

[18] X Xue ldquoExistence of solutions for semilinear nonlocal Cauchyproblems in Banach spacesrdquo Electronic Journal of DifferentialEquations vol 64 p 7 2005

[19] X Xue ldquoNonlocal nonlinear differential equations with a mea-sure of noncompactness in Banach spacesrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 70 no 7 pp 2593ndash26012009

[20] L Olszowy ldquoExistence ofmild solutions for the semilinear non-local problem in Banach spacesrdquo Nonlinear Analysis TheoryMethods amp Applications A vol 81 pp 211ndash223 2013

[21] D Bothe ldquoMultivalued perturbations of119898-accretive differentialinclusionsrdquo Israel Journal of Mathematics vol 108 pp 109ndash1381998

[22] H-S Ding J Liang G M NrsquoGuerekata and T-J XiaoldquoMild pseudo-almost periodic solutions of nonautonomoussemilinear evolution equationsrdquo Mathematical and ComputerModelling vol 45 no 5-6 pp 579ndash584 2007

[23] J Garcıa-Falset ldquoThe asymptotic behavior of the solutions of theCauchy problem generated by 120593-accretive operatorsrdquo Journal ofMathematical Analysis and Applications vol 310 no 2 pp 594ndash608 2005

[24] Z Fan and G Li ldquoAsymptotic behavior of the solutions of non-autonomous systems in Banach spacesrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 68 no 12 pp 3733ndash37412008

[25] J Banas ldquoOn existence theorems for differential equations inBanach spacesrdquo Bulletin of the Australian Mathematical Societyvol 32 no 1 pp 73ndash82 1985

[26] J Banas ldquoMeasures of noncompactness in the space of contin-uous tempered functionsrdquo Demonstratio Mathematica vol 14no 1 pp 127ndash133 1981

Journal of Function Spaces and Applications 11

[27] J Banas and K GoebelMeasures of Noncompactness in BanachSpaces vol 60 ofLectureNotes in Pure andAppliedMathematicsMarcel Dekker New York NY USA 1980

[28] J Banas and M Lecko ldquoSolvability of infinite systems ofdifferential equations in Banach sequence spacesrdquo Journal ofComputational andAppliedMathematics vol 137 no 2 pp 363ndash375 2001

[29] J Banas and K Sadarangani ldquoCompactness conditions inthe study of functional differential and integral equationsrdquoAbstract and Applied Analysis vol 2013 Article ID 819315 14pages 2013

[30] M Mursaleen and S A Mohiuddine ldquoApplications of mea-sures of noncompactness to the infinite system of differentialequations in ℓ

119901spacesrdquo Nonlinear Analysis Theory Methods amp

Applications A vol 75 no 4 pp 2111ndash2115 2012[31] L Olszowy ldquoSolvability of some functional integral equationrdquo

Dynamic Systems and Applications vol 18 no 3-4 pp 667ndash6762009

[32] L Olszowy ldquoOn some measures of noncompactness in theFrechet spaces of continuous functionsrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 72 pp 2119ndash2126 2010

[33] L Olszowy ldquoOn existence of solutions of a quadratic Urysohnintegral equation on an unbounded intervalrdquo CommentationesMathematicae vol 48 no 1 pp 103ndash112 2008

[34] L Olszowy ldquoExistence ofmild solutions for semilinear nonlocalCauchy problems in separable Banach spacesrdquo Journal of Anal-ysis and its Applications vol 32 no 2 pp 215ndash232 2013

[35] L Olszowy ldquoFixed point theorems in the Frechet space 119862(119877+)

and functional integral equations on an unbounded intervalrdquoApplied Mathematics and Computation vol 218 no 18 pp9066ndash9074 2012

[36] J Banas and D OrsquoRegan ldquoOn existence and local attractivity ofsolutions of a quadratic Volterra integral equation of fractionalorderrdquo Journal of Mathematical Analysis and Applications vol345 no 1 pp 573ndash582 2008

[37] L D Landau EM Lifshitz and L P Pitaevski Electrodynamicsof Continuous Media vol 8 Butterworth-Heinemann 2ndedition 1984

[38] L D Landau and E M Lifshitz Fluid Mechanics vol bButterworth-Heinemann 2nd edition 1987

[39] R Temam Infinite-Dimensional Dynamical Systems inMechan-ics and Physics vol 68 Springer New York NY USA 2ndedition 1997

[40] J K Hale Asymptotic Behavior of Dissipative Systems vol 25 ofMathematical Surveys and Monographs American Mathemati-cal Society Providence RI USA 1988

[41] R A Adams and J J F Fournier Sobolev Spaces vol 140Elsevier 2nd edition 2003

[42] V Mazrsquoya Sobolev Spaces with Applications to Elliptic PartialDifferential Equations Springer Heidelberg Germany 2ndedition 2011

[43] V Georgescu ldquoSome boundary value problems for differentialforms on compact Riemannian manifoldsrdquo Annali di Matemat-ica Pura ed Applicata vol 122 pp 159ndash198 1979

[44] D Henry Geometric Theory of Semilinear Parabolic Equationsvol 840 of Lecture Notes in Mathematics Springer New YorkNY USA 1981

[45] V Girault and P-A Raviart Finite Element Methods for Navier-Stokes Equations Springer New York NY USA 1986

[46] J Bardeen L N Cooper and J R Schrieffer ldquoTheory ofsuperconductivityrdquo Physical Review vol 108 pp 1175ndash12041957

[47] A A Abrikosov Fundamental of the Theory of Metals North-Holland Amstedram The Netherlands 1988

[48] M Tinkham Introduction To Superconductivity McGraw-HillNew York NY USA 2nd edition 1996

[49] P DeGennes Superconductivity inMetals and Alloys BenjaminNew York NY USA 1966

[50] H Brezis Functional Analysis Sobolev Spaces and PartialDifferential Equations Springer New York NY USA 2011

[51] E Zeidler Nonlinear Functional Analysis and its Applicationsvol 2 Springer New York NY USA 1990

[52] A Schmid ldquoA time dependent Ginzburg-Landau equation andits application to the problem of resistivity in the mixed staterdquoPhysik der KondensiertenMaterie vol 5 no 4 pp 302ndash317 1966

[53] C M Elliot and Q Tang Existence Theorems For an Evolution-ary Superconductivity Model University of Sussex BrightonUK 1992

[54] Q Tang ldquoOn an evolutionary system of Ginzburg-Landauequations with fixed total magnetic fluxrdquo Communications inPartial Differential Equations vol 20 no 1-2 pp 1ndash36 1995

[55] Q Du ldquoGlobal existence and uniqueness of solutions of thetime-dependent Ginzburg-Landau model for superconductiv-ityrdquo Applicable Analysis vol 53 no 1-2 pp 1ndash18 1994

[56] Z M Chen K-H Hoffmann and J Liang ldquoOn a nonstation-ary Ginzburg-Landau superconductivity modelrdquoMathematicalMethods in the Applied Sciences vol 16 no 12 pp 855ndash875 1993

[57] J Liang and Q Tang ldquoAsymptotic behaviour of the solutionsof an evolutionaryGinzburg-Landau superconductivitymodelrdquoJournal of Mathematical Analysis and Applications vol 195 no1 pp 92ndash107 1995

[58] L P Gorrsquokov and G M Eliasherberg ldquoGeneralization of theGinzburg-Landau equations for an nonstationary problemsin the case of alloys with paramagnetic impartiesrdquo ZhurnalEksperimentalrsquonoi i Teoreticheskoi Fiziki vol 54 pp 612ndash6261968 Soviet Physics vol 27 pp 328ndash334 1968

[59] J Fleckinger-Pelle and H G Kaper ldquoGauges for the Ginzburg-Landau equations of superconductivityrdquo Zeitschrift fur Ange-wandte Mathematik und Mechanik vol 76 no 2 pp 345ndash3481996

[60] P Takac ldquoOn the dynamical process generated by supercon-ductivity modelrdquo Zeitschrift fur Angewandte Mathematik undMechanik vol 76 no 52 pp 349ndash352 1995

[61] W D Gropp H G Kaper DM Levine M Palumbo and VMVinokur ldquoNumerical simulation of vortex dynamics in type-IIsuperconductorsrdquo Journal of Computational Physics vol 123 no2 pp 254ndash266 1996

[62] D W Braun G W Crabtree H G Kaper et al ldquoStructure of amoving vortex latticerdquo Physical Review Letters vol 76 no 5 pp831ndash834 1996

[63] G W Crabtree G K Leaf H G Kaper et al ldquoTime-dependent Ginzburg-Landau simulations of vortex guidance bytwin boundariesrdquo Physica C vol 263 no 1ndash4 pp 401ndash408 1996

[64] G W Carabtree G K Leaf H G Kaper D W Braun V MVinokur andA E Koshelev ldquoDynamics vortex phases in super-conductors with correlated disorderrdquo Preprint ANLMCS-P590-0496 Mathematics and Computer Science DivisionAragonne National Laboratory 1996

[65] Th Gallay ldquoA center-stable manifold theorem for differentialequations in Banach spacesrdquo Communications in MathematicalPhysics vol 152 no 2 pp 249ndash268 1993

[66] J-M Ghidaglia and B Heron ldquoDimension of the attractorsassociated to the Ginzburg-Landau partial differential equa-tionrdquo Physica D vol 28 no 3 pp 282ndash304 1987

12 Journal of Function Spaces and Applications

[67] P Mattila Geometry of Sets and Measures in Euclidean Spacesvol 44 Cambridge University Press Cambridge UK 1995

[68] L HormanderThe Analysis of Linear Partial Differential Oper-ators vol 1 Springer Berlin Germany 2nd edition 1990

[69] H Amann Linear and Quasilinear Parabolic ProblemBikhauser Basel Switzerland 1995

[70] H-P Heinz ldquoOn the behaviour ofmeasures of noncompactnesswith respect to differentiation and integration of vector-valuedfunctionsrdquo Nonlinear Analysis Theory Methods amp ApplicationsA vol 7 no 12 pp 1351ndash1371 1983

[71] R P Agarwal M Meehan and D OrsquoRegan Fixed Point Theoryand Applications vol 141 of Cambridge Tracts in MathematicsCambridge University Press Cambridge UK 2001

[72] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer New York NY USA1983

[73] J Fleckinger-Pelle H G Kaper and P Takac ldquoDynamics of theGinzburg-Landau equations of superconductivityrdquo NonlinearAnalysis Theory Methods amp Applications A vol 32 no 5 pp647ndash665 1998

[74] H G Kaper and P Takac ldquoAn equivalence relation for theGinzburg-Landau equations of superconductivityrdquo Journal ofApplied Mathematics and Physics vol 48 no 4 pp 665ndash6751997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces and Applications 7

other solution V(119905) with initial condition V(0) = V0 obviously

described by similar formula as 119906(119905) we would have acontradiction

|119906 (119905) minus V (119905)| =

10038161003816100381610038161199060 minus V0

1003816100381610038161003816 119890(119898minus120588)119905

997888rarr infin

or10038161003816100381610038161199060 minus V

0

1003816100381610038161003816

(72)

5 Asymptotic Behaviour

In this section we will give a theorem describing asymptoticbehaviour of mild solutions of (6) with condition (7) Thistheorem generalizes the result included in [72Theorem 44]First we formulate the assumptions

(1198671015840

3) This condition is almost identical with (119867

3) and the

only difference is that we assume the functions 119886 and119887 are locally essentially bounded on R

+

(1198671015840

5) There exists 119906

infinisin X such that there exists the limit

lim119905rarrinfin

F(119905 119906infin) and

lim119905rarrinfin

F (119905 119906infin) = A

minus1119906infin (73)

Moreover there exists a number119898 lt 120588119872 such that1003817100381710038171003817F (119905 119906) minusF (119905 119906

infin)1003817100381710038171003817 le 119898

1003817100381710038171003817119906 minus 119906infin1003817100381710038171003817 (74)

for 119905 ge 0 and 119906 isin X

Remark 17 The condition (11986710158405) in conjunction with (1198671015840

1)

ensures the existence ofAminus1 (see [72]) Clearly (11986710158403) implies

(1198673)

Theorem 18 Under assumptions (11986710158401) (1198672) (11986710158403) (1198674) and

(11986710158405) (6) with condition (7) has a mild solution 119906 = 119906(119905) for

each 1199060isin X such that lim

119905rarrinfin119906(119905) = 119906

infin

Proof The existence of a mild solution 119906(119905) is guaranteed byTheorem 10 Let us put

119892 (119905) =1003817100381710038171003817119906 (119905) minus 119906infin

1003817100381710038171003817

Finfin= lim119905rarrinfin

F (119905 119906infin)

(75)

We show that

lim119905rarrinfin

119892 (119905) = 0 (76)

Recall that if assumption (11986710158401) is fulfilled then for each 119911 isin X

we have (see [72])

int

infin

0

119890minusA119904

119911 119889119904 = minusAminus1119911 (77)

Using the above fact and (11986710158405) we get

119906infin= (minusA

minus1(minusAminus1119906infin)) = minusA

minus1Finfin

= int

infin

0

119890minusA119904

Finfin119889119904

(78)

Linking the above equality with (11986710158401) we obtain

119892 (119905)

=

10038171003817100381710038171003817100381710038171003817119890minusA1199051199060+ int

119905

0

119890minusA(119905minus119904)

F (119904 119906 (119904)) 119889119904 minus int

infin

0

119890minusA119904

Finfin119889119904

10038171003817100381710038171003817100381710038171003817

le 119872119890minus120588119905 10038171003817100381710038171199060

1003817100381710038171003817 +

10038171003817100381710038171003817100381710038171003817int

infin

119905

119890minusA119904

Finfin119889119904

10038171003817100381710038171003817100381710038171003817

+

10038171003817100381710038171003817100381710038171003817int

119905

0

119890minusA(119905minus119904)

F (119904 119906 (119904)) 119889119904 minus int

119905

0

119890minusA119904

Finfin119889119904

10038171003817100381710038171003817100381710038171003817

le 119872119890minus120588119905 10038171003817100381710038171199060

1003817100381710038171003817 +119872

119890minus1205881199051205881003817100381710038171003817Finfin

1003817100381710038171003817

+

10038171003817100381710038171003817100381710038171003817int

119905

0

119890minusA(119905minus119904)

F (119904 119906 (119904)) 119889119904 minus int

119905

0

119890minusA(119905minus119904)

Finfin119889119904

10038171003817100381710038171003817100381710038171003817

le 119872119890minus120588119905(100381710038171003817100381711990601003817100381710038171003817 +

1003817100381710038171003817Finfin1003817100381710038171003817

120588)

+ int

119905

0

119872119890minus120588(119905minus119904) 1003817100381710038171003817F (119904 119906 (119904)) minusF

infin

1003817100381710038171003817 119889119904

le 119872119890minus120588119905(100381710038171003817100381711990601003817100381710038171003817 +

1003817100381710038171003817Finfin1003817100381710038171003817

120588)

+119872119890minus120588119905int

119905

0

119890120588119904 1003817100381710038171003817F (119904 119906

infin) minusF

infin

1003817100381710038171003817 119889119904

+119872119890minus120588119905int

119905

0

119890120588119904 1003817100381710038171003817F (119904 119906 (119904)) minusF (119904 119906

infin)1003817100381710038171003817 119889119904

(79)

Next putting

119888 (119905) = 119872119890minus120588(119905)

(100381710038171003817100381711990601003817100381710038171003817 +

1003817100381710038171003817Finfin1003817100381710038171003817

120588)

+119872119890minus120588119905int

119905

0

119890120588119904 1003817100381710038171003817F (119904 119906

infin) minusF

infin

1003817100381710038171003817 119889119904

(80)

and applying (11986710158405) we derive the following inequality

119892 (119905) le 119888 (119905) + 119898119872119890minus120588119905int

119905

0

119890120588119904119892 (119904) 119889119904 (81)

Hence

119890120588119905119892 (119905) le 119890

120588(119905)119888 (119905) + 119898119872int

119905

0

119890120588119904119892 (119904) 119889119904 (82)

The above inequality in conjunction with Lemma 5 gives

119890120588119905119892 (119905) le 119890

120588119905119888 (119905)

+ int

119905

0

119890120588119904119888 (119904)119898119872 sdot exp (119898119872(119905 minus 119904)) 119889119904

= 119890120588119905119888 (119905)

+ 119898119872119890119898119872119905

int

119905

0

119888 (119904) exp ((120588 minus 119898119872) 119904) 119889119904

(83)

8 Journal of Function Spaces and Applications

Hence

119892 (119905) le 119888 (119905) +119898119872int

119905

0119888 (119904) 119890(119892minus119898119872)119904

119889119904

exp ((120588 minus 119898119872) 119905) (84)

Before proving (76) we first show that

lim119905rarrinfin

119888 (119905) = 0 (85)

To prove this equality it is sufficient to show that the lastcomponent in the formula expressing 119888(119905) that is 119872119890minus120588119905

int119905

0119890120588119904F(119904 119906

infin) minusF

infin119889119904 tends to zero as 119905 rarr infin

To this end let us fix 120576 gt 0 Assumption (11986710158405) implies that

there exists 1199050such that

sup 1003817100381710038171003817F (119904 119906infin) minusF

infin

1003817100381710038171003817 119904 ge 1199050 le120576120588

2119872 (86)

This inequality together with (11986710158403) implies

ess sup 1003817100381710038171003817F (119904 119906infin) minusF

infin

1003817100381710038171003817 119904 isin R lt infin (87)

Now fix 1199051isin R such that 119905

1ge 119905 and

ess sup 1003817100381710038171003817F (119904 119906infin) minusF

infin

1003817100381710038171003817 119904 isin R

le120576120588 exp (120588119905

1)

2119872 (exp (1205881199050) minus 1)

(88)

Then using (86) and (88) we conclude that for 119905 ge 1199051we have

119872119890minus120588119905int

119905

0

119890120588119904 1003817100381710038171003817F (119904 119906

infin) minusF

infin

1003817100381710038171003817 119889119904

le 119872119890minus120588119905int

1199050

0

119890120588119904 1003817100381710038171003817F (119904 119906

infin) minusF

infin

1003817100381710038171003817 119889119904

+119872119890minus120588119905int

119905

1199050

119890120588119904 1003817100381710038171003817F (119904 119906

infin) minusF

infin

1003817100381710038171003817 119889119904 le120576

2+120576

2= 120576

(89)

This fact proves (85)Further using (1198671015840

5) and (85) and employing de

LrsquoHospitalrsquos rule for the fraction on the right-hand sideof inequality (84) we obtain that condition (76) is satisfiedThis fact completes the proof

6 An Application to the Ginzburg-LandauEquations of Superconductivity

In this section we formulate the gauged time-dependentGinzburg-Landau (TDGL) equations as an abstract evolutionequation in a Hilbert space Moreover we show applicationsof the above theorems to TDGL equations

We assume that Ω is a bounded domain in R119899 withboundary 120597Ω of class11986211That isΩ is an open and connectedset whose boundary 120597Ω is a compact (119899 minus 1)-manifolddescribed by Lipschitz continuous differentiable charts Weconsider two- and three-dimensional problems (119899 = 2 and119899 = 3 resp) Assume that the vector potential A takes its

values in R119899 The vector H will represent the (externally)applied magnetic field which is a function of space andtime similarly to A it takes its values in R119899 The function120574 is defined and satisfies Lipschitz condition on 120597Ω and120574(119909) ge 0 for 119909 isin 120597Ω The parameters in the TDGLequations are 120578 a (dimensionless) friction coefficient and 120581the (dimensionless) Ginzburg-Landau parameter

The order parameter should be thought of as the wavefunction of the center-of-massmotion of the ldquosuperelectronsrdquo(Cooper pairs) whose density is 119899

119904= |120595|

2 and whose fluxis J119904 The vector potential A determines the electromagnetic

field E = minus120597119905A minus nabla120601 is the electric field and B = nabla times A

is the magnetic induction where J the total current is thesum of a ldquonormalrdquo current J

119899= E the supercurrent J

119904 and

the transport current J119905= nabla times H The normal current obeys

Ohmrsquos law J119899= 120590119899E the ldquonormal conductivityrdquo coefficient 120590

119899

is equal to one in the adopted system of units The differenceM = BminusH is known as themagnetizationThe trivial solution(120595 = 0 B = H E = 0) represents the normal state where allsuperconducting properties have been lost

Nowwe accept the following notion all Banach spaces arereal the (real) dual of a Banach space119883 is denoted by1198831015840Thesymbol 119871119901(Ω) for 1 le 119901 le infin denotes the usual Lebesguespace with norm sdot

119871119901 (sdot sdot) is the inner product in 1198712(Ω)

1198821198982(Ω) for nonnegative integer 119898 is the usual Sobolev

space with norm sdot 1198821198982 1198821198982(Ω) is a Hilbert space for the

inner product (sdot sdot)1198982

given by (119906 V)1198982

= sum|120572|le119898

(120597120572119906 120597120572V)

for 119906 V isin 1198821198982(Ω) Fractional Sobolev space1198821199042(Ω) with afractional 119904 is defined by interpolation ([40 Chap VII] and[41 49 50]) 119862]

(Ω) for ] ge 0 ] = 119898 + 120582 with 0 le 120582 lt 1 isthe space of 119898 times continuous differentiable functions onΩ those 119898th-order derivatives satisfy the Holder conditionwith exponent 120582 if ] is a proper fraction the norm sdot

119862] is

defined in the usual wayThe definitions extend to the space of vector-valued

functions in the standard way with the caveat that the innerproduct in [1198712(Ω)]119899 is defined by (119906 V) = int

Ω119906 sdot V where the

symbol (119906 V) indicates the scalar product in R119899 Complex-valued functions are interpreted as vector-valued functionswith two real components

Functions that vary in space and time like the orderparameter and the vector potential are considered as map-pings from the time domain which is a subinterval of [0infin)into spaces of complex- or vector-valued functions definedin Ω Let 119883 = (119883 sdot

119883) be a Banach space of functions

defined inΩ Then the functions are defined in Ω Then thefunctions of space and time defined on Ω times (0 119879) for 119879 gt 0may be considered as elements of 119871119901(0 119879119883) for 1 le 119901 le infinor1198822119898(0 119879119883) for nonnegative119898 or119862]

(0 119879119883) for ] ge 0] = 119898 + 120582 with 0 le 120582 lt 1 Detailed definitions can be foundfor example in [43]

Obviously function spaces of ordered pairs (120595A) where120595 Ω rarr R2 and A Ω rarr R119899 (119899 = 2 3) play an importantrole in the study of the gauged TDGL equationsWe thereforeadapt the following special notationH = [119883(Ω)]

2times[119883(Ω)]

119899for any Banach space for the order parameter 120595 and thevector potential A respectively A suitable framework forthe functional analysis of the gauged TDGL equations is the

Journal of Function Spaces and Applications 9

Cartesian product W1+1205722 = [1198821+1205722

(Ω)]2times [1198821+1205722

(Ω)]119899

where 120572 isin (12 1) This space is continuously imbedded inW12 capLinfin

Assume H = 0 and H isin [1198712(Ω)]119899 Let AH be a minimizer

of the convex quadratic form 119869120596equiv J120596[119860]

119869120596 [A] = int

Ω

[120596(nabla sdot A)2 + |nabla times A minusH|2] 119889119909 (90)

on the domain

D (119869120596) = A isin [119882

12(Ω)]119899

n sdot A = 0 on 120597Ω (91)

We now introduce the reduced vector potential A1015840

A1015840 = A minus AH (92)

In terms of 120595 and A1015840 the gauged TDGL equations have thefollowing form

120597120595

120597119905minus

1

1205781205812Δ120595 = 120593 in Ω times (0infin)

120597A1015840

120597119905+ nabla times nabla times A1015840 minus 120596nabla (nabla sdot A1015840) = F in Ω times (0infin)

n sdot nabla + 120574120595 = 0 n sdot A1015840 = 0

n times (nabla sdot A1015840) = 0 on 120597Ω times (0infin)

(93)

Here 120593 and F are nonlinear functions of 120595 and A1015840

120593 equiv 120593 (119905 120595A1015840) = 1

120578[minus2119894

120581(nabla120595) sdot (A1015840 + AH)

minus119894

120581(1 minus 120578120581

2120596)120595 (nabla sdot A1015840)

minus12059510038161003816100381610038161003816A1015840 + AH

10038161003816100381610038161003816

2

+ (1 minus100381610038161003816100381612059510038161003816100381610038162

) 120595]

(94)

F equiv F (119905 120595A1015840) = 1

2119894119896(120595lowastnabla120595 minus 120595nabla120595

lowast)

minus100381610038161003816100381612059510038161003816100381610038162

(A1015840 + AH) minus120597AH120597119905

(95)

The equations are supplemented by initial data which is inthe followimg form

120595 = 1205950 A = A

0on Ω times 0 (96)

where 1205950and A

0are given and by (92) we have

120595 = 1205950 A1015840 = A

0minus AH (0) on Ω times 0 (97)

In the next part we connect the evolution of the solution(120595A1015840) of the system of (93) with the initial data (120595

0A0minus

AH(0)) with the dynamics of a vector 119906 in a Hilbert space

L2= [119871119901(Ω)]2

times [119871119901(Ω)]119899

(98)

The following analysis is restricted to the case 120596 gt 0 andthe case 120596 = 0 (see [73])

Let vector 119906 [0infin) rarr L2 represent the pair (120595A1015840)

119906 = (120595A1015840) equiv (120595A minus AH) (99)

and letA be the linear self-adjoint operator inL2 associatedwith the quadratic form 119876

120596equiv 119876120596[119906] given by the following

formula

119876120596 [119906] = int

Ω

[1

12057812058121003816100381610038161003816nabla120595

10038161003816100381610038162

+ 120596(nabla sdot A1015840)2

+10038161003816100381610038161003816nabla times A101584010038161003816100381610038161003816

2

] 119889119909

+ int120597Ω

120574

1205781205812100381610038161003816100381612059510038161003816100381610038162

119889120590 (119909)

(100)

on the domain

D (119876120596) = D (A

12)

= 119906 = (120595A1015840) isinW12 n sdot A1015840 = 0 on 120597Ω

(101)

The quadratic form 119876120596is nonnegative Furthermore since

120596 gt 0119876120596[120595A1015840]+119888 120595

1198712 is coercive onW12 for any constant

119888 gt 0 Hence A is positively definite in L2 ([44 Chap 1]equation (545)) If it does not lead to confusion we use thesame symbol A for the restriction A

120595and AA of A to the

respective linear subspace [1198712(Ω)]2 = [1198712(Ω)]2 times 0 (for 120595)and [1198712(Ω)]119899 equiv 0 times [1198712(Ω)]119899 (for A) ofL2

Now consider the initial-value problems (6) and (7) inL2 whereF(119905 119906) = (120593 F) 120593 and F is given by (94) and (95)and 119906

0= (1205950A0minus AH(0))

With 120582 isin (12 1) and 1199060isin W1+1205822 we say that 119906 are a

solution of (6) and (7) on the interval [0 119879] for some 119879 gt 0if 119906 [0 119879] rarr W1+1205822 is continuous and

119906 (119905) = 119890minusA(119905)

+ int

119905

0

119890minusA(119905minus119904)

F (119904 119906 (119904)) 119889119904 for 119905 isin [0 119879]

(102)

in L2 A mild solution of the initial-value problems (6) and(7) defines a weak solution (120595A1015840) of the boundary valueproblem (93) which in turn defines a weak solution (120595A)of the gauged TDGL equations provided AH is sufficientlyregular

Namely let us assume that X = L2 and 1199060isin L2 and

119906 R+times R119899 rarr R119899 for 119899 isin N is an unknown function 119906 =

119906(119905 119909) In order to applyTheorems 10 and 14 we are not goingto consider 119906 as a function of 119905 and 119909 together but rather asa mapping 119906 of variable 119905 into the spaceX =L2 of functions119909 that is 119906 R

+rarr L2 119906(119905)(119909) = 119906(119905 119909) 119909 isin R119899 119905 isin R

+

Remark 19 Since A is the linear self-adjoint operator inL2

associated with the quadratic form (100) thenA generates asemigroup 119890minusA119905

119905ge0(see [74])

Below we formulate the principal theorem of this paperThis theorem is a simple consequence of Theorem 10

10 Journal of Function Spaces and Applications

Theorem 20 Under assumptions (1198671)ndash(1198674) problem (6)-(7)

is equivalent to the system of (1)ndash(5) (TDGL equations) andhas for each 119906

0isin L2 at least one mild solution in sense of

Definition 4 119906 = 119906(119905) Moreover each solution 119906(119905) satisfiesestimate (25)

We observe that in the case when 119890minusA119905119905ge0

is the semi-group of contractions and 119886(119905) equiv 119886 119887(119905) equiv 119887 for 119905 isin

R+are constant after simple calculations based on estimate

(25) we get that the solution 119906 = 119906(119905) has the asymptoticcharacterization

119906 (119905) = 119874 (1199053) as 119905 997888rarr infin (103)

For further purposes let us formulate the followingassumption

(A) A is the infinitesimal generator of an exponentiallystable 119862

0-semigroup 119890minusA119905

119905ge0

The next result of this chapter is shown in the giventheorem below

Theorem21 Under assumptions (A) and (1198672)ndash(1198674) problem

(6)-(7) which is equivalent to the systems equations (1)ndash(5)(TDGL equations) has a mild solution 119906 = 119906(119905) for each1199060isin L2 which is globally attractive and locally uniformly

attractive

References

[1] V L Ginzburg and L D Landau ldquoOn the theory of supercon-ductivityrdquo Zhurnal Eksperimentalrsquonoi i Teoreticheskoi Fiziki vol20 pp 1064ndash1082 1950 English translation in L D LandauMen of Physics D ter Haar ed Vol I Pergamon Press OxfordUK pp 138ndash167 1965

[2] S Aizicovici and H Lee ldquoNonlinear nonlocal Cauchy problemsin Banach spacesrdquo Applied Mathematics Letters vol 18 no 4pp 401ndash407 2005

[3] S Aizicovici and M McKibben ldquoExistence results for a classof abstract nonlocal Cauchy problemsrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 39 no 5 pp 649ndash6682000

[4] MMcKibbenDiscovering Evolution Equation with ApplicationCRC Press Taylor amp Francis Group 2011

[5] S Aizicovici and V Staicu ldquoMultivalued evolution equationswith nonlocal initial conditions in Banach spacesrdquo NonlinearDifferential Equations and Applications vol 14 no 3-4 pp 361ndash376 2007

[6] R P Agarwal and D OrsquoRegan Infinite Interval Problems for Dif-ferential Difference and Integral Equations Kluwer AcademicPublishers Dordrecht The Netherlands 2001

[7] Z Fan ldquoExistence of nondensely defined evolution equationswith nonlocal conditionsrdquo Nonlinear Analysis Theory Methodsamp Applications A vol 70 no 11 pp 3829ndash3836 2009

[8] Z Fan ldquoImpulsive problems for semilinear differential equa-tions with nonlocal conditionsrdquo Nonlinear Analysis TheoryMethods amp Applications A vol 72 no 2 pp 1104ndash1109 2010

[9] Z Fan and G Li ldquoExistence results for semilinear differentialequations with nonlocal and impulsive conditionsrdquo Journal ofFunctional Analysis vol 258 no 5 pp 1709ndash1727 2010

[10] J Garcıa-Falset ldquoExistence results and asymptotic behavior fornonlocal abstract Cauchy problemsrdquo Journal of MathematicalAnalysis and Applications vol 338 no 1 pp 639ndash652 2008

[11] J Liang J H Liu and T-J Xiao ldquoNonlocal impulsive problemsfor nonlinear differential equations in Banach spacesrdquo Mathe-matical and Computer Modelling vol 49 no 3-4 pp 798ndash8042009

[12] L Zhu and G Li ldquoOn a nonlocal problem for semilinear differ-ential equations with upper semicontinuous nonlinearities ingeneral Banach spacesrdquo Journal of Mathematical Analysis andApplications vol 341 no 1 pp 660ndash675 2008

[13] Q Liu and R Yuan ldquoExistence of mild solutions for semilinearevolution equations with non-local initial conditionsrdquo Nonlin-ear AnalysisTheoryMethodsampApplications A vol 71 no 9 pp4177ndash4184 2009

[14] C Xie W Zhong and Z Fan ldquoExistence results for nonlinearnonlocal problems in Banach spacesrdquo Applied MathematicsLetters vol 22 no 7 pp 998ndash1002 2009

[15] T-J Xiao and J Liang ldquoExistence of classical solutions tononautonomous nonlocal parabolic problemsrdquoNonlinear Anal-ysisTheoryMethods and Applications vol 63 no 5-7 pp e225ndashe232 2005

[16] X Xue ldquo119871119901 theory for semilinear nonlocal problems with mea-sure of noncompactness in separable Banach spacesrdquo Journal ofFixed Point Theory and Applications vol 5 no 1 pp 129ndash1442009

[17] X Xue ldquoNonlinear differential equations with nonlocal condi-tions in Banach spacesrdquo Nonlinear Analysis Theory Methods ampApplications A vol 63 no 4 pp 575ndash586 2005

[18] X Xue ldquoExistence of solutions for semilinear nonlocal Cauchyproblems in Banach spacesrdquo Electronic Journal of DifferentialEquations vol 64 p 7 2005

[19] X Xue ldquoNonlocal nonlinear differential equations with a mea-sure of noncompactness in Banach spacesrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 70 no 7 pp 2593ndash26012009

[20] L Olszowy ldquoExistence ofmild solutions for the semilinear non-local problem in Banach spacesrdquo Nonlinear Analysis TheoryMethods amp Applications A vol 81 pp 211ndash223 2013

[21] D Bothe ldquoMultivalued perturbations of119898-accretive differentialinclusionsrdquo Israel Journal of Mathematics vol 108 pp 109ndash1381998

[22] H-S Ding J Liang G M NrsquoGuerekata and T-J XiaoldquoMild pseudo-almost periodic solutions of nonautonomoussemilinear evolution equationsrdquo Mathematical and ComputerModelling vol 45 no 5-6 pp 579ndash584 2007

[23] J Garcıa-Falset ldquoThe asymptotic behavior of the solutions of theCauchy problem generated by 120593-accretive operatorsrdquo Journal ofMathematical Analysis and Applications vol 310 no 2 pp 594ndash608 2005

[24] Z Fan and G Li ldquoAsymptotic behavior of the solutions of non-autonomous systems in Banach spacesrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 68 no 12 pp 3733ndash37412008

[25] J Banas ldquoOn existence theorems for differential equations inBanach spacesrdquo Bulletin of the Australian Mathematical Societyvol 32 no 1 pp 73ndash82 1985

[26] J Banas ldquoMeasures of noncompactness in the space of contin-uous tempered functionsrdquo Demonstratio Mathematica vol 14no 1 pp 127ndash133 1981

Journal of Function Spaces and Applications 11

[27] J Banas and K GoebelMeasures of Noncompactness in BanachSpaces vol 60 ofLectureNotes in Pure andAppliedMathematicsMarcel Dekker New York NY USA 1980

[28] J Banas and M Lecko ldquoSolvability of infinite systems ofdifferential equations in Banach sequence spacesrdquo Journal ofComputational andAppliedMathematics vol 137 no 2 pp 363ndash375 2001

[29] J Banas and K Sadarangani ldquoCompactness conditions inthe study of functional differential and integral equationsrdquoAbstract and Applied Analysis vol 2013 Article ID 819315 14pages 2013

[30] M Mursaleen and S A Mohiuddine ldquoApplications of mea-sures of noncompactness to the infinite system of differentialequations in ℓ

119901spacesrdquo Nonlinear Analysis Theory Methods amp

Applications A vol 75 no 4 pp 2111ndash2115 2012[31] L Olszowy ldquoSolvability of some functional integral equationrdquo

Dynamic Systems and Applications vol 18 no 3-4 pp 667ndash6762009

[32] L Olszowy ldquoOn some measures of noncompactness in theFrechet spaces of continuous functionsrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 72 pp 2119ndash2126 2010

[33] L Olszowy ldquoOn existence of solutions of a quadratic Urysohnintegral equation on an unbounded intervalrdquo CommentationesMathematicae vol 48 no 1 pp 103ndash112 2008

[34] L Olszowy ldquoExistence ofmild solutions for semilinear nonlocalCauchy problems in separable Banach spacesrdquo Journal of Anal-ysis and its Applications vol 32 no 2 pp 215ndash232 2013

[35] L Olszowy ldquoFixed point theorems in the Frechet space 119862(119877+)

and functional integral equations on an unbounded intervalrdquoApplied Mathematics and Computation vol 218 no 18 pp9066ndash9074 2012

[36] J Banas and D OrsquoRegan ldquoOn existence and local attractivity ofsolutions of a quadratic Volterra integral equation of fractionalorderrdquo Journal of Mathematical Analysis and Applications vol345 no 1 pp 573ndash582 2008

[37] L D Landau EM Lifshitz and L P Pitaevski Electrodynamicsof Continuous Media vol 8 Butterworth-Heinemann 2ndedition 1984

[38] L D Landau and E M Lifshitz Fluid Mechanics vol bButterworth-Heinemann 2nd edition 1987

[39] R Temam Infinite-Dimensional Dynamical Systems inMechan-ics and Physics vol 68 Springer New York NY USA 2ndedition 1997

[40] J K Hale Asymptotic Behavior of Dissipative Systems vol 25 ofMathematical Surveys and Monographs American Mathemati-cal Society Providence RI USA 1988

[41] R A Adams and J J F Fournier Sobolev Spaces vol 140Elsevier 2nd edition 2003

[42] V Mazrsquoya Sobolev Spaces with Applications to Elliptic PartialDifferential Equations Springer Heidelberg Germany 2ndedition 2011

[43] V Georgescu ldquoSome boundary value problems for differentialforms on compact Riemannian manifoldsrdquo Annali di Matemat-ica Pura ed Applicata vol 122 pp 159ndash198 1979

[44] D Henry Geometric Theory of Semilinear Parabolic Equationsvol 840 of Lecture Notes in Mathematics Springer New YorkNY USA 1981

[45] V Girault and P-A Raviart Finite Element Methods for Navier-Stokes Equations Springer New York NY USA 1986

[46] J Bardeen L N Cooper and J R Schrieffer ldquoTheory ofsuperconductivityrdquo Physical Review vol 108 pp 1175ndash12041957

[47] A A Abrikosov Fundamental of the Theory of Metals North-Holland Amstedram The Netherlands 1988

[48] M Tinkham Introduction To Superconductivity McGraw-HillNew York NY USA 2nd edition 1996

[49] P DeGennes Superconductivity inMetals and Alloys BenjaminNew York NY USA 1966

[50] H Brezis Functional Analysis Sobolev Spaces and PartialDifferential Equations Springer New York NY USA 2011

[51] E Zeidler Nonlinear Functional Analysis and its Applicationsvol 2 Springer New York NY USA 1990

[52] A Schmid ldquoA time dependent Ginzburg-Landau equation andits application to the problem of resistivity in the mixed staterdquoPhysik der KondensiertenMaterie vol 5 no 4 pp 302ndash317 1966

[53] C M Elliot and Q Tang Existence Theorems For an Evolution-ary Superconductivity Model University of Sussex BrightonUK 1992

[54] Q Tang ldquoOn an evolutionary system of Ginzburg-Landauequations with fixed total magnetic fluxrdquo Communications inPartial Differential Equations vol 20 no 1-2 pp 1ndash36 1995

[55] Q Du ldquoGlobal existence and uniqueness of solutions of thetime-dependent Ginzburg-Landau model for superconductiv-ityrdquo Applicable Analysis vol 53 no 1-2 pp 1ndash18 1994

[56] Z M Chen K-H Hoffmann and J Liang ldquoOn a nonstation-ary Ginzburg-Landau superconductivity modelrdquoMathematicalMethods in the Applied Sciences vol 16 no 12 pp 855ndash875 1993

[57] J Liang and Q Tang ldquoAsymptotic behaviour of the solutionsof an evolutionaryGinzburg-Landau superconductivitymodelrdquoJournal of Mathematical Analysis and Applications vol 195 no1 pp 92ndash107 1995

[58] L P Gorrsquokov and G M Eliasherberg ldquoGeneralization of theGinzburg-Landau equations for an nonstationary problemsin the case of alloys with paramagnetic impartiesrdquo ZhurnalEksperimentalrsquonoi i Teoreticheskoi Fiziki vol 54 pp 612ndash6261968 Soviet Physics vol 27 pp 328ndash334 1968

[59] J Fleckinger-Pelle and H G Kaper ldquoGauges for the Ginzburg-Landau equations of superconductivityrdquo Zeitschrift fur Ange-wandte Mathematik und Mechanik vol 76 no 2 pp 345ndash3481996

[60] P Takac ldquoOn the dynamical process generated by supercon-ductivity modelrdquo Zeitschrift fur Angewandte Mathematik undMechanik vol 76 no 52 pp 349ndash352 1995

[61] W D Gropp H G Kaper DM Levine M Palumbo and VMVinokur ldquoNumerical simulation of vortex dynamics in type-IIsuperconductorsrdquo Journal of Computational Physics vol 123 no2 pp 254ndash266 1996

[62] D W Braun G W Crabtree H G Kaper et al ldquoStructure of amoving vortex latticerdquo Physical Review Letters vol 76 no 5 pp831ndash834 1996

[63] G W Crabtree G K Leaf H G Kaper et al ldquoTime-dependent Ginzburg-Landau simulations of vortex guidance bytwin boundariesrdquo Physica C vol 263 no 1ndash4 pp 401ndash408 1996

[64] G W Carabtree G K Leaf H G Kaper D W Braun V MVinokur andA E Koshelev ldquoDynamics vortex phases in super-conductors with correlated disorderrdquo Preprint ANLMCS-P590-0496 Mathematics and Computer Science DivisionAragonne National Laboratory 1996

[65] Th Gallay ldquoA center-stable manifold theorem for differentialequations in Banach spacesrdquo Communications in MathematicalPhysics vol 152 no 2 pp 249ndash268 1993

[66] J-M Ghidaglia and B Heron ldquoDimension of the attractorsassociated to the Ginzburg-Landau partial differential equa-tionrdquo Physica D vol 28 no 3 pp 282ndash304 1987

12 Journal of Function Spaces and Applications

[67] P Mattila Geometry of Sets and Measures in Euclidean Spacesvol 44 Cambridge University Press Cambridge UK 1995

[68] L HormanderThe Analysis of Linear Partial Differential Oper-ators vol 1 Springer Berlin Germany 2nd edition 1990

[69] H Amann Linear and Quasilinear Parabolic ProblemBikhauser Basel Switzerland 1995

[70] H-P Heinz ldquoOn the behaviour ofmeasures of noncompactnesswith respect to differentiation and integration of vector-valuedfunctionsrdquo Nonlinear Analysis Theory Methods amp ApplicationsA vol 7 no 12 pp 1351ndash1371 1983

[71] R P Agarwal M Meehan and D OrsquoRegan Fixed Point Theoryand Applications vol 141 of Cambridge Tracts in MathematicsCambridge University Press Cambridge UK 2001

[72] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer New York NY USA1983

[73] J Fleckinger-Pelle H G Kaper and P Takac ldquoDynamics of theGinzburg-Landau equations of superconductivityrdquo NonlinearAnalysis Theory Methods amp Applications A vol 32 no 5 pp647ndash665 1998

[74] H G Kaper and P Takac ldquoAn equivalence relation for theGinzburg-Landau equations of superconductivityrdquo Journal ofApplied Mathematics and Physics vol 48 no 4 pp 665ndash6751997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Mathematical PhysicsAdvances in

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Function Spaces and Applications

Hence

119892 (119905) le 119888 (119905) +119898119872int

119905

0119888 (119904) 119890(119892minus119898119872)119904

119889119904

exp ((120588 minus 119898119872) 119905) (84)

Before proving (76) we first show that

lim119905rarrinfin

119888 (119905) = 0 (85)

To prove this equality it is sufficient to show that the lastcomponent in the formula expressing 119888(119905) that is 119872119890minus120588119905

int119905

0119890120588119904F(119904 119906

infin) minusF

infin119889119904 tends to zero as 119905 rarr infin

To this end let us fix 120576 gt 0 Assumption (11986710158405) implies that

there exists 1199050such that

sup 1003817100381710038171003817F (119904 119906infin) minusF

infin

1003817100381710038171003817 119904 ge 1199050 le120576120588

2119872 (86)

This inequality together with (11986710158403) implies

ess sup 1003817100381710038171003817F (119904 119906infin) minusF

infin

1003817100381710038171003817 119904 isin R lt infin (87)

Now fix 1199051isin R such that 119905

1ge 119905 and

ess sup 1003817100381710038171003817F (119904 119906infin) minusF

infin

1003817100381710038171003817 119904 isin R

le120576120588 exp (120588119905

1)

2119872 (exp (1205881199050) minus 1)

(88)

Then using (86) and (88) we conclude that for 119905 ge 1199051we have

119872119890minus120588119905int

119905

0

119890120588119904 1003817100381710038171003817F (119904 119906

infin) minusF

infin

1003817100381710038171003817 119889119904

le 119872119890minus120588119905int

1199050

0

119890120588119904 1003817100381710038171003817F (119904 119906

infin) minusF

infin

1003817100381710038171003817 119889119904

+119872119890minus120588119905int

119905

1199050

119890120588119904 1003817100381710038171003817F (119904 119906

infin) minusF

infin

1003817100381710038171003817 119889119904 le120576

2+120576

2= 120576

(89)

This fact proves (85)Further using (1198671015840

5) and (85) and employing de

LrsquoHospitalrsquos rule for the fraction on the right-hand sideof inequality (84) we obtain that condition (76) is satisfiedThis fact completes the proof

6 An Application to the Ginzburg-LandauEquations of Superconductivity

In this section we formulate the gauged time-dependentGinzburg-Landau (TDGL) equations as an abstract evolutionequation in a Hilbert space Moreover we show applicationsof the above theorems to TDGL equations

We assume that Ω is a bounded domain in R119899 withboundary 120597Ω of class11986211That isΩ is an open and connectedset whose boundary 120597Ω is a compact (119899 minus 1)-manifolddescribed by Lipschitz continuous differentiable charts Weconsider two- and three-dimensional problems (119899 = 2 and119899 = 3 resp) Assume that the vector potential A takes its

values in R119899 The vector H will represent the (externally)applied magnetic field which is a function of space andtime similarly to A it takes its values in R119899 The function120574 is defined and satisfies Lipschitz condition on 120597Ω and120574(119909) ge 0 for 119909 isin 120597Ω The parameters in the TDGLequations are 120578 a (dimensionless) friction coefficient and 120581the (dimensionless) Ginzburg-Landau parameter

The order parameter should be thought of as the wavefunction of the center-of-massmotion of the ldquosuperelectronsrdquo(Cooper pairs) whose density is 119899

119904= |120595|

2 and whose fluxis J119904 The vector potential A determines the electromagnetic

field E = minus120597119905A minus nabla120601 is the electric field and B = nabla times A

is the magnetic induction where J the total current is thesum of a ldquonormalrdquo current J

119899= E the supercurrent J

119904 and

the transport current J119905= nabla times H The normal current obeys

Ohmrsquos law J119899= 120590119899E the ldquonormal conductivityrdquo coefficient 120590

119899

is equal to one in the adopted system of units The differenceM = BminusH is known as themagnetizationThe trivial solution(120595 = 0 B = H E = 0) represents the normal state where allsuperconducting properties have been lost

Nowwe accept the following notion all Banach spaces arereal the (real) dual of a Banach space119883 is denoted by1198831015840Thesymbol 119871119901(Ω) for 1 le 119901 le infin denotes the usual Lebesguespace with norm sdot

119871119901 (sdot sdot) is the inner product in 1198712(Ω)

1198821198982(Ω) for nonnegative integer 119898 is the usual Sobolev

space with norm sdot 1198821198982 1198821198982(Ω) is a Hilbert space for the

inner product (sdot sdot)1198982

given by (119906 V)1198982

= sum|120572|le119898

(120597120572119906 120597120572V)

for 119906 V isin 1198821198982(Ω) Fractional Sobolev space1198821199042(Ω) with afractional 119904 is defined by interpolation ([40 Chap VII] and[41 49 50]) 119862]

(Ω) for ] ge 0 ] = 119898 + 120582 with 0 le 120582 lt 1 isthe space of 119898 times continuous differentiable functions onΩ those 119898th-order derivatives satisfy the Holder conditionwith exponent 120582 if ] is a proper fraction the norm sdot

119862] is

defined in the usual wayThe definitions extend to the space of vector-valued

functions in the standard way with the caveat that the innerproduct in [1198712(Ω)]119899 is defined by (119906 V) = int

Ω119906 sdot V where the

symbol (119906 V) indicates the scalar product in R119899 Complex-valued functions are interpreted as vector-valued functionswith two real components

Functions that vary in space and time like the orderparameter and the vector potential are considered as map-pings from the time domain which is a subinterval of [0infin)into spaces of complex- or vector-valued functions definedin Ω Let 119883 = (119883 sdot

119883) be a Banach space of functions

defined inΩ Then the functions are defined in Ω Then thefunctions of space and time defined on Ω times (0 119879) for 119879 gt 0may be considered as elements of 119871119901(0 119879119883) for 1 le 119901 le infinor1198822119898(0 119879119883) for nonnegative119898 or119862]

(0 119879119883) for ] ge 0] = 119898 + 120582 with 0 le 120582 lt 1 Detailed definitions can be foundfor example in [43]

Obviously function spaces of ordered pairs (120595A) where120595 Ω rarr R2 and A Ω rarr R119899 (119899 = 2 3) play an importantrole in the study of the gauged TDGL equationsWe thereforeadapt the following special notationH = [119883(Ω)]

2times[119883(Ω)]

119899for any Banach space for the order parameter 120595 and thevector potential A respectively A suitable framework forthe functional analysis of the gauged TDGL equations is the

Journal of Function Spaces and Applications 9

Cartesian product W1+1205722 = [1198821+1205722

(Ω)]2times [1198821+1205722

(Ω)]119899

where 120572 isin (12 1) This space is continuously imbedded inW12 capLinfin

Assume H = 0 and H isin [1198712(Ω)]119899 Let AH be a minimizer

of the convex quadratic form 119869120596equiv J120596[119860]

119869120596 [A] = int

Ω

[120596(nabla sdot A)2 + |nabla times A minusH|2] 119889119909 (90)

on the domain

D (119869120596) = A isin [119882

12(Ω)]119899

n sdot A = 0 on 120597Ω (91)

We now introduce the reduced vector potential A1015840

A1015840 = A minus AH (92)

In terms of 120595 and A1015840 the gauged TDGL equations have thefollowing form

120597120595

120597119905minus

1

1205781205812Δ120595 = 120593 in Ω times (0infin)

120597A1015840

120597119905+ nabla times nabla times A1015840 minus 120596nabla (nabla sdot A1015840) = F in Ω times (0infin)

n sdot nabla + 120574120595 = 0 n sdot A1015840 = 0

n times (nabla sdot A1015840) = 0 on 120597Ω times (0infin)

(93)

Here 120593 and F are nonlinear functions of 120595 and A1015840

120593 equiv 120593 (119905 120595A1015840) = 1

120578[minus2119894

120581(nabla120595) sdot (A1015840 + AH)

minus119894

120581(1 minus 120578120581

2120596)120595 (nabla sdot A1015840)

minus12059510038161003816100381610038161003816A1015840 + AH

10038161003816100381610038161003816

2

+ (1 minus100381610038161003816100381612059510038161003816100381610038162

) 120595]

(94)

F equiv F (119905 120595A1015840) = 1

2119894119896(120595lowastnabla120595 minus 120595nabla120595

lowast)

minus100381610038161003816100381612059510038161003816100381610038162

(A1015840 + AH) minus120597AH120597119905

(95)

The equations are supplemented by initial data which is inthe followimg form

120595 = 1205950 A = A

0on Ω times 0 (96)

where 1205950and A

0are given and by (92) we have

120595 = 1205950 A1015840 = A

0minus AH (0) on Ω times 0 (97)

In the next part we connect the evolution of the solution(120595A1015840) of the system of (93) with the initial data (120595

0A0minus

AH(0)) with the dynamics of a vector 119906 in a Hilbert space

L2= [119871119901(Ω)]2

times [119871119901(Ω)]119899

(98)

The following analysis is restricted to the case 120596 gt 0 andthe case 120596 = 0 (see [73])

Let vector 119906 [0infin) rarr L2 represent the pair (120595A1015840)

119906 = (120595A1015840) equiv (120595A minus AH) (99)

and letA be the linear self-adjoint operator inL2 associatedwith the quadratic form 119876

120596equiv 119876120596[119906] given by the following

formula

119876120596 [119906] = int

Ω

[1

12057812058121003816100381610038161003816nabla120595

10038161003816100381610038162

+ 120596(nabla sdot A1015840)2

+10038161003816100381610038161003816nabla times A101584010038161003816100381610038161003816

2

] 119889119909

+ int120597Ω

120574

1205781205812100381610038161003816100381612059510038161003816100381610038162

119889120590 (119909)

(100)

on the domain

D (119876120596) = D (A

12)

= 119906 = (120595A1015840) isinW12 n sdot A1015840 = 0 on 120597Ω

(101)

The quadratic form 119876120596is nonnegative Furthermore since

120596 gt 0119876120596[120595A1015840]+119888 120595

1198712 is coercive onW12 for any constant

119888 gt 0 Hence A is positively definite in L2 ([44 Chap 1]equation (545)) If it does not lead to confusion we use thesame symbol A for the restriction A

120595and AA of A to the

respective linear subspace [1198712(Ω)]2 = [1198712(Ω)]2 times 0 (for 120595)and [1198712(Ω)]119899 equiv 0 times [1198712(Ω)]119899 (for A) ofL2

Now consider the initial-value problems (6) and (7) inL2 whereF(119905 119906) = (120593 F) 120593 and F is given by (94) and (95)and 119906

0= (1205950A0minus AH(0))

With 120582 isin (12 1) and 1199060isin W1+1205822 we say that 119906 are a

solution of (6) and (7) on the interval [0 119879] for some 119879 gt 0if 119906 [0 119879] rarr W1+1205822 is continuous and

119906 (119905) = 119890minusA(119905)

+ int

119905

0

119890minusA(119905minus119904)

F (119904 119906 (119904)) 119889119904 for 119905 isin [0 119879]

(102)

in L2 A mild solution of the initial-value problems (6) and(7) defines a weak solution (120595A1015840) of the boundary valueproblem (93) which in turn defines a weak solution (120595A)of the gauged TDGL equations provided AH is sufficientlyregular

Namely let us assume that X = L2 and 1199060isin L2 and

119906 R+times R119899 rarr R119899 for 119899 isin N is an unknown function 119906 =

119906(119905 119909) In order to applyTheorems 10 and 14 we are not goingto consider 119906 as a function of 119905 and 119909 together but rather asa mapping 119906 of variable 119905 into the spaceX =L2 of functions119909 that is 119906 R

+rarr L2 119906(119905)(119909) = 119906(119905 119909) 119909 isin R119899 119905 isin R

+

Remark 19 Since A is the linear self-adjoint operator inL2

associated with the quadratic form (100) thenA generates asemigroup 119890minusA119905

119905ge0(see [74])

Below we formulate the principal theorem of this paperThis theorem is a simple consequence of Theorem 10

10 Journal of Function Spaces and Applications

Theorem 20 Under assumptions (1198671)ndash(1198674) problem (6)-(7)

is equivalent to the system of (1)ndash(5) (TDGL equations) andhas for each 119906

0isin L2 at least one mild solution in sense of

Definition 4 119906 = 119906(119905) Moreover each solution 119906(119905) satisfiesestimate (25)

We observe that in the case when 119890minusA119905119905ge0

is the semi-group of contractions and 119886(119905) equiv 119886 119887(119905) equiv 119887 for 119905 isin

R+are constant after simple calculations based on estimate

(25) we get that the solution 119906 = 119906(119905) has the asymptoticcharacterization

119906 (119905) = 119874 (1199053) as 119905 997888rarr infin (103)

For further purposes let us formulate the followingassumption

(A) A is the infinitesimal generator of an exponentiallystable 119862

0-semigroup 119890minusA119905

119905ge0

The next result of this chapter is shown in the giventheorem below

Theorem21 Under assumptions (A) and (1198672)ndash(1198674) problem

(6)-(7) which is equivalent to the systems equations (1)ndash(5)(TDGL equations) has a mild solution 119906 = 119906(119905) for each1199060isin L2 which is globally attractive and locally uniformly

attractive

References

[1] V L Ginzburg and L D Landau ldquoOn the theory of supercon-ductivityrdquo Zhurnal Eksperimentalrsquonoi i Teoreticheskoi Fiziki vol20 pp 1064ndash1082 1950 English translation in L D LandauMen of Physics D ter Haar ed Vol I Pergamon Press OxfordUK pp 138ndash167 1965

[2] S Aizicovici and H Lee ldquoNonlinear nonlocal Cauchy problemsin Banach spacesrdquo Applied Mathematics Letters vol 18 no 4pp 401ndash407 2005

[3] S Aizicovici and M McKibben ldquoExistence results for a classof abstract nonlocal Cauchy problemsrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 39 no 5 pp 649ndash6682000

[4] MMcKibbenDiscovering Evolution Equation with ApplicationCRC Press Taylor amp Francis Group 2011

[5] S Aizicovici and V Staicu ldquoMultivalued evolution equationswith nonlocal initial conditions in Banach spacesrdquo NonlinearDifferential Equations and Applications vol 14 no 3-4 pp 361ndash376 2007

[6] R P Agarwal and D OrsquoRegan Infinite Interval Problems for Dif-ferential Difference and Integral Equations Kluwer AcademicPublishers Dordrecht The Netherlands 2001

[7] Z Fan ldquoExistence of nondensely defined evolution equationswith nonlocal conditionsrdquo Nonlinear Analysis Theory Methodsamp Applications A vol 70 no 11 pp 3829ndash3836 2009

[8] Z Fan ldquoImpulsive problems for semilinear differential equa-tions with nonlocal conditionsrdquo Nonlinear Analysis TheoryMethods amp Applications A vol 72 no 2 pp 1104ndash1109 2010

[9] Z Fan and G Li ldquoExistence results for semilinear differentialequations with nonlocal and impulsive conditionsrdquo Journal ofFunctional Analysis vol 258 no 5 pp 1709ndash1727 2010

[10] J Garcıa-Falset ldquoExistence results and asymptotic behavior fornonlocal abstract Cauchy problemsrdquo Journal of MathematicalAnalysis and Applications vol 338 no 1 pp 639ndash652 2008

[11] J Liang J H Liu and T-J Xiao ldquoNonlocal impulsive problemsfor nonlinear differential equations in Banach spacesrdquo Mathe-matical and Computer Modelling vol 49 no 3-4 pp 798ndash8042009

[12] L Zhu and G Li ldquoOn a nonlocal problem for semilinear differ-ential equations with upper semicontinuous nonlinearities ingeneral Banach spacesrdquo Journal of Mathematical Analysis andApplications vol 341 no 1 pp 660ndash675 2008

[13] Q Liu and R Yuan ldquoExistence of mild solutions for semilinearevolution equations with non-local initial conditionsrdquo Nonlin-ear AnalysisTheoryMethodsampApplications A vol 71 no 9 pp4177ndash4184 2009

[14] C Xie W Zhong and Z Fan ldquoExistence results for nonlinearnonlocal problems in Banach spacesrdquo Applied MathematicsLetters vol 22 no 7 pp 998ndash1002 2009

[15] T-J Xiao and J Liang ldquoExistence of classical solutions tononautonomous nonlocal parabolic problemsrdquoNonlinear Anal-ysisTheoryMethods and Applications vol 63 no 5-7 pp e225ndashe232 2005

[16] X Xue ldquo119871119901 theory for semilinear nonlocal problems with mea-sure of noncompactness in separable Banach spacesrdquo Journal ofFixed Point Theory and Applications vol 5 no 1 pp 129ndash1442009

[17] X Xue ldquoNonlinear differential equations with nonlocal condi-tions in Banach spacesrdquo Nonlinear Analysis Theory Methods ampApplications A vol 63 no 4 pp 575ndash586 2005

[18] X Xue ldquoExistence of solutions for semilinear nonlocal Cauchyproblems in Banach spacesrdquo Electronic Journal of DifferentialEquations vol 64 p 7 2005

[19] X Xue ldquoNonlocal nonlinear differential equations with a mea-sure of noncompactness in Banach spacesrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 70 no 7 pp 2593ndash26012009

[20] L Olszowy ldquoExistence ofmild solutions for the semilinear non-local problem in Banach spacesrdquo Nonlinear Analysis TheoryMethods amp Applications A vol 81 pp 211ndash223 2013

[21] D Bothe ldquoMultivalued perturbations of119898-accretive differentialinclusionsrdquo Israel Journal of Mathematics vol 108 pp 109ndash1381998

[22] H-S Ding J Liang G M NrsquoGuerekata and T-J XiaoldquoMild pseudo-almost periodic solutions of nonautonomoussemilinear evolution equationsrdquo Mathematical and ComputerModelling vol 45 no 5-6 pp 579ndash584 2007

[23] J Garcıa-Falset ldquoThe asymptotic behavior of the solutions of theCauchy problem generated by 120593-accretive operatorsrdquo Journal ofMathematical Analysis and Applications vol 310 no 2 pp 594ndash608 2005

[24] Z Fan and G Li ldquoAsymptotic behavior of the solutions of non-autonomous systems in Banach spacesrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 68 no 12 pp 3733ndash37412008

[25] J Banas ldquoOn existence theorems for differential equations inBanach spacesrdquo Bulletin of the Australian Mathematical Societyvol 32 no 1 pp 73ndash82 1985

[26] J Banas ldquoMeasures of noncompactness in the space of contin-uous tempered functionsrdquo Demonstratio Mathematica vol 14no 1 pp 127ndash133 1981

Journal of Function Spaces and Applications 11

[27] J Banas and K GoebelMeasures of Noncompactness in BanachSpaces vol 60 ofLectureNotes in Pure andAppliedMathematicsMarcel Dekker New York NY USA 1980

[28] J Banas and M Lecko ldquoSolvability of infinite systems ofdifferential equations in Banach sequence spacesrdquo Journal ofComputational andAppliedMathematics vol 137 no 2 pp 363ndash375 2001

[29] J Banas and K Sadarangani ldquoCompactness conditions inthe study of functional differential and integral equationsrdquoAbstract and Applied Analysis vol 2013 Article ID 819315 14pages 2013

[30] M Mursaleen and S A Mohiuddine ldquoApplications of mea-sures of noncompactness to the infinite system of differentialequations in ℓ

119901spacesrdquo Nonlinear Analysis Theory Methods amp

Applications A vol 75 no 4 pp 2111ndash2115 2012[31] L Olszowy ldquoSolvability of some functional integral equationrdquo

Dynamic Systems and Applications vol 18 no 3-4 pp 667ndash6762009

[32] L Olszowy ldquoOn some measures of noncompactness in theFrechet spaces of continuous functionsrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 72 pp 2119ndash2126 2010

[33] L Olszowy ldquoOn existence of solutions of a quadratic Urysohnintegral equation on an unbounded intervalrdquo CommentationesMathematicae vol 48 no 1 pp 103ndash112 2008

[34] L Olszowy ldquoExistence ofmild solutions for semilinear nonlocalCauchy problems in separable Banach spacesrdquo Journal of Anal-ysis and its Applications vol 32 no 2 pp 215ndash232 2013

[35] L Olszowy ldquoFixed point theorems in the Frechet space 119862(119877+)

and functional integral equations on an unbounded intervalrdquoApplied Mathematics and Computation vol 218 no 18 pp9066ndash9074 2012

[36] J Banas and D OrsquoRegan ldquoOn existence and local attractivity ofsolutions of a quadratic Volterra integral equation of fractionalorderrdquo Journal of Mathematical Analysis and Applications vol345 no 1 pp 573ndash582 2008

[37] L D Landau EM Lifshitz and L P Pitaevski Electrodynamicsof Continuous Media vol 8 Butterworth-Heinemann 2ndedition 1984

[38] L D Landau and E M Lifshitz Fluid Mechanics vol bButterworth-Heinemann 2nd edition 1987

[39] R Temam Infinite-Dimensional Dynamical Systems inMechan-ics and Physics vol 68 Springer New York NY USA 2ndedition 1997

[40] J K Hale Asymptotic Behavior of Dissipative Systems vol 25 ofMathematical Surveys and Monographs American Mathemati-cal Society Providence RI USA 1988

[41] R A Adams and J J F Fournier Sobolev Spaces vol 140Elsevier 2nd edition 2003

[42] V Mazrsquoya Sobolev Spaces with Applications to Elliptic PartialDifferential Equations Springer Heidelberg Germany 2ndedition 2011

[43] V Georgescu ldquoSome boundary value problems for differentialforms on compact Riemannian manifoldsrdquo Annali di Matemat-ica Pura ed Applicata vol 122 pp 159ndash198 1979

[44] D Henry Geometric Theory of Semilinear Parabolic Equationsvol 840 of Lecture Notes in Mathematics Springer New YorkNY USA 1981

[45] V Girault and P-A Raviart Finite Element Methods for Navier-Stokes Equations Springer New York NY USA 1986

[46] J Bardeen L N Cooper and J R Schrieffer ldquoTheory ofsuperconductivityrdquo Physical Review vol 108 pp 1175ndash12041957

[47] A A Abrikosov Fundamental of the Theory of Metals North-Holland Amstedram The Netherlands 1988

[48] M Tinkham Introduction To Superconductivity McGraw-HillNew York NY USA 2nd edition 1996

[49] P DeGennes Superconductivity inMetals and Alloys BenjaminNew York NY USA 1966

[50] H Brezis Functional Analysis Sobolev Spaces and PartialDifferential Equations Springer New York NY USA 2011

[51] E Zeidler Nonlinear Functional Analysis and its Applicationsvol 2 Springer New York NY USA 1990

[52] A Schmid ldquoA time dependent Ginzburg-Landau equation andits application to the problem of resistivity in the mixed staterdquoPhysik der KondensiertenMaterie vol 5 no 4 pp 302ndash317 1966

[53] C M Elliot and Q Tang Existence Theorems For an Evolution-ary Superconductivity Model University of Sussex BrightonUK 1992

[54] Q Tang ldquoOn an evolutionary system of Ginzburg-Landauequations with fixed total magnetic fluxrdquo Communications inPartial Differential Equations vol 20 no 1-2 pp 1ndash36 1995

[55] Q Du ldquoGlobal existence and uniqueness of solutions of thetime-dependent Ginzburg-Landau model for superconductiv-ityrdquo Applicable Analysis vol 53 no 1-2 pp 1ndash18 1994

[56] Z M Chen K-H Hoffmann and J Liang ldquoOn a nonstation-ary Ginzburg-Landau superconductivity modelrdquoMathematicalMethods in the Applied Sciences vol 16 no 12 pp 855ndash875 1993

[57] J Liang and Q Tang ldquoAsymptotic behaviour of the solutionsof an evolutionaryGinzburg-Landau superconductivitymodelrdquoJournal of Mathematical Analysis and Applications vol 195 no1 pp 92ndash107 1995

[58] L P Gorrsquokov and G M Eliasherberg ldquoGeneralization of theGinzburg-Landau equations for an nonstationary problemsin the case of alloys with paramagnetic impartiesrdquo ZhurnalEksperimentalrsquonoi i Teoreticheskoi Fiziki vol 54 pp 612ndash6261968 Soviet Physics vol 27 pp 328ndash334 1968

[59] J Fleckinger-Pelle and H G Kaper ldquoGauges for the Ginzburg-Landau equations of superconductivityrdquo Zeitschrift fur Ange-wandte Mathematik und Mechanik vol 76 no 2 pp 345ndash3481996

[60] P Takac ldquoOn the dynamical process generated by supercon-ductivity modelrdquo Zeitschrift fur Angewandte Mathematik undMechanik vol 76 no 52 pp 349ndash352 1995

[61] W D Gropp H G Kaper DM Levine M Palumbo and VMVinokur ldquoNumerical simulation of vortex dynamics in type-IIsuperconductorsrdquo Journal of Computational Physics vol 123 no2 pp 254ndash266 1996

[62] D W Braun G W Crabtree H G Kaper et al ldquoStructure of amoving vortex latticerdquo Physical Review Letters vol 76 no 5 pp831ndash834 1996

[63] G W Crabtree G K Leaf H G Kaper et al ldquoTime-dependent Ginzburg-Landau simulations of vortex guidance bytwin boundariesrdquo Physica C vol 263 no 1ndash4 pp 401ndash408 1996

[64] G W Carabtree G K Leaf H G Kaper D W Braun V MVinokur andA E Koshelev ldquoDynamics vortex phases in super-conductors with correlated disorderrdquo Preprint ANLMCS-P590-0496 Mathematics and Computer Science DivisionAragonne National Laboratory 1996

[65] Th Gallay ldquoA center-stable manifold theorem for differentialequations in Banach spacesrdquo Communications in MathematicalPhysics vol 152 no 2 pp 249ndash268 1993

[66] J-M Ghidaglia and B Heron ldquoDimension of the attractorsassociated to the Ginzburg-Landau partial differential equa-tionrdquo Physica D vol 28 no 3 pp 282ndash304 1987

12 Journal of Function Spaces and Applications

[67] P Mattila Geometry of Sets and Measures in Euclidean Spacesvol 44 Cambridge University Press Cambridge UK 1995

[68] L HormanderThe Analysis of Linear Partial Differential Oper-ators vol 1 Springer Berlin Germany 2nd edition 1990

[69] H Amann Linear and Quasilinear Parabolic ProblemBikhauser Basel Switzerland 1995

[70] H-P Heinz ldquoOn the behaviour ofmeasures of noncompactnesswith respect to differentiation and integration of vector-valuedfunctionsrdquo Nonlinear Analysis Theory Methods amp ApplicationsA vol 7 no 12 pp 1351ndash1371 1983

[71] R P Agarwal M Meehan and D OrsquoRegan Fixed Point Theoryand Applications vol 141 of Cambridge Tracts in MathematicsCambridge University Press Cambridge UK 2001

[72] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer New York NY USA1983

[73] J Fleckinger-Pelle H G Kaper and P Takac ldquoDynamics of theGinzburg-Landau equations of superconductivityrdquo NonlinearAnalysis Theory Methods amp Applications A vol 32 no 5 pp647ndash665 1998

[74] H G Kaper and P Takac ldquoAn equivalence relation for theGinzburg-Landau equations of superconductivityrdquo Journal ofApplied Mathematics and Physics vol 48 no 4 pp 665ndash6751997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces and Applications 9

Cartesian product W1+1205722 = [1198821+1205722

(Ω)]2times [1198821+1205722

(Ω)]119899

where 120572 isin (12 1) This space is continuously imbedded inW12 capLinfin

Assume H = 0 and H isin [1198712(Ω)]119899 Let AH be a minimizer

of the convex quadratic form 119869120596equiv J120596[119860]

119869120596 [A] = int

Ω

[120596(nabla sdot A)2 + |nabla times A minusH|2] 119889119909 (90)

on the domain

D (119869120596) = A isin [119882

12(Ω)]119899

n sdot A = 0 on 120597Ω (91)

We now introduce the reduced vector potential A1015840

A1015840 = A minus AH (92)

In terms of 120595 and A1015840 the gauged TDGL equations have thefollowing form

120597120595

120597119905minus

1

1205781205812Δ120595 = 120593 in Ω times (0infin)

120597A1015840

120597119905+ nabla times nabla times A1015840 minus 120596nabla (nabla sdot A1015840) = F in Ω times (0infin)

n sdot nabla + 120574120595 = 0 n sdot A1015840 = 0

n times (nabla sdot A1015840) = 0 on 120597Ω times (0infin)

(93)

Here 120593 and F are nonlinear functions of 120595 and A1015840

120593 equiv 120593 (119905 120595A1015840) = 1

120578[minus2119894

120581(nabla120595) sdot (A1015840 + AH)

minus119894

120581(1 minus 120578120581

2120596)120595 (nabla sdot A1015840)

minus12059510038161003816100381610038161003816A1015840 + AH

10038161003816100381610038161003816

2

+ (1 minus100381610038161003816100381612059510038161003816100381610038162

) 120595]

(94)

F equiv F (119905 120595A1015840) = 1

2119894119896(120595lowastnabla120595 minus 120595nabla120595

lowast)

minus100381610038161003816100381612059510038161003816100381610038162

(A1015840 + AH) minus120597AH120597119905

(95)

The equations are supplemented by initial data which is inthe followimg form

120595 = 1205950 A = A

0on Ω times 0 (96)

where 1205950and A

0are given and by (92) we have

120595 = 1205950 A1015840 = A

0minus AH (0) on Ω times 0 (97)

In the next part we connect the evolution of the solution(120595A1015840) of the system of (93) with the initial data (120595

0A0minus

AH(0)) with the dynamics of a vector 119906 in a Hilbert space

L2= [119871119901(Ω)]2

times [119871119901(Ω)]119899

(98)

The following analysis is restricted to the case 120596 gt 0 andthe case 120596 = 0 (see [73])

Let vector 119906 [0infin) rarr L2 represent the pair (120595A1015840)

119906 = (120595A1015840) equiv (120595A minus AH) (99)

and letA be the linear self-adjoint operator inL2 associatedwith the quadratic form 119876

120596equiv 119876120596[119906] given by the following

formula

119876120596 [119906] = int

Ω

[1

12057812058121003816100381610038161003816nabla120595

10038161003816100381610038162

+ 120596(nabla sdot A1015840)2

+10038161003816100381610038161003816nabla times A101584010038161003816100381610038161003816

2

] 119889119909

+ int120597Ω

120574

1205781205812100381610038161003816100381612059510038161003816100381610038162

119889120590 (119909)

(100)

on the domain

D (119876120596) = D (A

12)

= 119906 = (120595A1015840) isinW12 n sdot A1015840 = 0 on 120597Ω

(101)

The quadratic form 119876120596is nonnegative Furthermore since

120596 gt 0119876120596[120595A1015840]+119888 120595

1198712 is coercive onW12 for any constant

119888 gt 0 Hence A is positively definite in L2 ([44 Chap 1]equation (545)) If it does not lead to confusion we use thesame symbol A for the restriction A

120595and AA of A to the

respective linear subspace [1198712(Ω)]2 = [1198712(Ω)]2 times 0 (for 120595)and [1198712(Ω)]119899 equiv 0 times [1198712(Ω)]119899 (for A) ofL2

Now consider the initial-value problems (6) and (7) inL2 whereF(119905 119906) = (120593 F) 120593 and F is given by (94) and (95)and 119906

0= (1205950A0minus AH(0))

With 120582 isin (12 1) and 1199060isin W1+1205822 we say that 119906 are a

solution of (6) and (7) on the interval [0 119879] for some 119879 gt 0if 119906 [0 119879] rarr W1+1205822 is continuous and

119906 (119905) = 119890minusA(119905)

+ int

119905

0

119890minusA(119905minus119904)

F (119904 119906 (119904)) 119889119904 for 119905 isin [0 119879]

(102)

in L2 A mild solution of the initial-value problems (6) and(7) defines a weak solution (120595A1015840) of the boundary valueproblem (93) which in turn defines a weak solution (120595A)of the gauged TDGL equations provided AH is sufficientlyregular

Namely let us assume that X = L2 and 1199060isin L2 and

119906 R+times R119899 rarr R119899 for 119899 isin N is an unknown function 119906 =

119906(119905 119909) In order to applyTheorems 10 and 14 we are not goingto consider 119906 as a function of 119905 and 119909 together but rather asa mapping 119906 of variable 119905 into the spaceX =L2 of functions119909 that is 119906 R

+rarr L2 119906(119905)(119909) = 119906(119905 119909) 119909 isin R119899 119905 isin R

+

Remark 19 Since A is the linear self-adjoint operator inL2

associated with the quadratic form (100) thenA generates asemigroup 119890minusA119905

119905ge0(see [74])

Below we formulate the principal theorem of this paperThis theorem is a simple consequence of Theorem 10

10 Journal of Function Spaces and Applications

Theorem 20 Under assumptions (1198671)ndash(1198674) problem (6)-(7)

is equivalent to the system of (1)ndash(5) (TDGL equations) andhas for each 119906

0isin L2 at least one mild solution in sense of

Definition 4 119906 = 119906(119905) Moreover each solution 119906(119905) satisfiesestimate (25)

We observe that in the case when 119890minusA119905119905ge0

is the semi-group of contractions and 119886(119905) equiv 119886 119887(119905) equiv 119887 for 119905 isin

R+are constant after simple calculations based on estimate

(25) we get that the solution 119906 = 119906(119905) has the asymptoticcharacterization

119906 (119905) = 119874 (1199053) as 119905 997888rarr infin (103)

For further purposes let us formulate the followingassumption

(A) A is the infinitesimal generator of an exponentiallystable 119862

0-semigroup 119890minusA119905

119905ge0

The next result of this chapter is shown in the giventheorem below

Theorem21 Under assumptions (A) and (1198672)ndash(1198674) problem

(6)-(7) which is equivalent to the systems equations (1)ndash(5)(TDGL equations) has a mild solution 119906 = 119906(119905) for each1199060isin L2 which is globally attractive and locally uniformly

attractive

References

[1] V L Ginzburg and L D Landau ldquoOn the theory of supercon-ductivityrdquo Zhurnal Eksperimentalrsquonoi i Teoreticheskoi Fiziki vol20 pp 1064ndash1082 1950 English translation in L D LandauMen of Physics D ter Haar ed Vol I Pergamon Press OxfordUK pp 138ndash167 1965

[2] S Aizicovici and H Lee ldquoNonlinear nonlocal Cauchy problemsin Banach spacesrdquo Applied Mathematics Letters vol 18 no 4pp 401ndash407 2005

[3] S Aizicovici and M McKibben ldquoExistence results for a classof abstract nonlocal Cauchy problemsrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 39 no 5 pp 649ndash6682000

[4] MMcKibbenDiscovering Evolution Equation with ApplicationCRC Press Taylor amp Francis Group 2011

[5] S Aizicovici and V Staicu ldquoMultivalued evolution equationswith nonlocal initial conditions in Banach spacesrdquo NonlinearDifferential Equations and Applications vol 14 no 3-4 pp 361ndash376 2007

[6] R P Agarwal and D OrsquoRegan Infinite Interval Problems for Dif-ferential Difference and Integral Equations Kluwer AcademicPublishers Dordrecht The Netherlands 2001

[7] Z Fan ldquoExistence of nondensely defined evolution equationswith nonlocal conditionsrdquo Nonlinear Analysis Theory Methodsamp Applications A vol 70 no 11 pp 3829ndash3836 2009

[8] Z Fan ldquoImpulsive problems for semilinear differential equa-tions with nonlocal conditionsrdquo Nonlinear Analysis TheoryMethods amp Applications A vol 72 no 2 pp 1104ndash1109 2010

[9] Z Fan and G Li ldquoExistence results for semilinear differentialequations with nonlocal and impulsive conditionsrdquo Journal ofFunctional Analysis vol 258 no 5 pp 1709ndash1727 2010

[10] J Garcıa-Falset ldquoExistence results and asymptotic behavior fornonlocal abstract Cauchy problemsrdquo Journal of MathematicalAnalysis and Applications vol 338 no 1 pp 639ndash652 2008

[11] J Liang J H Liu and T-J Xiao ldquoNonlocal impulsive problemsfor nonlinear differential equations in Banach spacesrdquo Mathe-matical and Computer Modelling vol 49 no 3-4 pp 798ndash8042009

[12] L Zhu and G Li ldquoOn a nonlocal problem for semilinear differ-ential equations with upper semicontinuous nonlinearities ingeneral Banach spacesrdquo Journal of Mathematical Analysis andApplications vol 341 no 1 pp 660ndash675 2008

[13] Q Liu and R Yuan ldquoExistence of mild solutions for semilinearevolution equations with non-local initial conditionsrdquo Nonlin-ear AnalysisTheoryMethodsampApplications A vol 71 no 9 pp4177ndash4184 2009

[14] C Xie W Zhong and Z Fan ldquoExistence results for nonlinearnonlocal problems in Banach spacesrdquo Applied MathematicsLetters vol 22 no 7 pp 998ndash1002 2009

[15] T-J Xiao and J Liang ldquoExistence of classical solutions tononautonomous nonlocal parabolic problemsrdquoNonlinear Anal-ysisTheoryMethods and Applications vol 63 no 5-7 pp e225ndashe232 2005

[16] X Xue ldquo119871119901 theory for semilinear nonlocal problems with mea-sure of noncompactness in separable Banach spacesrdquo Journal ofFixed Point Theory and Applications vol 5 no 1 pp 129ndash1442009

[17] X Xue ldquoNonlinear differential equations with nonlocal condi-tions in Banach spacesrdquo Nonlinear Analysis Theory Methods ampApplications A vol 63 no 4 pp 575ndash586 2005

[18] X Xue ldquoExistence of solutions for semilinear nonlocal Cauchyproblems in Banach spacesrdquo Electronic Journal of DifferentialEquations vol 64 p 7 2005

[19] X Xue ldquoNonlocal nonlinear differential equations with a mea-sure of noncompactness in Banach spacesrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 70 no 7 pp 2593ndash26012009

[20] L Olszowy ldquoExistence ofmild solutions for the semilinear non-local problem in Banach spacesrdquo Nonlinear Analysis TheoryMethods amp Applications A vol 81 pp 211ndash223 2013

[21] D Bothe ldquoMultivalued perturbations of119898-accretive differentialinclusionsrdquo Israel Journal of Mathematics vol 108 pp 109ndash1381998

[22] H-S Ding J Liang G M NrsquoGuerekata and T-J XiaoldquoMild pseudo-almost periodic solutions of nonautonomoussemilinear evolution equationsrdquo Mathematical and ComputerModelling vol 45 no 5-6 pp 579ndash584 2007

[23] J Garcıa-Falset ldquoThe asymptotic behavior of the solutions of theCauchy problem generated by 120593-accretive operatorsrdquo Journal ofMathematical Analysis and Applications vol 310 no 2 pp 594ndash608 2005

[24] Z Fan and G Li ldquoAsymptotic behavior of the solutions of non-autonomous systems in Banach spacesrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 68 no 12 pp 3733ndash37412008

[25] J Banas ldquoOn existence theorems for differential equations inBanach spacesrdquo Bulletin of the Australian Mathematical Societyvol 32 no 1 pp 73ndash82 1985

[26] J Banas ldquoMeasures of noncompactness in the space of contin-uous tempered functionsrdquo Demonstratio Mathematica vol 14no 1 pp 127ndash133 1981

Journal of Function Spaces and Applications 11

[27] J Banas and K GoebelMeasures of Noncompactness in BanachSpaces vol 60 ofLectureNotes in Pure andAppliedMathematicsMarcel Dekker New York NY USA 1980

[28] J Banas and M Lecko ldquoSolvability of infinite systems ofdifferential equations in Banach sequence spacesrdquo Journal ofComputational andAppliedMathematics vol 137 no 2 pp 363ndash375 2001

[29] J Banas and K Sadarangani ldquoCompactness conditions inthe study of functional differential and integral equationsrdquoAbstract and Applied Analysis vol 2013 Article ID 819315 14pages 2013

[30] M Mursaleen and S A Mohiuddine ldquoApplications of mea-sures of noncompactness to the infinite system of differentialequations in ℓ

119901spacesrdquo Nonlinear Analysis Theory Methods amp

Applications A vol 75 no 4 pp 2111ndash2115 2012[31] L Olszowy ldquoSolvability of some functional integral equationrdquo

Dynamic Systems and Applications vol 18 no 3-4 pp 667ndash6762009

[32] L Olszowy ldquoOn some measures of noncompactness in theFrechet spaces of continuous functionsrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 72 pp 2119ndash2126 2010

[33] L Olszowy ldquoOn existence of solutions of a quadratic Urysohnintegral equation on an unbounded intervalrdquo CommentationesMathematicae vol 48 no 1 pp 103ndash112 2008

[34] L Olszowy ldquoExistence ofmild solutions for semilinear nonlocalCauchy problems in separable Banach spacesrdquo Journal of Anal-ysis and its Applications vol 32 no 2 pp 215ndash232 2013

[35] L Olszowy ldquoFixed point theorems in the Frechet space 119862(119877+)

and functional integral equations on an unbounded intervalrdquoApplied Mathematics and Computation vol 218 no 18 pp9066ndash9074 2012

[36] J Banas and D OrsquoRegan ldquoOn existence and local attractivity ofsolutions of a quadratic Volterra integral equation of fractionalorderrdquo Journal of Mathematical Analysis and Applications vol345 no 1 pp 573ndash582 2008

[37] L D Landau EM Lifshitz and L P Pitaevski Electrodynamicsof Continuous Media vol 8 Butterworth-Heinemann 2ndedition 1984

[38] L D Landau and E M Lifshitz Fluid Mechanics vol bButterworth-Heinemann 2nd edition 1987

[39] R Temam Infinite-Dimensional Dynamical Systems inMechan-ics and Physics vol 68 Springer New York NY USA 2ndedition 1997

[40] J K Hale Asymptotic Behavior of Dissipative Systems vol 25 ofMathematical Surveys and Monographs American Mathemati-cal Society Providence RI USA 1988

[41] R A Adams and J J F Fournier Sobolev Spaces vol 140Elsevier 2nd edition 2003

[42] V Mazrsquoya Sobolev Spaces with Applications to Elliptic PartialDifferential Equations Springer Heidelberg Germany 2ndedition 2011

[43] V Georgescu ldquoSome boundary value problems for differentialforms on compact Riemannian manifoldsrdquo Annali di Matemat-ica Pura ed Applicata vol 122 pp 159ndash198 1979

[44] D Henry Geometric Theory of Semilinear Parabolic Equationsvol 840 of Lecture Notes in Mathematics Springer New YorkNY USA 1981

[45] V Girault and P-A Raviart Finite Element Methods for Navier-Stokes Equations Springer New York NY USA 1986

[46] J Bardeen L N Cooper and J R Schrieffer ldquoTheory ofsuperconductivityrdquo Physical Review vol 108 pp 1175ndash12041957

[47] A A Abrikosov Fundamental of the Theory of Metals North-Holland Amstedram The Netherlands 1988

[48] M Tinkham Introduction To Superconductivity McGraw-HillNew York NY USA 2nd edition 1996

[49] P DeGennes Superconductivity inMetals and Alloys BenjaminNew York NY USA 1966

[50] H Brezis Functional Analysis Sobolev Spaces and PartialDifferential Equations Springer New York NY USA 2011

[51] E Zeidler Nonlinear Functional Analysis and its Applicationsvol 2 Springer New York NY USA 1990

[52] A Schmid ldquoA time dependent Ginzburg-Landau equation andits application to the problem of resistivity in the mixed staterdquoPhysik der KondensiertenMaterie vol 5 no 4 pp 302ndash317 1966

[53] C M Elliot and Q Tang Existence Theorems For an Evolution-ary Superconductivity Model University of Sussex BrightonUK 1992

[54] Q Tang ldquoOn an evolutionary system of Ginzburg-Landauequations with fixed total magnetic fluxrdquo Communications inPartial Differential Equations vol 20 no 1-2 pp 1ndash36 1995

[55] Q Du ldquoGlobal existence and uniqueness of solutions of thetime-dependent Ginzburg-Landau model for superconductiv-ityrdquo Applicable Analysis vol 53 no 1-2 pp 1ndash18 1994

[56] Z M Chen K-H Hoffmann and J Liang ldquoOn a nonstation-ary Ginzburg-Landau superconductivity modelrdquoMathematicalMethods in the Applied Sciences vol 16 no 12 pp 855ndash875 1993

[57] J Liang and Q Tang ldquoAsymptotic behaviour of the solutionsof an evolutionaryGinzburg-Landau superconductivitymodelrdquoJournal of Mathematical Analysis and Applications vol 195 no1 pp 92ndash107 1995

[58] L P Gorrsquokov and G M Eliasherberg ldquoGeneralization of theGinzburg-Landau equations for an nonstationary problemsin the case of alloys with paramagnetic impartiesrdquo ZhurnalEksperimentalrsquonoi i Teoreticheskoi Fiziki vol 54 pp 612ndash6261968 Soviet Physics vol 27 pp 328ndash334 1968

[59] J Fleckinger-Pelle and H G Kaper ldquoGauges for the Ginzburg-Landau equations of superconductivityrdquo Zeitschrift fur Ange-wandte Mathematik und Mechanik vol 76 no 2 pp 345ndash3481996

[60] P Takac ldquoOn the dynamical process generated by supercon-ductivity modelrdquo Zeitschrift fur Angewandte Mathematik undMechanik vol 76 no 52 pp 349ndash352 1995

[61] W D Gropp H G Kaper DM Levine M Palumbo and VMVinokur ldquoNumerical simulation of vortex dynamics in type-IIsuperconductorsrdquo Journal of Computational Physics vol 123 no2 pp 254ndash266 1996

[62] D W Braun G W Crabtree H G Kaper et al ldquoStructure of amoving vortex latticerdquo Physical Review Letters vol 76 no 5 pp831ndash834 1996

[63] G W Crabtree G K Leaf H G Kaper et al ldquoTime-dependent Ginzburg-Landau simulations of vortex guidance bytwin boundariesrdquo Physica C vol 263 no 1ndash4 pp 401ndash408 1996

[64] G W Carabtree G K Leaf H G Kaper D W Braun V MVinokur andA E Koshelev ldquoDynamics vortex phases in super-conductors with correlated disorderrdquo Preprint ANLMCS-P590-0496 Mathematics and Computer Science DivisionAragonne National Laboratory 1996

[65] Th Gallay ldquoA center-stable manifold theorem for differentialequations in Banach spacesrdquo Communications in MathematicalPhysics vol 152 no 2 pp 249ndash268 1993

[66] J-M Ghidaglia and B Heron ldquoDimension of the attractorsassociated to the Ginzburg-Landau partial differential equa-tionrdquo Physica D vol 28 no 3 pp 282ndash304 1987

12 Journal of Function Spaces and Applications

[67] P Mattila Geometry of Sets and Measures in Euclidean Spacesvol 44 Cambridge University Press Cambridge UK 1995

[68] L HormanderThe Analysis of Linear Partial Differential Oper-ators vol 1 Springer Berlin Germany 2nd edition 1990

[69] H Amann Linear and Quasilinear Parabolic ProblemBikhauser Basel Switzerland 1995

[70] H-P Heinz ldquoOn the behaviour ofmeasures of noncompactnesswith respect to differentiation and integration of vector-valuedfunctionsrdquo Nonlinear Analysis Theory Methods amp ApplicationsA vol 7 no 12 pp 1351ndash1371 1983

[71] R P Agarwal M Meehan and D OrsquoRegan Fixed Point Theoryand Applications vol 141 of Cambridge Tracts in MathematicsCambridge University Press Cambridge UK 2001

[72] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer New York NY USA1983

[73] J Fleckinger-Pelle H G Kaper and P Takac ldquoDynamics of theGinzburg-Landau equations of superconductivityrdquo NonlinearAnalysis Theory Methods amp Applications A vol 32 no 5 pp647ndash665 1998

[74] H G Kaper and P Takac ldquoAn equivalence relation for theGinzburg-Landau equations of superconductivityrdquo Journal ofApplied Mathematics and Physics vol 48 no 4 pp 665ndash6751997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Journal of Function Spaces and Applications

Theorem 20 Under assumptions (1198671)ndash(1198674) problem (6)-(7)

is equivalent to the system of (1)ndash(5) (TDGL equations) andhas for each 119906

0isin L2 at least one mild solution in sense of

Definition 4 119906 = 119906(119905) Moreover each solution 119906(119905) satisfiesestimate (25)

We observe that in the case when 119890minusA119905119905ge0

is the semi-group of contractions and 119886(119905) equiv 119886 119887(119905) equiv 119887 for 119905 isin

R+are constant after simple calculations based on estimate

(25) we get that the solution 119906 = 119906(119905) has the asymptoticcharacterization

119906 (119905) = 119874 (1199053) as 119905 997888rarr infin (103)

For further purposes let us formulate the followingassumption

(A) A is the infinitesimal generator of an exponentiallystable 119862

0-semigroup 119890minusA119905

119905ge0

The next result of this chapter is shown in the giventheorem below

Theorem21 Under assumptions (A) and (1198672)ndash(1198674) problem

(6)-(7) which is equivalent to the systems equations (1)ndash(5)(TDGL equations) has a mild solution 119906 = 119906(119905) for each1199060isin L2 which is globally attractive and locally uniformly

attractive

References

[1] V L Ginzburg and L D Landau ldquoOn the theory of supercon-ductivityrdquo Zhurnal Eksperimentalrsquonoi i Teoreticheskoi Fiziki vol20 pp 1064ndash1082 1950 English translation in L D LandauMen of Physics D ter Haar ed Vol I Pergamon Press OxfordUK pp 138ndash167 1965

[2] S Aizicovici and H Lee ldquoNonlinear nonlocal Cauchy problemsin Banach spacesrdquo Applied Mathematics Letters vol 18 no 4pp 401ndash407 2005

[3] S Aizicovici and M McKibben ldquoExistence results for a classof abstract nonlocal Cauchy problemsrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 39 no 5 pp 649ndash6682000

[4] MMcKibbenDiscovering Evolution Equation with ApplicationCRC Press Taylor amp Francis Group 2011

[5] S Aizicovici and V Staicu ldquoMultivalued evolution equationswith nonlocal initial conditions in Banach spacesrdquo NonlinearDifferential Equations and Applications vol 14 no 3-4 pp 361ndash376 2007

[6] R P Agarwal and D OrsquoRegan Infinite Interval Problems for Dif-ferential Difference and Integral Equations Kluwer AcademicPublishers Dordrecht The Netherlands 2001

[7] Z Fan ldquoExistence of nondensely defined evolution equationswith nonlocal conditionsrdquo Nonlinear Analysis Theory Methodsamp Applications A vol 70 no 11 pp 3829ndash3836 2009

[8] Z Fan ldquoImpulsive problems for semilinear differential equa-tions with nonlocal conditionsrdquo Nonlinear Analysis TheoryMethods amp Applications A vol 72 no 2 pp 1104ndash1109 2010

[9] Z Fan and G Li ldquoExistence results for semilinear differentialequations with nonlocal and impulsive conditionsrdquo Journal ofFunctional Analysis vol 258 no 5 pp 1709ndash1727 2010

[10] J Garcıa-Falset ldquoExistence results and asymptotic behavior fornonlocal abstract Cauchy problemsrdquo Journal of MathematicalAnalysis and Applications vol 338 no 1 pp 639ndash652 2008

[11] J Liang J H Liu and T-J Xiao ldquoNonlocal impulsive problemsfor nonlinear differential equations in Banach spacesrdquo Mathe-matical and Computer Modelling vol 49 no 3-4 pp 798ndash8042009

[12] L Zhu and G Li ldquoOn a nonlocal problem for semilinear differ-ential equations with upper semicontinuous nonlinearities ingeneral Banach spacesrdquo Journal of Mathematical Analysis andApplications vol 341 no 1 pp 660ndash675 2008

[13] Q Liu and R Yuan ldquoExistence of mild solutions for semilinearevolution equations with non-local initial conditionsrdquo Nonlin-ear AnalysisTheoryMethodsampApplications A vol 71 no 9 pp4177ndash4184 2009

[14] C Xie W Zhong and Z Fan ldquoExistence results for nonlinearnonlocal problems in Banach spacesrdquo Applied MathematicsLetters vol 22 no 7 pp 998ndash1002 2009

[15] T-J Xiao and J Liang ldquoExistence of classical solutions tononautonomous nonlocal parabolic problemsrdquoNonlinear Anal-ysisTheoryMethods and Applications vol 63 no 5-7 pp e225ndashe232 2005

[16] X Xue ldquo119871119901 theory for semilinear nonlocal problems with mea-sure of noncompactness in separable Banach spacesrdquo Journal ofFixed Point Theory and Applications vol 5 no 1 pp 129ndash1442009

[17] X Xue ldquoNonlinear differential equations with nonlocal condi-tions in Banach spacesrdquo Nonlinear Analysis Theory Methods ampApplications A vol 63 no 4 pp 575ndash586 2005

[18] X Xue ldquoExistence of solutions for semilinear nonlocal Cauchyproblems in Banach spacesrdquo Electronic Journal of DifferentialEquations vol 64 p 7 2005

[19] X Xue ldquoNonlocal nonlinear differential equations with a mea-sure of noncompactness in Banach spacesrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 70 no 7 pp 2593ndash26012009

[20] L Olszowy ldquoExistence ofmild solutions for the semilinear non-local problem in Banach spacesrdquo Nonlinear Analysis TheoryMethods amp Applications A vol 81 pp 211ndash223 2013

[21] D Bothe ldquoMultivalued perturbations of119898-accretive differentialinclusionsrdquo Israel Journal of Mathematics vol 108 pp 109ndash1381998

[22] H-S Ding J Liang G M NrsquoGuerekata and T-J XiaoldquoMild pseudo-almost periodic solutions of nonautonomoussemilinear evolution equationsrdquo Mathematical and ComputerModelling vol 45 no 5-6 pp 579ndash584 2007

[23] J Garcıa-Falset ldquoThe asymptotic behavior of the solutions of theCauchy problem generated by 120593-accretive operatorsrdquo Journal ofMathematical Analysis and Applications vol 310 no 2 pp 594ndash608 2005

[24] Z Fan and G Li ldquoAsymptotic behavior of the solutions of non-autonomous systems in Banach spacesrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 68 no 12 pp 3733ndash37412008

[25] J Banas ldquoOn existence theorems for differential equations inBanach spacesrdquo Bulletin of the Australian Mathematical Societyvol 32 no 1 pp 73ndash82 1985

[26] J Banas ldquoMeasures of noncompactness in the space of contin-uous tempered functionsrdquo Demonstratio Mathematica vol 14no 1 pp 127ndash133 1981

Journal of Function Spaces and Applications 11

[27] J Banas and K GoebelMeasures of Noncompactness in BanachSpaces vol 60 ofLectureNotes in Pure andAppliedMathematicsMarcel Dekker New York NY USA 1980

[28] J Banas and M Lecko ldquoSolvability of infinite systems ofdifferential equations in Banach sequence spacesrdquo Journal ofComputational andAppliedMathematics vol 137 no 2 pp 363ndash375 2001

[29] J Banas and K Sadarangani ldquoCompactness conditions inthe study of functional differential and integral equationsrdquoAbstract and Applied Analysis vol 2013 Article ID 819315 14pages 2013

[30] M Mursaleen and S A Mohiuddine ldquoApplications of mea-sures of noncompactness to the infinite system of differentialequations in ℓ

119901spacesrdquo Nonlinear Analysis Theory Methods amp

Applications A vol 75 no 4 pp 2111ndash2115 2012[31] L Olszowy ldquoSolvability of some functional integral equationrdquo

Dynamic Systems and Applications vol 18 no 3-4 pp 667ndash6762009

[32] L Olszowy ldquoOn some measures of noncompactness in theFrechet spaces of continuous functionsrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 72 pp 2119ndash2126 2010

[33] L Olszowy ldquoOn existence of solutions of a quadratic Urysohnintegral equation on an unbounded intervalrdquo CommentationesMathematicae vol 48 no 1 pp 103ndash112 2008

[34] L Olszowy ldquoExistence ofmild solutions for semilinear nonlocalCauchy problems in separable Banach spacesrdquo Journal of Anal-ysis and its Applications vol 32 no 2 pp 215ndash232 2013

[35] L Olszowy ldquoFixed point theorems in the Frechet space 119862(119877+)

and functional integral equations on an unbounded intervalrdquoApplied Mathematics and Computation vol 218 no 18 pp9066ndash9074 2012

[36] J Banas and D OrsquoRegan ldquoOn existence and local attractivity ofsolutions of a quadratic Volterra integral equation of fractionalorderrdquo Journal of Mathematical Analysis and Applications vol345 no 1 pp 573ndash582 2008

[37] L D Landau EM Lifshitz and L P Pitaevski Electrodynamicsof Continuous Media vol 8 Butterworth-Heinemann 2ndedition 1984

[38] L D Landau and E M Lifshitz Fluid Mechanics vol bButterworth-Heinemann 2nd edition 1987

[39] R Temam Infinite-Dimensional Dynamical Systems inMechan-ics and Physics vol 68 Springer New York NY USA 2ndedition 1997

[40] J K Hale Asymptotic Behavior of Dissipative Systems vol 25 ofMathematical Surveys and Monographs American Mathemati-cal Society Providence RI USA 1988

[41] R A Adams and J J F Fournier Sobolev Spaces vol 140Elsevier 2nd edition 2003

[42] V Mazrsquoya Sobolev Spaces with Applications to Elliptic PartialDifferential Equations Springer Heidelberg Germany 2ndedition 2011

[43] V Georgescu ldquoSome boundary value problems for differentialforms on compact Riemannian manifoldsrdquo Annali di Matemat-ica Pura ed Applicata vol 122 pp 159ndash198 1979

[44] D Henry Geometric Theory of Semilinear Parabolic Equationsvol 840 of Lecture Notes in Mathematics Springer New YorkNY USA 1981

[45] V Girault and P-A Raviart Finite Element Methods for Navier-Stokes Equations Springer New York NY USA 1986

[46] J Bardeen L N Cooper and J R Schrieffer ldquoTheory ofsuperconductivityrdquo Physical Review vol 108 pp 1175ndash12041957

[47] A A Abrikosov Fundamental of the Theory of Metals North-Holland Amstedram The Netherlands 1988

[48] M Tinkham Introduction To Superconductivity McGraw-HillNew York NY USA 2nd edition 1996

[49] P DeGennes Superconductivity inMetals and Alloys BenjaminNew York NY USA 1966

[50] H Brezis Functional Analysis Sobolev Spaces and PartialDifferential Equations Springer New York NY USA 2011

[51] E Zeidler Nonlinear Functional Analysis and its Applicationsvol 2 Springer New York NY USA 1990

[52] A Schmid ldquoA time dependent Ginzburg-Landau equation andits application to the problem of resistivity in the mixed staterdquoPhysik der KondensiertenMaterie vol 5 no 4 pp 302ndash317 1966

[53] C M Elliot and Q Tang Existence Theorems For an Evolution-ary Superconductivity Model University of Sussex BrightonUK 1992

[54] Q Tang ldquoOn an evolutionary system of Ginzburg-Landauequations with fixed total magnetic fluxrdquo Communications inPartial Differential Equations vol 20 no 1-2 pp 1ndash36 1995

[55] Q Du ldquoGlobal existence and uniqueness of solutions of thetime-dependent Ginzburg-Landau model for superconductiv-ityrdquo Applicable Analysis vol 53 no 1-2 pp 1ndash18 1994

[56] Z M Chen K-H Hoffmann and J Liang ldquoOn a nonstation-ary Ginzburg-Landau superconductivity modelrdquoMathematicalMethods in the Applied Sciences vol 16 no 12 pp 855ndash875 1993

[57] J Liang and Q Tang ldquoAsymptotic behaviour of the solutionsof an evolutionaryGinzburg-Landau superconductivitymodelrdquoJournal of Mathematical Analysis and Applications vol 195 no1 pp 92ndash107 1995

[58] L P Gorrsquokov and G M Eliasherberg ldquoGeneralization of theGinzburg-Landau equations for an nonstationary problemsin the case of alloys with paramagnetic impartiesrdquo ZhurnalEksperimentalrsquonoi i Teoreticheskoi Fiziki vol 54 pp 612ndash6261968 Soviet Physics vol 27 pp 328ndash334 1968

[59] J Fleckinger-Pelle and H G Kaper ldquoGauges for the Ginzburg-Landau equations of superconductivityrdquo Zeitschrift fur Ange-wandte Mathematik und Mechanik vol 76 no 2 pp 345ndash3481996

[60] P Takac ldquoOn the dynamical process generated by supercon-ductivity modelrdquo Zeitschrift fur Angewandte Mathematik undMechanik vol 76 no 52 pp 349ndash352 1995

[61] W D Gropp H G Kaper DM Levine M Palumbo and VMVinokur ldquoNumerical simulation of vortex dynamics in type-IIsuperconductorsrdquo Journal of Computational Physics vol 123 no2 pp 254ndash266 1996

[62] D W Braun G W Crabtree H G Kaper et al ldquoStructure of amoving vortex latticerdquo Physical Review Letters vol 76 no 5 pp831ndash834 1996

[63] G W Crabtree G K Leaf H G Kaper et al ldquoTime-dependent Ginzburg-Landau simulations of vortex guidance bytwin boundariesrdquo Physica C vol 263 no 1ndash4 pp 401ndash408 1996

[64] G W Carabtree G K Leaf H G Kaper D W Braun V MVinokur andA E Koshelev ldquoDynamics vortex phases in super-conductors with correlated disorderrdquo Preprint ANLMCS-P590-0496 Mathematics and Computer Science DivisionAragonne National Laboratory 1996

[65] Th Gallay ldquoA center-stable manifold theorem for differentialequations in Banach spacesrdquo Communications in MathematicalPhysics vol 152 no 2 pp 249ndash268 1993

[66] J-M Ghidaglia and B Heron ldquoDimension of the attractorsassociated to the Ginzburg-Landau partial differential equa-tionrdquo Physica D vol 28 no 3 pp 282ndash304 1987

12 Journal of Function Spaces and Applications

[67] P Mattila Geometry of Sets and Measures in Euclidean Spacesvol 44 Cambridge University Press Cambridge UK 1995

[68] L HormanderThe Analysis of Linear Partial Differential Oper-ators vol 1 Springer Berlin Germany 2nd edition 1990

[69] H Amann Linear and Quasilinear Parabolic ProblemBikhauser Basel Switzerland 1995

[70] H-P Heinz ldquoOn the behaviour ofmeasures of noncompactnesswith respect to differentiation and integration of vector-valuedfunctionsrdquo Nonlinear Analysis Theory Methods amp ApplicationsA vol 7 no 12 pp 1351ndash1371 1983

[71] R P Agarwal M Meehan and D OrsquoRegan Fixed Point Theoryand Applications vol 141 of Cambridge Tracts in MathematicsCambridge University Press Cambridge UK 2001

[72] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer New York NY USA1983

[73] J Fleckinger-Pelle H G Kaper and P Takac ldquoDynamics of theGinzburg-Landau equations of superconductivityrdquo NonlinearAnalysis Theory Methods amp Applications A vol 32 no 5 pp647ndash665 1998

[74] H G Kaper and P Takac ldquoAn equivalence relation for theGinzburg-Landau equations of superconductivityrdquo Journal ofApplied Mathematics and Physics vol 48 no 4 pp 665ndash6751997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces and Applications 11

[27] J Banas and K GoebelMeasures of Noncompactness in BanachSpaces vol 60 ofLectureNotes in Pure andAppliedMathematicsMarcel Dekker New York NY USA 1980

[28] J Banas and M Lecko ldquoSolvability of infinite systems ofdifferential equations in Banach sequence spacesrdquo Journal ofComputational andAppliedMathematics vol 137 no 2 pp 363ndash375 2001

[29] J Banas and K Sadarangani ldquoCompactness conditions inthe study of functional differential and integral equationsrdquoAbstract and Applied Analysis vol 2013 Article ID 819315 14pages 2013

[30] M Mursaleen and S A Mohiuddine ldquoApplications of mea-sures of noncompactness to the infinite system of differentialequations in ℓ

119901spacesrdquo Nonlinear Analysis Theory Methods amp

Applications A vol 75 no 4 pp 2111ndash2115 2012[31] L Olszowy ldquoSolvability of some functional integral equationrdquo

Dynamic Systems and Applications vol 18 no 3-4 pp 667ndash6762009

[32] L Olszowy ldquoOn some measures of noncompactness in theFrechet spaces of continuous functionsrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 72 pp 2119ndash2126 2010

[33] L Olszowy ldquoOn existence of solutions of a quadratic Urysohnintegral equation on an unbounded intervalrdquo CommentationesMathematicae vol 48 no 1 pp 103ndash112 2008

[34] L Olszowy ldquoExistence ofmild solutions for semilinear nonlocalCauchy problems in separable Banach spacesrdquo Journal of Anal-ysis and its Applications vol 32 no 2 pp 215ndash232 2013

[35] L Olszowy ldquoFixed point theorems in the Frechet space 119862(119877+)

and functional integral equations on an unbounded intervalrdquoApplied Mathematics and Computation vol 218 no 18 pp9066ndash9074 2012

[36] J Banas and D OrsquoRegan ldquoOn existence and local attractivity ofsolutions of a quadratic Volterra integral equation of fractionalorderrdquo Journal of Mathematical Analysis and Applications vol345 no 1 pp 573ndash582 2008

[37] L D Landau EM Lifshitz and L P Pitaevski Electrodynamicsof Continuous Media vol 8 Butterworth-Heinemann 2ndedition 1984

[38] L D Landau and E M Lifshitz Fluid Mechanics vol bButterworth-Heinemann 2nd edition 1987

[39] R Temam Infinite-Dimensional Dynamical Systems inMechan-ics and Physics vol 68 Springer New York NY USA 2ndedition 1997

[40] J K Hale Asymptotic Behavior of Dissipative Systems vol 25 ofMathematical Surveys and Monographs American Mathemati-cal Society Providence RI USA 1988

[41] R A Adams and J J F Fournier Sobolev Spaces vol 140Elsevier 2nd edition 2003

[42] V Mazrsquoya Sobolev Spaces with Applications to Elliptic PartialDifferential Equations Springer Heidelberg Germany 2ndedition 2011

[43] V Georgescu ldquoSome boundary value problems for differentialforms on compact Riemannian manifoldsrdquo Annali di Matemat-ica Pura ed Applicata vol 122 pp 159ndash198 1979

[44] D Henry Geometric Theory of Semilinear Parabolic Equationsvol 840 of Lecture Notes in Mathematics Springer New YorkNY USA 1981

[45] V Girault and P-A Raviart Finite Element Methods for Navier-Stokes Equations Springer New York NY USA 1986

[46] J Bardeen L N Cooper and J R Schrieffer ldquoTheory ofsuperconductivityrdquo Physical Review vol 108 pp 1175ndash12041957

[47] A A Abrikosov Fundamental of the Theory of Metals North-Holland Amstedram The Netherlands 1988

[48] M Tinkham Introduction To Superconductivity McGraw-HillNew York NY USA 2nd edition 1996

[49] P DeGennes Superconductivity inMetals and Alloys BenjaminNew York NY USA 1966

[50] H Brezis Functional Analysis Sobolev Spaces and PartialDifferential Equations Springer New York NY USA 2011

[51] E Zeidler Nonlinear Functional Analysis and its Applicationsvol 2 Springer New York NY USA 1990

[52] A Schmid ldquoA time dependent Ginzburg-Landau equation andits application to the problem of resistivity in the mixed staterdquoPhysik der KondensiertenMaterie vol 5 no 4 pp 302ndash317 1966

[53] C M Elliot and Q Tang Existence Theorems For an Evolution-ary Superconductivity Model University of Sussex BrightonUK 1992

[54] Q Tang ldquoOn an evolutionary system of Ginzburg-Landauequations with fixed total magnetic fluxrdquo Communications inPartial Differential Equations vol 20 no 1-2 pp 1ndash36 1995

[55] Q Du ldquoGlobal existence and uniqueness of solutions of thetime-dependent Ginzburg-Landau model for superconductiv-ityrdquo Applicable Analysis vol 53 no 1-2 pp 1ndash18 1994

[56] Z M Chen K-H Hoffmann and J Liang ldquoOn a nonstation-ary Ginzburg-Landau superconductivity modelrdquoMathematicalMethods in the Applied Sciences vol 16 no 12 pp 855ndash875 1993

[57] J Liang and Q Tang ldquoAsymptotic behaviour of the solutionsof an evolutionaryGinzburg-Landau superconductivitymodelrdquoJournal of Mathematical Analysis and Applications vol 195 no1 pp 92ndash107 1995

[58] L P Gorrsquokov and G M Eliasherberg ldquoGeneralization of theGinzburg-Landau equations for an nonstationary problemsin the case of alloys with paramagnetic impartiesrdquo ZhurnalEksperimentalrsquonoi i Teoreticheskoi Fiziki vol 54 pp 612ndash6261968 Soviet Physics vol 27 pp 328ndash334 1968

[59] J Fleckinger-Pelle and H G Kaper ldquoGauges for the Ginzburg-Landau equations of superconductivityrdquo Zeitschrift fur Ange-wandte Mathematik und Mechanik vol 76 no 2 pp 345ndash3481996

[60] P Takac ldquoOn the dynamical process generated by supercon-ductivity modelrdquo Zeitschrift fur Angewandte Mathematik undMechanik vol 76 no 52 pp 349ndash352 1995

[61] W D Gropp H G Kaper DM Levine M Palumbo and VMVinokur ldquoNumerical simulation of vortex dynamics in type-IIsuperconductorsrdquo Journal of Computational Physics vol 123 no2 pp 254ndash266 1996

[62] D W Braun G W Crabtree H G Kaper et al ldquoStructure of amoving vortex latticerdquo Physical Review Letters vol 76 no 5 pp831ndash834 1996

[63] G W Crabtree G K Leaf H G Kaper et al ldquoTime-dependent Ginzburg-Landau simulations of vortex guidance bytwin boundariesrdquo Physica C vol 263 no 1ndash4 pp 401ndash408 1996

[64] G W Carabtree G K Leaf H G Kaper D W Braun V MVinokur andA E Koshelev ldquoDynamics vortex phases in super-conductors with correlated disorderrdquo Preprint ANLMCS-P590-0496 Mathematics and Computer Science DivisionAragonne National Laboratory 1996

[65] Th Gallay ldquoA center-stable manifold theorem for differentialequations in Banach spacesrdquo Communications in MathematicalPhysics vol 152 no 2 pp 249ndash268 1993

[66] J-M Ghidaglia and B Heron ldquoDimension of the attractorsassociated to the Ginzburg-Landau partial differential equa-tionrdquo Physica D vol 28 no 3 pp 282ndash304 1987

12 Journal of Function Spaces and Applications

[67] P Mattila Geometry of Sets and Measures in Euclidean Spacesvol 44 Cambridge University Press Cambridge UK 1995

[68] L HormanderThe Analysis of Linear Partial Differential Oper-ators vol 1 Springer Berlin Germany 2nd edition 1990

[69] H Amann Linear and Quasilinear Parabolic ProblemBikhauser Basel Switzerland 1995

[70] H-P Heinz ldquoOn the behaviour ofmeasures of noncompactnesswith respect to differentiation and integration of vector-valuedfunctionsrdquo Nonlinear Analysis Theory Methods amp ApplicationsA vol 7 no 12 pp 1351ndash1371 1983

[71] R P Agarwal M Meehan and D OrsquoRegan Fixed Point Theoryand Applications vol 141 of Cambridge Tracts in MathematicsCambridge University Press Cambridge UK 2001

[72] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer New York NY USA1983

[73] J Fleckinger-Pelle H G Kaper and P Takac ldquoDynamics of theGinzburg-Landau equations of superconductivityrdquo NonlinearAnalysis Theory Methods amp Applications A vol 32 no 5 pp647ndash665 1998

[74] H G Kaper and P Takac ldquoAn equivalence relation for theGinzburg-Landau equations of superconductivityrdquo Journal ofApplied Mathematics and Physics vol 48 no 4 pp 665ndash6751997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Journal of Function Spaces and Applications

[67] P Mattila Geometry of Sets and Measures in Euclidean Spacesvol 44 Cambridge University Press Cambridge UK 1995

[68] L HormanderThe Analysis of Linear Partial Differential Oper-ators vol 1 Springer Berlin Germany 2nd edition 1990

[69] H Amann Linear and Quasilinear Parabolic ProblemBikhauser Basel Switzerland 1995

[70] H-P Heinz ldquoOn the behaviour ofmeasures of noncompactnesswith respect to differentiation and integration of vector-valuedfunctionsrdquo Nonlinear Analysis Theory Methods amp ApplicationsA vol 7 no 12 pp 1351ndash1371 1983

[71] R P Agarwal M Meehan and D OrsquoRegan Fixed Point Theoryand Applications vol 141 of Cambridge Tracts in MathematicsCambridge University Press Cambridge UK 2001

[72] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer New York NY USA1983

[73] J Fleckinger-Pelle H G Kaper and P Takac ldquoDynamics of theGinzburg-Landau equations of superconductivityrdquo NonlinearAnalysis Theory Methods amp Applications A vol 32 no 5 pp647ndash665 1998

[74] H G Kaper and P Takac ldquoAn equivalence relation for theGinzburg-Landau equations of superconductivityrdquo Journal ofApplied Mathematics and Physics vol 48 no 4 pp 665ndash6751997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of