Comportement asymptotique de modèles

THÈSE

Pour l'obtention du grade deDOCTEUR DE L'UNIVERSITÉ DE POITIERS

UFR des sciences fondamentales et appliquéesLaboratoire de mathématiques et applications - LMA (Poitiers)

(Diplôme National - Arrêté du 7 août 2006)

École doctorale : Sciences et ingénierie pour l'information, mathématiques - S2IM(Poitiers)

Secteur de recherche : Mathématiques et leurs interactions

Présentée par :Haydi Israel

Comportement asymptotique de modèlesen séparation de phases

Directeur(s) de Thèse :Alain Miranville, Madalina Petcu

Soutenue le 05 décembre 2013 devant le jury

Jury :

Président Mikhael Balabane Professeur des Universités, Université de Paris 13

Rapporteur Pavel Krej i Professor, Prague Institute of Mathematics AS CR, Czech Republic

Rapporteur Maurizio Grasselli Professore Politecnique di Milano, Italia

Membre Alain Miranville Professeur des Universités, Université de Poitiers

Membre Madalina Petcu Maître de conférences, Université de Poitiers

Membre Jean-Michel Rakotoson Professeur des Universités, Université de Poitiers

Membre Frédéric Pascal Professeur des Universités, ENS de Cachan

Membre Laurence Cherfils Maître de conférences, Université de La Rochelle

Pour citer cette thèse :Haydi Israel. Comportement asymptotique de modèles en séparation de phases [En ligne]. Thèse Mathématiques etleurs interactions. Poitiers : Université de Poitiers, 2013. Disponible sur Internet <http://theses.univ-poitiers.fr>

false

Université de Poitiers

THÈSE

pour l’obtention du Grade de

Docteur de l’Université de Poitiers

(Faculté des Sciences Fondamentales et Appliquées)

(Diplôme National - Arrêté du 7 Août 2006)

École Doctorale: Sciences et Ingénierie pour l’Information,Mathématiques (S2IM)

Secteur de Recherche: Mathématiques et leurs Interactions

présentée par:

Haydi ISRAEL

**************************************************************************Comportement asymptotique de modèles en séparation de

phase**************************************************************************

Directeur de thèse : Alain MiranvilleCo-directeur de thèse : Madalina Petcu

Soutenue le Jeudi 05 Décembre 2013

devant la commission d’Examen

Jury

Maurizo Grasselli Professeur, Politecnique de Milan RapporteurPavel Krejci Professeur, Institut AS CR RapporteurMikhael Balabane Professeur, Université Paris 13 ExaminateurLaurence Cherfils MCF Habilité, Université de La Rochelle ExaminateurFrédéric Pascal Professeur, ENS de Cachan ExaminateurJean-Michel Rakotoson Professeur, Université de Poitiers ExaminateurAlain Miranville Professeur, Université de Poitiers Directeur de thèseMadalina Petcu MCF Habilité, Université de Poitiers co-directeur de thèse

À tous ceux que j’aime et qui m’aiment

Remerciements

J’adresse ici mes vifs remerciements à toutes les personnes qui m’ont accompa-gnée pendant ces quelques années par leur soutien et leurs encouragements et quiont contribué de près comme de loin à la réalisation de ce rapport de thèse.

Je voudrais commencer par mon Père et ma Mère à qui je serai toujours incapabled’exprimer ma reconnaissance. Je vous remercie chaleureusement pour vos encou-ragements, vos conseils, l’espoir que vous portez en moi et surtout d’avoir toujoursété là pour moi.

Merci également à Alain et Madalina, mes deux directeurs de thèse, pour m’avoirtoujours soutenue depuis mon arrivée à Poitiers, m’avoir initiée à la recherche en di-rigeant mon mémoire de Master, m’avoir fait confiance en me proposant cette thèseet m’avoir guidée jusqu’à son aboutissement. Je vous remercie aussi pour votre dis-ponibilité, vos conseils, votre soutien et votre patience que vous m’avez témoignéetout au long de la réalisation de ce mémoire et surtout de me faire bénéficier devotre grand savoir.

Je voudrais aussi remercier les Professeurs Maurizo Grasselli et Pavel Krejcid’avoir eu l’extrême gentillesse d’être rapporteurs de cette thèse. Je remercieégalement Mikhael Balabane, Laurence Cherfils, Frédéric Pascal et Jean-MichelRakotoson d’avoir accepté de faire partie de mon jury.

Je tiens à remercier tous les membres du Laboratoire de Mathématiques etApplications qui ont contribué à créer une ambiance conviviale et un cadre de tra-vail agréable. Je remercie en particulier le directeur du laboratoire Pol Vanhaecke,les professeurs Pierre Torasso et Abderrazak Bouaziz pour leur excellent accueil etleur écoute. J’en profite aussi pour remercier Professeur Alain cimetière, tous mesprofesseurs de l’Ecole Nationale Orthodoxe et tous mes enseignants de l’UniversitéLibanaise.

Je souhaite également remercier les personnels dont la bonne humeur constanteest un plaisir quotidien : Brigitte Brault au secrétariat, Nathalie Mongin à la comp-tabilité, Nathalie Marlet à la bibliothèque, Jocelyne Attab à la bibliothèque et à lareprographie et Benoît Métrot au service informatique. Merci de m’avoir remontéele moral et donnée un mot gentil quand j’en avais besoin.

Je tiens aussi à remercier Mme Barbara Mérigeault, Mme Sylvie Perez, Mme

v

Nathalie Fofana et Mme Marie-Thérèse Péguin pour tous les services rendus.

À tous mes amis Docteurs ou futurs docteurs au laboratoire, je vous dit aussimerci.

À Rim, Abdallah, Firas, Roukaya, Nazek, Ziad, Ali, Shiraz, Khaoula et particu-lièrement Mme Yvette Moret, je vous remercie d’avoir été pour moi une deuxièmefamille en France et encouragée pendant les moments les plus difficiles.

Pour compléter mes remerciements, je me dois de remercier mes deux frèresAyman et Mazen ainsi que ma belle famille et tous les autres membres de ma famille.Je vous remercie pour votre soutien, vos encouragements et vos prières.

Du fond du coeur, un grand MERCI à Houssam et à ma belle Naya. Je suisincapable d’exprimer toute la reconnaissance, la fierté et le profond amour que jevous porte pour tout ce que vous avez fait pour moi. Sachez que c’est une grandechance d’avoir une famille comme vous.

Merci à vous tous !

vi

Table des matières

Remerciements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable des matières . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Introduction 1

1 Well-Posedness and Long Time Behavior of an Allen-Cahn TypeEquation 71.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2 Existence and uniqueness results . . . . . . . . . . . . . . . . . . . . . 11

1.2.1 A priori estimates . . . . . . . . . . . . . . . . . . . . . . . . . 121.2.2 Passage to the limit . . . . . . . . . . . . . . . . . . . . . . . . 141.2.3 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.3 Additional regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.4 Existence of the global attractor . . . . . . . . . . . . . . . . . . . . . 191.5 Existence of an exponential attractor . . . . . . . . . . . . . . . . . . 24

2 Well-Posedness and Long Time Behavior of a Perturbed Cahn-Hilliard System with Regular Potentials 292.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.2 Notation and assumptions . . . . . . . . . . . . . . . . . . . . . . . . 322.3 Uniform a priori estimates. Existence and uniqueness of solutions. . . 352.4 Additional regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.5 Existence of the global attractor . . . . . . . . . . . . . . . . . . . . . 472.6 Existence of an exponential attractor . . . . . . . . . . . . . . . . . . 512.7 Construction of a robust family of exponential attractors . . . . . . . 55

3 A Cahn-Hilliard Type Equation With Dynamic Boundary Condi-tions and Regular Potentials 633.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.1.1 Assumptions and notations . . . . . . . . . . . . . . . . . . . 643.2 Uniform a priori estimates . . . . . . . . . . . . . . . . . . . . . . . . 653.3 Existence of solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . 733.4 Global and exponential attractor . . . . . . . . . . . . . . . . . . . . 75

4 Numerical Analysis of a Cahn-Hilliard Type Equation With Dyna-mic Boundary Conditions 814.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.2 Assumptions and notation . . . . . . . . . . . . . . . . . . . . . . . . 82

vii

TABLE DES MATIÈRES

4.3 The semi-discrete scheme . . . . . . . . . . . . . . . . . . . . . . . . . 834.4 Error estimates for the space semi-discrete scheme . . . . . . . . . . . 874.5 Stability of the backward Euler scheme . . . . . . . . . . . . . . . . . 964.6 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5 Long Time Behavior of an Allen-Cahn Type Equation With a Sin-gular Potential and Dynamic Boundary Conditions 1035.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.2 Approximations and uniform a priori estimates . . . . . . . . . . . . . 1065.3 Variational formulation and well-posedness . . . . . . . . . . . . . . . 1165.4 Additional regularity results and separation from the singularities . . 1235.5 Attractors and exponential attractors . . . . . . . . . . . . . . . . . . 127

Bibliographie 135

viii

Introduction

Cette thèse réunit un certain nombre de résultats théoriques et numériques relatifs àun problème de type Cahn-Hilliard. Ces résultats portent sur le caractère bien posédu problème ainsi que sur l’étude du comportement asymptotique des solutions entermes d’existence de l’attracteur global et d’attracteurs exponentiels.

• Présentation du problème :

Dans cette thèse, on considère le problème de type Cahn-Hilliard suivant :

∂tu+∆2u−∆u−∆f(u) + f(u) = 0,u(0, x) = u0(x),

(1)

où Ω est un domaine régulier et borné, f est une fonction non linéaire et u est leparamètre d’ordre qui représente la concentration locale de l’un des deux composantsdu mélange.En posant w = −∆u + f(u), le problème (1) peut être formulé comme étant unsystème de deux équations de second ordre :

∂tu = ∆w − w,w = −∆u+ f(u),u(0, x) = u0(x).

(2)

Ce problème a été introduit récemment et étudié dans [30], [31], [32], [37]. Ilreprésente une simplification d’un modèle mésoscopique des mécanismes micro-scopiques dans les processus de surface tels que la diffusion de surface et l’adsorption-désorption. Le modèle mésoscopique correspondant est une combinaison de la dy-namique d’Arrhenius adsorption/désorption, de la diffusion Metropolis de surface etd’une simple réaction unimoléculaire. Il est décrit par :

∂tu−D∇· [∇u−βu(1−u)∇Jm∗u]− [kap(1−u)−kdu exp(−βJd∗u)]+kru = 0, (3)

où D > 0 représente la constante de diffusion, kr, kd et ka désignent respectivementles constantes de réaction, de desorption et d’adsorption, p est la pression partiellede l’espèce gazeuse, Jd et Jm sont les potentiels inter-moléculaires de la désorptionet la migration de surface. Pour plus de détails, on réfère [30].

On note que l’on peut voir comme une combinaison de l’équation de Cahn-Hilliard

∂tu = −∆2u+∆f(u), u(0, x) = u0(x)

1

INTRODUCTION

et de l’équation d’Allen-Cahn

∂tu = ∆u− f(u), u(0, x) = u0(x).

L’équation de Cahn-Hilliard a été très étudiée, tant d’un point de vue math-ématique que d’un point de vue numérique (voir [7], [10], [14], [42]). Ce modèledécrit le processus de séparation de phase dans un alliage binaire, en particulier,la décomposition spinodale. Un tel processus peut être observé lorsqu’un alliagebinaire, de composition homogène à température élevée, avec une concentrationuniforme de chacune des deux phases, est brutalement refroidi; le matériau devientnon homogène, les phases se séparant en domaines de concentration relativementplus élevée en l’une ou l’autre des phases.

De même, l’équation d’Allen-Cahn, aussi connue des mathématiciens sous le nomd’équation de la chaleur semi-linéaire, a été étudiée du point de vue mathématiqueet numérique (voir [1], [53], [57]). Cette équation est fondamentale en science desmatériaux et décrit un modèle simplifié d’adsorption et de désorption à partir d’unesurface.

Dans cette thèse, deux différents types de conditions au bord ont été proposés etpour chaque type on donne deux conditions sur le bord vu que (1) est une équation duquatrième ordre en espace. On propose également différents types de nonlinéarités.

D’abord, on considère des conditions de type Dirichlet, typiquement :

u = ∆u = 0, sur le bord ∂Ω.

On désigne par E l’énergie de l’équation, elle est donnée par :

E(u) =∫

Ω

(1

2|∇u|2 + F (u))dx,

où le potentiel F est une primitive de f avec f(0) = 0. On remarque que E est unefonctionnelle de Lyapunov c’est à dire que E(u(t)) décroît avec le temps t. En effet,

d

dtE(u(t)) = −

∫

Ω

(|∇w|2 + |w|2)dx, t > 0.

Ensuite, on considère des conditions dynamiques sur le bord, typiquement,

∂nw = 0, x ∈ ∂Ω, (4)

∂tu = ∆Γu− g(u)− λu− ∂nu, x ∈ ∂Ω, (5)

où ∆Γ est l’opérateur de Laplace-Beltrami sur le bord et ∂n est la dérivée normaleextérieure. Cette appelation vient du fait que la dérivée de u par rapport au temps,∂tu, apparaît explicitement dans (5).La condition (5) a été récemment introduite par les physiciens afin de tenir comptedes interactions entre les deux constituants du mélange et la paroi (le bord ∂Ω)dans les systèmes confinés et décrire l’influence de cette dernière sur le processusde séparation de phases. Cependant, la condition (4) signifie qu’il ne peut y avoiraucun échange des constituants du mélange à travers la paroi. Pour plus de détails

2

INTRODUCTION

sur l’équation de Cahn-Hilliard classique avec des conditions dynamiques sur le bord,on réfère [19], [33], [48], [49]. Avec ce type de conditions, l’énergie libre de l’équationest donnée par :

E(u) =∫

Ω

(

1

2|∇u|2 + F (u)

)

dx+

∫

∂Ω

(

1

2|∇Γu|2 +

λ

2|u|2 +G(u)

)

dσ,

oùG est une primitive de g. La première intégrale est l’énergie de volume du matériauet la seconde intégrale est l’énergie de surface. De même, on a une dissipationd’énergie,

d

dtE(u(t)) = −

∫

Ω

(|∇w|2 + |w|2)dx−∫

∂Ω

|∂tu|2dσ, t > 0.

Outre l’existence et l’unicité des solutions, on établit le comportement asympto-tique des solutions. On cherche l’attracteur global qui, lorsqu’il existe, est l’uniqueensemble compact, invariant par le semi-groupe, qui attire toutes les solutions duproblème lorsque le temps t tend vers +∞ et qui est minimal parmi les ensemblesvérifiant cette définition.Cependant, l’attracteur global est sensible aux perturbations et cela est lié à lavitesse d’attraction de trajectoires qui peut être lente. Dans le but de corriger cedéfaut, on a introduit dans [9] la notion d’attracteur exponentiel.Par définition, un attracteur exponentiel est un ensemble compact qui contientl’attracteur global, qui est beaucoup plus robuste face aux perturbations, qui estde dimension fractale finie et qui attire à vitesse exponentielle toutes les solutionsdu problème. Il a été d’abord construit en démontrant la propriété de laminage quin’est valable que dans des espaces de Hilbert et n’est pas vraie dans des espaces deBanach, voir [1], [9]. Récement, on donne dans [10] une construction d’attracteursexponentiels plus générale, pour des espaces de Banach, construction qui consiste àvérifier une propriété de régularisation sur la différence de deux solutions.Vu qu’un attracteur exponentiel est robuste face aux perturbations, on proposedans cette thèse d’étudier également son existence. Une fois l’existence établie, onen déduit que la dimension fractale de l’attracteur global est finie.

• Présentation des résultats et plan de la thèse :

La thèse est organisée comme suit :Dans le premier chapitre, on considère le problème (1) avec des conditions aux

limites de type Dirichlet et une nonlinéarité f régulier, typiquement,

f(s) =

2p−1∑

i=1

aisi, p ∈ N, p > 2, a2p−1 > 0.

On démontre l’existence et l’unicité de la solution faible qui correspond à une donnéeinitiale dansH1

0 (Ω)∩L2p(Ω) et l’existence de la solution forte quand la donnée initialeest dans H2(Ω) ∩ H1

0 (Ω). De plus, on étudie le comportement asymptotique dessolutions et on démontre l’existence de l’attracteur global dans H−1(Ω) qui est borné

3

INTRODUCTION

dans H2(Ω) ∩H10 (Ω). On prouve également l’existence d’un attracteur exponentiel

en vérifiant une propriété de régularisation H−1(Ω)−L2(Ω) sur la difference de deuxsolutions.

Dans le deuxième chapitre, on considère une équation perturbée du problème (1)avec le même type de conditions aux limites du deuxième chapitre :

∂tu+∆2u−∆f(u)− ε∆u+ εf(u) = 0, dans Ω,u = ∆u = 0 sur ∂Ω,u|t=0 = u0,

(6)

où ε > 0. La fonction f satisfait cette fois ci les conditions suivantes :

f ∈ C2(R),f(0) = 0,f ′(s) > −κ,f(s)s > pb2ps

2p − c > c(s2 + s2p)− c,|f(s)| 6 c(|s|2p−1 + 1).

(7)

En ce qui concerne le caractère bien posé du problème (6), la régularité des solutionset l’existence de l’attracteur global, on a des résultats analogues à ceux du premierchapitre. Cependant, on prouve l’existence d’un attracteur exponentiel en vérifiantune propriété de régularisation H−1(Ω)−H2(Ω) sur la difference de deux solutions.À la fin de ce chapitre, on démontre l’existence d’une famille robuste d’attracteursexponentiels, on étudie la limite lorsque ε tend vers 0 et on prouve la continuité desattracteurs exponentiels du problème perturbé vers un attracteur exponentiel duproblème non perturbé qui n’est autre que le problème de Cahn-Hilliard classique.

Dans les chapitres 3 et 4, on donne une étude théorique et numérique pourle problème (1) avec des conditions aux limites dynamiques et des nonliéaritésrégulières :

∂tu = ∆w − w, ∂nw = 0, x ∈ ∂Ωw = −∆u+ f(u), u|t=0 = u0,∂tv = ∆Γv − g(v)− λv − ∂nu, x ∈ ∂Ω, v|t=0 = v0u|∂Ω = v.

(8)

Les fonctions f et g sont dans C2(R) satisfaisant les conditions de dissipativitésuivantes :

lim inf|s|→+∞

f ′(s) > 0, lim inf|s|→+∞

g′(s) > 0. (9)

D’abord, on démontre l’existence et l’unicité de la solution (u(t), v(t)) dansL∞([0, T ],W) avec (∂tu, ∂tv) dans L2([0, T ],V) où

W := (u, v) ∈ H2(Ω)×H2(Γ), w = −∆u+ f(u) ∈ H1(Ω), u|Γ = v, ∂nw|Γ = 0

etV := (u, v) ∈ H1(Ω)×H1(Γ), u|Γ = v.

De plus, on démontre l’existence de l’attracteur global A ⊂ W ∩ H3(Ω) × H3(Γ)et également l’existence d’un attracteur exponentiel en vérifiant une propriété derégularisation H−1(Ω)× L2(Γ)−H2(Ω)×H2(Γ).

4

INTRODUCTION

Ensuite, on effectue l’analyse numérique du problème. On considère un domaineΩ correspondant à des conditions aux limites périodiques telles que :

Ω = Πd−1i=1 (IR/(LiZ))× (0, Ld), Li > 0, i = 1, ..., d, d = 2 or 3,

avec la frontière régulière

Γ = ∂Ω = Πd−1i=1 (IR/(LiZ))× 0, Ld .

On propose une discrétisation en espace par des éléments finis et on démontrel’existence et l’unicité de la solution de la version discrète associée à la formula-tion variationnelle de (8). Puis, on prouve des estimations d’erreur optimales danscertaines normes pour la différence uh − u entre la solution approchée de la versiondiscrète et la solution exacte du problème continu lorsque le pas de maillage h tendvers 0.On aborde aussi l’étude du problème totalement discrétisé. Pour la discrétisation entemps et en espace, on utilise un schéma d’Euler implicite en temps et semi-discrétiséen espace et on démontre l’existence, l’unicité et la stabilité de la solution. Ces ré-sultats sont illustrés par des simulations numériques en dimension deux d’espaceréalisées avec FreeFem++, simulations qui permettent d’étudier l’influence des dif-férents paramètres.

Dans le dernier chapitre, on considère (1) avec des conditions dynamiques sur lebord et f singulier :

∂tu = ∆w − w, ∂nw = 0, x ∈ ∂Ω,w = −∆u+ f(u)− λu, u|t=0 = u0,∂tv = ∆Γv − g(v)− ∂nu, x ∈ ∂Ω, v|t=0 = v0,u|Γ = v.

(10)

Les fonctions f et g satisfont respectivement

f ∈ C2((−1, 1)),f(0) = 0, lim

s→±1f(s) = ±∞,

f ′(s) > 0, lims→±1

f ′(s) = +∞,

f ′′(s) sgn s > 0,

(11)

g(σ) = σ + g0(σ), ∀σ ∈ R, où ‖g0‖C2(R) := C0 < +∞. (12)

Tout d’abord, on approche f par des fonctions régulières et on obtient des esti-mations à priori uniformes sur les solutions correspondantes. Ensuite, on définit lasolution variationnelle pour le problème et on étudie l’existence de la solution au sensusuel. On termine ce chapitre par le comportement asymptotique des solutions.

5

INTRODUCTION

6

Chapitre 1

Well-Posedness and Long Time

Behavior of an Allen-Cahn Type

Equation

Sur le caractère bien posé et le com-portement asymptotique d’une équa-tion de type Allen-Cahn

Ce chapitre est constitué de l’article Well-posedness and long time behaviorof an Allen-Cahn type equation, paru en 2013 dans Communication on Pureand Applied Analysis, volume 12, numero 6, pages 2811-2827.

Communications on Pure and Applied Analysis

Volume 12, Number 6, pp. 2811-2827, November 2013

Well-posedness and long time behavior of an

Allen-Cahn type equation

Haydi ISRAEL

UMR 7348 CNRS. Laboratoire de Mathématiques et Applications - Université de Poitiers

SP2MI - Boulevard Marie et Pierre Curie - Téléport 2

BP30179 - 86962 Futuroscope Chasseneuil Cedex - FRANCE.

Abstract. The aim of this article is to study the existence and uniqueness of so-lutions for an equation of Allen-Cahn type and to prove the existence of the finite-dimensional global attractor as well as the existence of exponential attractors.

1.1 Introduction

In this article we are interested in the study of the following partial differentialequation, considered in a smooth and bounded domain Ω ⊂ R

n with boundary ∂Ω:

∂tu+∆2u−∆f(u)−∆u+ f(u) = 0 in Ω,u = ∆u = 0 on ∂Ω,

(1.1)

where f is a polynomial of order 2p− 1

f(s) =

2p−1∑

i=1

aisi, p ∈ N, p > 2.

This equation is associated with the effect of multiple microscopic mechanisms suchas surface diffusion and adsorption/desorption and it was recently derived and stud-ied in [30], [32], [37]. We note that equation (1.1) may be viewed as a combinationof the well-known Cahn-Hilliard equation

∂tu = −∆(∆u− f(u)), u(0, x) = u0(x)

and of the Allen-Cahn equation

∂tu = ∆u− f(u), u(0, x) = u0(x).

We recall that the Cahn-Hilliard equation describes the behavior of two-phase sys-tems, in particular in spinodal decomposition, meaning in the case of rapid sepa-rations of phases when the material is cooled down sufficiently. The Allen-Cahn

1 . Well-Posedness and Long Time Behavior of an Allen-Cahn Type Equation

equation is also used in the study of two-phase systems and it describes the orderingof atoms within unit cells on a lattice.We denote by F the primitive of f vanishing at u = 0,

F (s) =

2p∑

j=2

bisj, jbj = aj−1, 2 6 j 6 2p,

and we assume that the leading coefficient of f (and g) is positive

a2p−1 = 2pb2p > 0.

Since a2p−1 > 0, it is easy to conclude that there exists two constants c1 and c2 suchthat

a2p−1

2s2p − c1 6 f(s)s 6 3

2a2p−1s

2p + c1, ∀s ∈ R;f ′(s) > a2p−1

2ps2p−2 − c2 > −κ, ∀s ∈ R.

(1.2)

Furthermore, there exists a constant c3 such that:

1

4pa2p−1s

2p − c3 6 F (s) 63

4pa2p−1s

2p + c3, ∀s ∈ R. (1.3)

For the mathematical setting of the problem, we introduce the following space H =L2(Ω), which we endow with the scalar product (., .) and the norm |.|.Let

A2 = −∆ : D(A2) ⊂ H → H and A1 = −∆+ I : D(A1) ⊂ H → H

with D(A1) = D(A2) = W :=

u ∈ H2, u = 0 on ∂Ω

. A1 is a strictly positive

self-adjoint linear operator with compact inverse A−11 . We set V = D(A

1/22 ) and we

endow V with the scalar product ((., .)) = (A1/22 ., A

1/22 .) and the norm ‖.‖ = |A1/2

2 .|.More generally, we endowD(As

2), s ∈ R, with the scalar product ((., .))s = (As2., A

s2.)

and the norm ‖.‖s = |As2.|.

Then, problem (1.1) can be reformulated as follows:

A−11 ∂tu+ A2u+ f(u) = 0 in Ω,

u = ∆u = 0 on ∂Ω.(1.4)

The variational formulation of problem (1.4) reads:Find u : [0, T ] → V such that

(

A−11 ∂tu, q

)

+ (A2u, q) + (f(u), q) = 0, ∀q ∈ V,u(0) = u0,

(1.5)

∀T > 0.Our aim is to prove the existence and uniqueness of solutions, as well as the exis-tence of a finite-dimensional attractor for the proposed model (1.1). This article isstructured as follows. In section 2 we consider problem (1.1) with Dirichlet bound-ary conditions and we prove the existence and uniqueness of solutions. Section 3is dedicated to the proof of the existence of the global attractor and in Section 4we prove a stronger result, namely the existence of an exponential attractor, whichimplies the finite-dimensionality of the global attractor.

10

1.2 Existence and uniqueness results


In this section, we establish the existence and uniqueness of a solution for problem(1.4). The proof is based on a priori estimates and on the Faedo-Galerkin scheme. Todo this, let us first consider the eigenfunctions ejj of the operator A1, A1ej = λjejwith ej ∈ V for all j ∈ N. We know that the eigenfunctions ej form an orthogonalbasis in H and V and the family of ejj may be assumed to be normalized in thenorm of H, i.e.,

(ei, ej) = δij,

where

δij =

1 if i = j,0 otherwise.

We denote by Em the space

Em = span e1, e2, . . . , em ,and by Pm the orthogonal projection from V onto Em:

Pmh =m∑

j=1

(h, ej) ej.

For any m ∈ N, we look for functions of the form

um (t) =m∑

j=1

umj(t) ej,

solving the approximate problem below:

d

dt

A−11

m∑

j=1

umj(t) ej , ei

+

A2

m∑

j=1

umj(t) ej , ei

+ (f (um) , ei) = 0, i = 1, ...,m,

um (0) = Pmu0.

(1.6)We rewrite problem (1.6) as follows:

M1dY

dt+M2Y + F (Y ) = 0,

where M1 =( (

A−11 (ei), ej

) )

i,j=1,...,m, M2 =

(

(A2(ei), ej))

i,j=1,...,m,

Y =

um1

...umm

and F (Y ) =

(f(um), e1)...

(f(um), em)

.

We can easily check that matrix M1 is invertible. Indeed, setting X =

x1...xm

and

X ′ =m∑

i=1

xiei, we have

(M1X,X) =m∑

j=1

m∑

i=1

(

A−11 ej, ei

)

xixj =(

A−11 X

′

, X′

)

> 0,

11


due to the fact that A−11 is a positive definite operator. Moreover, the matrix M2 is

positive definite and F (Y ) depends continuously on Y. Applying Cauchy’s theoremfor a system of ordinary differential equations, we find that there exists a time

tm ∈ (0, T ) and a unique solution Y for the equationdY

dt+M−1

1 M2Y +M−11 F (Y ) = 0

on the time interval t ∈ [0, tm[. Based on the a priori estimates with respect to tthat will be derived below for the solution um(t), we obtain that any local solutionof (1.6) is actually a global solution that is defined on the whole interval [0, T ].Now, we give the a priori estimates for the solution um(t):

1.2.1 A priori estimates

The solution um(t) verifies the following approximate problem:

A−11 ∂tum + A2um + f(um) = 0,

um (0) = Pmu0.(1.7)

Multiplying equation (1.7) by A1um(t) and integrating over Ω, we obtain:

1

2

d

dt|um(t)|2 + |A2um(t)|2 + |A1/2

2 um(t)|2 +∫

Ω

f(um) · A1um(t)dx = 0.

Using the fact that∫

Ω

f(um(t)) · um(t)dx >a2p−1

2

∫

Ω

u2pm (t)dx− c1|Ω|,

and

(f(um(t)), A2um(t)) =(f ′(um(t))A1/22 um(t), A

1/22 um(t))

>a2p−1

2p

∫

Ω

u2pm (t)|A1/22 um(t)|2dx− c2

∫

Ω

|A1/22 um(t)|2dx,

we find:

1

2

d

dt|um(t)|2+|A2um(t)|2 + |A1/2

2 um(t)|2 +a2p−1

2

∫

Ω

u2pm (t)dx

+a2p−1

2p

∫

Ω

u2pm (t)|A1/22 um(t)|2dx

6c1|Ω|+ c2

∫

Ω

|A1/22 um(t)|2dx 6 c1|Ω|+ c2 (A2um(t), um(t))

6c1|Ω|+1

2|A2um(t)|2 + c

′

3|um(t)|2.

(1.8)

From (1.8), we deduce:

1

2

d

dt|um(t)|2+

1

2|A2um(t)|2 + |A1/2

2 um|2 +a2p−1

2

∫

Ω

u2pm (t)dx

+a2p−1

2p

∫

Ω

u2pm (t)|A1/22 um(t)|2dx

6c1|Ω|+ c′

3|um(t)|2.

(1.9)

12


In particular, we have

1

2

d

dt|um(t)|2 6 c1|Ω|+ c

′

3|um(t)|2. (1.10)

Consequently, applying Gronwall’s inequality to (1.10), we find:

|um(t)|2 6

(

|u0|2 +c1c′

3

|Ω|)

ec′

3T

6 c(u0, T ),

where c(u0, T ) denotes a constant depending on the time T and the initial datumu0 but independent of m. Then, sup

t|um(t)|2 is bounded independently of m, which

implies the global existence of the solutions.Integrating equation (1.9) with respect to t, from 0 to t, we obtain:

1

2|um(t)|2 +

1

2

∫ t

0

|A2um(s)|2ds+∫ t

0

|A1

2

2 um(s)|2ds+a2p−1

2

∫ t

0

∫

Ω

u2pm (s)dxds

+a2p−1

2p

∫ t

0

∫

Ω

u2pm (s)|A1/22 um(s)|2dxds

6c1|Ω|T + c′

3

∫ t

0

|um(s)|2ds+1

2|um(0)|2

6c1|Ω|T + c′

3T sup |um(t)|2 +1

2|u(0)|2.

(1.11)

Due to equation (1.11), we conclude that umm is bounded in L2(0, T ;W ),L∞ (0, T ;H) and L2p (0, T ;L2p (Ω)), independently of m.Multiplying (1.7) by ∂tum and integrating over Ω, we have:

|A−1/21 ∂tum(t)|2 +

1

2∂t|A1/2

2 um(t)|2 + ∂t

∫

Ω

F (um(t))dx = 0. (1.12)

Integrating (1.12) with respect of t, we find:∫ t

0

|A−1/21 ∂tum(s)|2ds+

1

2|A1/2

2 um(t)|2 +∫

Ω

F (um(t))dx

=1

2|A1/2

2 um(0)|2 +∫

Ω

F (um(0))dx.

(1.13)

Using (1.3),(1.13) implies:∫ t

0

|A−1/21 ∂tum(s)|2ds+

1

2|A1/2

2 um(t)|2 +a2p−1

4p

∫

Ω

u2pm (t)dx

62c3|Ω|+1

2|A1/2

2 um(0)|2 +3a2p−1

4p

∫

Ω

u2pm (0)dx.

Consequently, if u0 ∈ V ∩ L2p(Ω), we deduce that:

um(t) ∈ L∞(0, T ;V ∩ L2p(Ω))

and∂tum(t) ∈ L2(0, T ;H−1(Ω)).

13


1.2.2 Passage to the limit

In this section, we intend to pass to the limit asm→ ∞ and study the convergence of

the sequence (um)m. Let q satisfy1

2p+1

q= 1, which implies q < 2. According to the

a priori estimates derived in the previous section, we have ‖um‖L2(0,T ;W ) uniformlybounded and consequently, (um)m is bounded in Lq(0, T ;Lq(Ω)). Moreover, we have‖f(um)‖Lq(0,T ;Lq(Ω)) uniformly bounded which implies that (A−1

1 ∂tum) is bounded inLq(0, T ;Lq(Ω)). It follows that:

‖∂tum‖Lq(0,T ;W−2,q(Ω)) 6 C.

Starting from this point, all convergence relations will be intended to hold up to theextraction of suitable subsequences, generally not relabelled. Thus, we observe thatweak and weak star compactness results applied to the sequence umm entail thatthere exists a function u such that the following properties hold:

um → u weakly in Lq(0, T ;W ), (1.14)

∂tum → ∂tu weakly in Lq(0, T ;W−2,q(Ω)), (1.15)

as m → ∞. It follows from (1.14), (1.15) and Aubin-Lions compactness theorem,that:

um → u strongly in Lq(0, T ;Lq(Ω)).

Consequently um(t, x) → u(t, x) a.e. (t, x) ∈ [0, T ]× Ω.Moreover, we have:

um(t, x) → u(t, x) a.e.f is a continuous polynomial function

=⇒ f(umi(t, x)) → f(u(t, x)) a.e.

f(umi(t, x)) → f(u(t, x)) a.e.

||f(umi)||Lq(ΩT ) 6 constant

=⇒ f(umi) → f(u) weakly in Lq(ΩT ).

Finally, we deduce that A−11 ∂tum A−1

1 ∂tu weakly in Lq(ΩT ). Thus, passing tothe limit in (1.7), we obtain:

A−11 ∂tu+ A2u+ f(u) = 0, in Lq(ΩT ). (1.16)

We also need to prove that u(0) = u0. To do this, we consider a test functionψ ∈ C1([0, T ];L2p(Ω)) such that ψ(T ) = 0. Multiplying (1.16) by ψ and integratingover Ω× [0, T ], we obtain:

∫ T

0

⟨

A−11 ∂tu, ψ

⟩

dt+

∫ T

0

〈A2u, ψ〉 dt+∫ T

0

〈f(u), ψ〉 dt = 0. (1.17)

Integrating by parts in (1.17), we have:

−∫ T

0

⟨

A−11 u, ∂tψ

⟩

dt−⟨

A−11 u(0), ψ(0)

⟩

+

∫ T

0


0

〈f(u), ψ〉 dt = 0.

(1.18)

14


Multiplying (1.7) by ψ and integrating over Ω× [0, T ], we obtain:

−∫ T

0

⟨

A−11 um, ∂tψ

⟩

dt−⟨

A−11 Pmu(0), ψ(0)

⟩

+

∫ T

0

〈A2um, ψ〉 dt+∫ T

0

〈f(um), ψ〉 dt = 0.

(1.19)

Having ψ ∈ L2p(0, T ;L2p(Ω)) and∂ψ

∂t∈ L2(0, T ;H) we deduce that:

∫ T

0

〈A2um, ψ〉 dt→∫ T

0

〈A2u, ψ〉 dt,

and∫ T

0

⟨

A−11 um, ∂tψ

⟩

dt→∫ T

0

⟨

A−11 u, ∂tψ

⟩

dt.

Then, passing to the limit in (1.19) as n→ ∞, we obtain:

−∫ T

0

⟨

A−11 u, ∂tψ

⟩

dt−⟨

A−11 u0, ψ(0)

⟩

+

∫ T

0


0

〈f(u), ψ〉 dt = 0. (1.20)

We deduce from (1.18) and (1.20) that A−11 u(0) = A−1

1 u0, which implies u(0) = u0.

1.2.3 Uniqueness

In what follows we need to prove the uniqueness of solutions for equation (1.4). Letu and v be two solutions of (1.4) on the time interval [0, T ]. We set w = u − v, wverifies the following equation:

(

A−11 ∂tw, q

)

+ (A2w, q) + (f(u)− f(v), q) = 0, ∀q ∈ W,w(0) = u(0)− v(0).

(1.21)

Multiplying (1.21) by w and integrating over Ω, we obtain:

1

2

d

dt|A−1/2

1 w|2 + |A1/22 w|2 + (f(u)− f(v), w) = 0. (1.22)

Using the fact that f ′(s) > −κ and f(u)− f(v) = l(t)w, with l defined as:

l(t) =

∫ 1

0

f ′ (su(t) + (1− s)v(t)) ds,

we have:(f(u)− f(v), w) > −κ|w|2.

Using the following interpolation inequality

|w|2L2(Ω) 6 c||w||H−1(Ω)|A1/22 w|,

we obtain:1

2

d

dt||w||2−1 +

1

2|A1/2

2 w|2 6 c2κ2

2||w||2−1. (1.23)

15


Applying Gronwall’s inequality to (1.23), we find:

||w(t)||2−1 6 ||u(0)− v(0)||2−1ec2κ2t. (1.24)

Relation (1.24) shows the continuous dependence of the solution on the initial dataand in particular, when u(0) = v(0), it implies the uniqueness of the solution.Due to previous estimates, we can conclude on the following result:

Theorem 1.2.1. Let us take u0 ∈ H. Then, there exists a unique solution u ofproblem (1.1) with initial datum u0 such that:

u ∈ L2([0, T ];W ) ∩ L∞([0, T ];H).

Furthermore, if u0 ∈ V ∩ L2p(Ω), then:

u ∈ C([0, T ];V ∩ L2p(Ω)) ∩ L2([0, T ];W ) and ∂tu ∈ L2([0, T ];H−1(Ω)).

1.3 Additional regularity

In this section, we will derive some additionnal regularity for the solution u(t).

Lemma 1.3.1. Let u(t) be a solution of (1.4) with u0 ∈ W . Then, there exists atime T0 = T0(‖u0‖H2(Ω)), 0 < T0 < 1/2, and a monotonic function Q such that:

|A2u(t)| 6 Q(‖u0‖H2(Ω)), t 6 T0(‖u0‖H2(Ω)). (1.25)

Proof:We rewrite equation (1.4) in the following equivalent form:

du

dt+ (I + A2)A2u+ (I + A2)f(u) = 0, u|∂Ω = ∆u|∂Ω = 0. (1.26)

Multiplying equation (1.26) by A22u(t), we obtain the following inequality:

1

2

d

dt|A2u(t)|2 + |A3/2

2 u(t)|2 + |A22u|2 =−

(

(I + A2)f(u(t)), A22u(t)

)

61

2|(I + A2)f(u(t))|2 +

1

2|A2

2u(t)|2

61

2‖f(u(t))‖2H2(Ω) +

1

2|A2

2u(t)|2.

(1.27)

We recall that f ∈ C2(Ω) and that H2(Ω) ⊂ C(Ω). Consequently, there exists amonotonic function Q (depending on f) such that:

‖f(u(t))‖2H2(Ω) 6 Q(

‖u(t)‖2H2(Ω)

)

6 Q1

(

|A2u(t)|2)

. (1.28)

Thus the function y(t) := |A2u(t)|2 satisfies the inequality:

y′(t) 6 Q1 (y(t)) . (1.29)

16


Let z(t) be a solution of the following equation:

z′(t) = Q1 (z(t)) , z(0) = y(0) = |A2u(0)|2. (1.30)

Due to the comparison principle, there exists a time T0(‖u0‖H2(Ω)) ∈ (0, 1/2) suchthat we have:

y(t) 6 z(t), ∀t 6 T0(‖u0‖H2(Ω)). (1.31)

Then, the lemma is an immediate consequence of (1.30) and (1.31).

Lemma 1.3.2. Let the above assumptions hold and let T0 be the same as in Lemma1.3.1. Then, the following estimate holds:

t‖∂tu(t)‖2−1 6 Q(‖u0‖H2(Ω)), t ∈ (0, T0], (1.32)

for some monotonic function Q.

Proof:Multiplying (1.4) by ∂tu(t) and integrating over Ω, we obtain:

‖∂tu(t)‖2−1 +1

2

d

dt|A1/2

2 u(t)|2 6 |(f(u(t)), ∂tu(t))| 6 c‖f(u(t))‖2H1(Ω) + 1/2‖∂tu(t)‖2−1,

(1.33)where c is a positive constant. Integrating (1.33) over [0, T0] and taking into account(1.25) and the fact that H2(Ω) ⊂ C(Ω), we have:

∫ T0

0

‖∂tu(t)‖2−1dt 6c

∫ T0

0

‖f(u(t))‖2H1(Ω)dt+ |A1/22 u(0)|2

6c′′T0Q(|A2u(0)|2) + |A1/22 u(0)|2

6Q(‖u0‖H2(Ω)).

(1.34)

Differentiating (1.4) with respect to t and setting θ(t) = ∂tu(t), we find:

A−11 ∂tθ + A2θ + f ′(u(t))θ = 0,

θ|∂Ω = 0.(1.35)

Multiplying (1.35) by tθ(t), integrating over Ω and using the fact that f ′(u) > −κ,we obtain:

d

dt(t‖θ(t)‖2−1) + 2t|A1/2

2 θ(t)|2 62κt|θ(t)|2 + ‖θ(t)‖2−1

62cκt‖θ(t)‖−1|A1/22 θ(t)|+ ‖θ(t)‖2−1

6t|A1/22 θ(t)|2 + C(t+ 1)‖θ(t)‖2−1,

(1.36)

where C is a positive constant. Hence, we have:

d

dt(t‖θ(t)‖2−1) 6 C(t+ 1)‖θ(t)‖2−1. (1.37)

Applying Gronwall inequality to estimate (1.37) and using (1.34), we deduce:

t‖θ(t)‖2−1 6

∫ t

0

‖θ(s)‖2−1ds 6 Q(‖u0‖H2(Ω)), (1.38)

and Lemma 1.3.2 is proven.

17


Lemma 1.3.3. Let u(t) be a solution of equation (1.4) and let t > T0, where T0 isthe same as in Lemma 1.3.1. Then, the following estimate holds:

‖∂tu(t)‖2−1 + ‖u(t)‖2H2(Ω) +

∫ t+1

t

‖∂tu(s)‖2H1(Ω)ds 6 eK1tQ(‖u0‖H2(Ω)), ∀t > T0,

where K1 is a positive constant and Q is some monotonic function.

Proof:Differentiating (1.4) with respect to t and setting θ(t) = ∂tu(t), we find:

A−11 ∂tθ + A2θ + f ′(u(t))θ = 0,

θ|∂Ω = 0, θ|t=T0= ∂tu(T0).

(1.39)

Multiplying (1.39) by θ(t), integrating over Ω and using the fact that f ′(u) > −κ,we have:

d

dt‖θ(t)‖2−1 + 2|A1/2

2 θ(t)|2 62κ|θ(t)|2

62cκ‖θ(t)‖−1|A1/22 θ(t)|

6|A1/22 θ(t)|2 + c2κ2‖θ(t)‖2−1,

(1.40)

for an appropriate positive constant c. Applying Gronwall inequality to estimate(1.40), we obtain:

‖θ(t)‖2−1 +

∫ t+1

t

|A1/22 θ(s)|2ds 6 eK1t‖θ(T0)‖2−1, t > T0, (1.41)

for some positive constant K1. Using Lemma 1.3.2, estimate (1.40) gives:

‖∂tu(t)‖2−1 +

∫ t+1

t

‖∂tu(s)‖2H1(Ω)ds 6 eK1t(Q(‖u0‖H2(Ω))). (1.42)

Interpreting the parabolic equation (1.4) as an elliptic boundary value problem:

−A2u(t)− f(u(t)) = h(t) := A−11 ∂tu(t), u(t)|∂Ω = 0, (1.43)

for every fixed t > T0, estimate (1.42) implies that:

|h(t)|2 6 ‖∂tu(t)‖2−1 6 eK1t(Q(‖u0‖H2(Ω))). (1.44)

We deduce from estimate (1.44) that:

‖u(t)‖2H2(Ω) 6 c|A2u(t)|2 6 CeK1t(Q(‖u0‖H2(Ω))), (1.45)

and the lemma is proven.

Corollary 1.3.4. Let the above assumptions hold and let u(t) be a solution of equa-tion (1.4). Then, the following estimate is valid, for every t > 0:

‖u(t)‖H2(Ω) 6 eKt(Q(‖u0‖H2(Ω))), (1.46)

for some positive constant K and a monotonic function Q.

18

1.4 Existence of the global attractor

Proof:Estimate (1.46) is an immediate consequence of Lemma 1.3.1 and Lemma 1.3.3.

Theorem 1.3.5. Let us take u0 ∈ W . Then, there exists a unique solution u ofproblem (1.1) with initial datum u0 such that:

u ∈ L∞([0, T ];H2(Ω)) ∩ L2([0, T ];H4(Ω)) and ∂tu ∈ L2([0, T ];H1(Ω)). (1.47)

Proof:This theorem is an immediate consequence of Lemma 1.3.1 and 1.3.3.

Remark 1.3.6. Theorem 1.3.5 is also true for every function f such that f ∈ C2(R),f(0) = 0 and f ′(s) > −κ, where κ > 0.


In this section, we are interested in proving the existence of a global attractor forproblem (1.1). We have the following:

Lemma 1.4.1. Problem (1.1) generates the following semigroup on the phase spaceH:

S(t) : H −→ H

u0 7−→ S(t)u0 = u(t), t > 0,

where u(t) is the unique solution of problem (1.1) with initial datum u0 at time t.Furthermore, this semigroup is Lipschitz continuous in the H−1(Ω)−topology,

‖S(t)u1 − S(t)u2‖2−1 +

∫ t+1

t

‖S(s)u1 − S(s)u2‖2H1(Ω)ds 6 cect‖u1 − u2‖2−1,

for any u1, u2 ∈ H, where c is a positive constant independent of t. Thus, S(t) canbe uniquely extended, by continuity, to a semigroup, still denoted by S(t) acting onH−1(Ω).

This Lemma is a direct consequence of (1.23), (1.24) and Theorem 1.2.1.To prove the existence of a global attractor we show the existence of absorbing setsin H−1(Ω), L2(Ω), H1

0 (Ω) and in H2(Ω).

An absorbing set in H−1(Ω)

Proposition 1.4.2. Problem (1.1) has an absorbing set in H−1(Ω). More precisely,there exists a constant ρ0 and a time t0(‖u0‖−1) such that, for the solution u(t) =S(t)u0, we have:

‖u(t)‖H−1(Ω) 6 ρ0 for all t > t0(||u0||−1).

Moreover,∫ t+r

t

|A1/22 u(s)|2ds+

∫ t+r

t

∫

Ω

F (u(s))dx ds 6 ρ10(r) for all t > t0(‖u0‖−1),

where ρ1H(r) is a positive constant depending on u0 but independent of t.

19


Proof:Multiplying equation (1.4) by u, we obtain:

1

2

d

dt|A−1/2

1 u(t)|2 + |A1/22 u(t)|2 + (f(u(t)), u(t)) = 0.

Since

(f(u), u) >a2p−1

2

∫

Ω

u2pdx− c1|Ω| (using (1.2))

>3

4pa2p−1

∫

Ω

u2pdx− c1|Ω|

>

∫

Ω

F (u)dx− (c1 + c3)|Ω| (using (1.3)),

we find:1

2

d

dt|A−1/2

1 u|2 + |A1/22 u|2 +

∫

Ω

F (u)dx 6 k0, (1.48)

where k0 = (c1+c3)|Ω|. Using the fact that |A−1/21 u|2 6 c′|A1/2

2 u|2 for all u ∈ H10 (Ω),

we obtain:1

2

d

dt|A−1/2

1 u|2 + 1

2c′|A−1/2

1 u|2 +∫

Ω

F (u)dx 6 k0.

Thus using (1.3), we have:

1

2

d

dt|A−1/2

1 u|2 + 1

2c′|A−1/2

1 u|2 + 1

4pa2p−1

∫

Ω

u2pdx 6 k1,

where k1 = k0 + c3|Ω|. The Gronwall’s inequality leads to:

|A−1/21 u(t)|2 6 |A−1/2

1 u0|2 exp(−1

c′t) + 2c′k1

(

1− exp(− 1

c′t)

)

, ∀t > 0.

It follows that if

t > t0 = t0(‖u0‖−1) = c1|Ω| ln(‖u0‖2−1

2c′k1

)

,

then:‖u‖2−1 = |A−1/2

1 u|2 6 4c′k1 = ρ20. (1.49)

To deduce the integral bound on |A1/22 u| and

∫

ΩF (u)dx, we return to (1.48) and we

integrate with respect to t from t to t+ r, for r > 0 a fixed constant. We find:

∫ t+r

t

|A1/22 u(s)|2ds+

∫ t+r

t

∫

Ω

F (u(s))dx ds 6 k1r + 1/2‖u(t)‖2H−1(Ω)

6 k1r + ρ20 := ρ10(r),(1.50)

for all t > t0.

20


An absorbing set in L2(Ω)

Proposition 1.4.3. Problem (1.1) has an absorbing set in L2(Ω). More precisely,there exists a constant ρH and a time t1(‖u0‖−1) such that, for the solution u(t) =S(t)u0, we have:

|u(t)| 6 ρH for all t > t1(||u0||−1).

Moreover,∫ t+r

t

|A2u(s)|2ds 6 ρ1H(r) for all t > t1(‖u0‖−1).

where ρ1H(r) is a positive constant depending on u0 but independent of t.

Proof:Multiplying equation (1.1) by u and using (1.2), we have:

1

2

d

dt|u|2 + |A2u|2 + |A1/2

2 u|2 + a2p−1

2

∫

Ω

u2pdx 6 c1|Ω|+ c|A1/22 u|2. (1.51)

Consequently, we obtain:

1

2

d

dt|u|2 6 c1|Ω|+ c|A1/2

2 u|2.

In what follows, we use the technical result:

Lemma 1.4.4. (The uniform Gronwall lemma). Let g, h, y, be three positive locallyintegrable functions on ]t0,+∞[ such that y′ is locally integrable on ]t0,+∞[, andwhich satisfy:

dy

dt6 gy + h for t > t0,

∫ t+r

t

g(s)ds 6 a1,

∫ t+r

t

h(s)ds 6 a2,

∫ t+r

t

y(s)ds 6 a3 for t > t0,

where r, a1, a2, a3, are positive constants. Then, we have:

y(t+ r) 6 (a3/r + a2)ea1 , ∀t > t0.

Applying the uniform Gronwall lemma , the following estimate holds:

|u(t+ r)|2 6 2

(

c′(r)

r+ c1|Ω|+ cρ10(r)

)

, ∀t > t0,

and

|u(t)|2 6 (ρH(r))2 =

(

c′(r)

r+ c1|Ω|+ cρ10(r)

)

, ∀t > t1 := t0 + r. (1.52)

Integrating (1.51) from t to t+ r and using (1.50), (1.52), we find:∫ t+r

t

|A2u|2ds 6c1|Ω|r + c

∫ t+r

t

|A1/22 u(s)|2ds+ 1

2|u(t)|2

6c1|Ω|r + cρ10(r) +1

2(ρH(r))

2 := ρ1H(r),

∀t > t1. This completes the proof of Proposition 1.4.2.

21


An absorbing set in H10 (Ω)

Proposition 1.4.5. Equation (1.1) has an absorbing set in H10 (Ω). More precisely,

there exists a constant ρV (r) and a time t1(‖u0‖−1) such that:

||u(t)|| 6 ρV (r) for all t > t1(‖u0‖−1).

Proof:We set

E(u) = 1

2|A1/2

2 u|2 +∫

Ω

F (u)dx and K(u) = −∆u+ f(u).

Multiplying (1.1) by K(u), we can prove that we have:

d

dtE(u) + |A1/2

2 K(u)|2 + |K(u)|2 = 0,

which shows that:E(u(t)) 6 E(u0), ∀t > 0.

Equation (1.48) can be written as follow:

1

2

d

dt|A−1/2

1 u|2 + E(u) 6 k0. (1.53)

Integrating (1.53) with respect to t from t to t+ r, we obtain:

1

2|A−1/2

1 u(t+ r)|2 +∫ t+r

t

E(u(s))ds 6 k0r +1

2|A−1/2

1 u(t)|2. (1.54)

Using the fact that E decays along the orbits and that |A−1/21 u(t)| 6 ρ0, ∀t > t0, we

deduce:

E(u(t+ r)) 6 k0 +1

2rρ20, ∀t > t0, (1.55)

and

|A1/22 u|2+ 1

2pa2p−1

∫

Ω

u2pdx 6 ρ2V (r) := 2k0+2c3|Ω|+1

rρ20, ∀t > t1 := t0+r. (1.56)

An absorbing set in H2(Ω) We can also prove the existence of an absorbing setin H2(Ω). We actually prove the following result:

Proposition 1.4.6. Problem (1.1) has an absorbing set in H2(Ω). More precisely,there exists a constant ρ and a time t2(‖u0‖−1) such that

|A2u(t)| 6 ρ(r) for all t > t2(‖u0‖−1). (1.57)

22


Proof:Multiplying (1.4) by ∂tu, we obtain:

1

2

d

dt|A1/2

2 u|2 + d

dt

∫

Ω

F (u)dx+ ‖∂tu‖2−1 = 0. (1.58)

Using (1.56), we find:

∫ t+r

t

‖∂tu(s)‖2−1ds 6 ρ′V (r), ∀t > t1. (1.59)

Differentiating (1.4) with respect to t, we have:

A−11 ∂t∂tu+ A2∂tu+ f ′(u)∂tu = 0. (1.60)

Multiplying (1.58) by ∂tu and using the fact that f ′(u) > −κ, we obtain:

1

2

d

dt‖∂tu‖2−1 + |A1/2

2 ∂tu|2 6 κ|∂tu|2 6 cκ‖∂tu‖2−1|A1/22 ∂tu|2, (1.61)

which yields:d

dt‖∂tu‖2−1 6 c‖∂tu‖2−1, (1.62)

for some positive constant c. Using (1.59), (1.61) and the uniform Gronwall lemma,we deduce that:

‖∂tu‖2−1 6 c(r), ∀t > t1 + r. (1.63)

We rewrite (1.4) in the following form:

A2u+ f(u) = −A−11

du

dt. (1.64)

Multiplying (1.64) by A2u and using that f ′(u) > −κ, we obtain:

|A2u|2 6κ|A1/22 u|2 + (A−1

1 ∂tu,A2u)

6κ|A1/22 u|2 + ‖∂tu‖−1|A2u|,

(1.65)

which yields:|A2u|2 6 2κ|A1/2

2 u|2 + ‖∂tu‖2−1. (1.66)

Using (1.56), (1.63), we deduce from (1.66) that:

|A2u| 6 ρ(r), ∀t > t2 := t1 + r,

for some positive constant ρ(r) which depend on r and Proposition 1.4.6 is proven.

Using the existence of an absorbing set in H2(Ω), we can deduce the existenceof the global attractor.The following theorem gives the existence of the global attractor A in H−1(Ω) forthe semigroup S(t). We recall that, by definition, a set A ⊂ H−1(Ω) is the globalattractor for the semigroup S(t) if the following properties are satisfied:

23


1. It is a compact subset of H−1(Ω);

2. It is strictly invariant, i.e., S(t)A = A, ∀t > 0;

3. It attracts all bounded sets in H−1(Ω) as t → ∞, i.e., for every bounded setX ⊂ H−1(Ω) there exists a neighborhood O(A) of A in H−1(Ω) and a timeT = T (O) such that:

S(t)X ⊂ O(A), t > T.

Theorem 1.4.7. The semigroup S(t) possesses the global attractor A in H−1(Ω)which is bounded in H2(Ω).

Proof:Let B1 be a bounded set in H−1(Ω) defined by

‖ϕ‖−1 6 ρ0,

with ρ0 as in (1.49) and B2 be a bounded set in H2(Ω) defined by

|A2ϕ| 6 ρ,

with ρ as in (1.57). B2 is a bounded absorbing set in H2(Ω), compact in H−1(Ω),which implies the existence of the global attractor in H−1(Ω).

1.5 Existence of an exponential attractor

In this section, we prove the existence of an exponential attractor which by definition,contains the global attractor and has finite fractal dimension. To do this, we firstrecall the definition of the exponential attractor where A is the global attractor forthe semigroup S(t)t>0:

Definition 1.5.1. Let X be a compact connected subset of a Banach space E. Acompact set M is called an exponential attractor for the semigroup S(t)t>0 ifA ⊂ M ⊂ X and

1. S(t)M ⊂ M, ∀t > 0.

2. M has finite fractal dimension, dF (M) <∞.

3. There exist positive constants c0 and c1 such that for every u0 ∈ X, we have:

distE(S(t)u0,M) 6 c0e−c1t, ∀t > 0, (1.67)

where the pseudo-distance dist is the standard Hausdorff pseudo-distance be-tween two sets, defined by distE(A,B) = supa∈A infb∈B ‖a− b‖E.

To prove the existence of an exponential attractor, we apply the following theo-rem (see [44]):

Theorem 1.5.2. Let E and E1 be two Hilbert spaces such that E1 is compactlyembedded into E and S(t) : X → X be a semigroup acting on a closed subset X ofE. We assume that:

24


1. ∀x1, x2 ∈ X, ∀t > 0,

‖S(t)x1 − S(t)x2‖E16 h(t)‖x1 − x2‖E,

where the function h is continuous;

2. (t, x) 7→ S(t)x is uniformly Hölder continuous on [0, T ]×B, ∀T > 0, ∀B ⊂ Xbounded.

Then, S(t) possesses an exponential attractor on X.

In order to apply this result to the semigroup S(t) associated with problem (1.1),we set E = H−1(Ω), E1 = L2(Ω) and we consider the set

X =⋃

t> t0+2r

S(t)B2,

where t0 + 2r is such that for all t > t0 + 2r, we have S(t)B2 ⊂ B2 (X is thuscompact in H−1(Ω), bounded in H2(Ω) and positively invariant by S(t)).Let u1 and u2 be respectively two solutions of (1.1) with initial data in X. We setw = u1 − u2. Then, w verifies:

A−11

dw

dt+ A2w + l(t)w = 0,

w(0) = u1(0)− u2(0),(1.68)

where l(t) =∫ 1

0

f ′(su1 + (1− s)u2)ds.

In order to complete the proof, we need to prove the following lemma:

Lemma 1.5.3. Let u1(t) and u2(t) be two solutions of (1.1) such that |A2ui(0)| 6ρ, i = 1, 2. Then, the following estimate is valid:

‖u1(t)− u2(t)‖2H−1(Ω) +

∫ t+1

t

‖w(s)‖2H1(Ω)ds 6 Cρeαρt‖u1(0)− u2(0)‖2H−1(Ω), (1.69)

where the positive constants Cρ and αρ depend on ρ.

Proof:We have:

‖l(t)‖L∞(Ω) 6 C(‖u1(0)‖H2(Ω), ‖u2(0)‖H2(Ω)) 6 Qρ, (1.70)

where the constant Qρ depends on the constant ρ. Multiplying (1.68) by w, inte-grating over Ω and using (1.70), we obtain the following inequality:

1

2

d

dt‖w‖2H−1(Ω) + |A1/2

2 w|2 =− (l(t)w,w)

6‖l(t)‖L∞(Ω)‖w‖2L2(Ω)

6cQρ‖w‖H−1(Ω)‖A1/22 w‖L2(Ω).

(1.71)

Applying Gronwall’s inequality to (1.71), we conclude the proof of the lemma.The next lemma gives the H−1(Ω) → L2(Ω)-smoothing property for the differenceof two solutions.

25


Lemma 1.5.4. Let u1(t) and u2(t) be two solutions of (1.1) such that |A2ui(0)| 6ρ, i = 1, 2. Then, the following estimate is valid:

t‖u1(t)− u2(t)‖2L2(Ω) 6 Rρeαρt‖u1(0)− u2(0)‖2H−1(Ω), t > 0, (1.72)

where Rρ is a positive constant depending on ρ.

Proof:Multiplying (1.72) by tA2w and integrating over Ω, we find:

1

2td

dt(A2A

−11 w,w) + t|A2w|2 + t (l(t)w,A2w) = 0. (1.73)

Setting (A2A−11 w,w) = ‖w‖2∗, where ‖.‖∗ ∼ |.|L2(Ω), equation (1.73) yields:

1

2

d

dt

(

t‖w‖2∗)

+ t|A2w|2 61

2‖w‖2∗ − t (l(t)w,A2w)

61

2‖w‖2∗ + t‖l(t)‖L∞(Ω)|w||A2w|

61

2‖w‖2∗ + tQρ|w||A2w|

61

2‖w‖2∗ +

t

2Q2

ρ|w|2 +t

2|A2w|2

61

2‖w‖2∗ +

t

2Q2

ρ|A1/22 w|2 + t

2|A2w|2

Integrating with respect to t from 0 to t, we obtain:

t‖w(t)‖2∗ +∫ t

0

s|A2w(s)|2ds 6∫ t

0

‖w‖2∗ds+Q2ρ

∫ t

0

s|A1/22 w(s)|2ds

6c

∫ t

0

|A1/22 w(s)|2ds+Q2

ρ

∫ t

0

s|A1/22 w(s)|2ds

6Rρeαρt‖u1(0)− u2(0)‖2H−1(Ω),

where we use the fact that ‖w‖∗ 6 c|A1/22 w| and Lemma 1.5.3.

Finally, we deduce:

t‖u1(t)− u2(t)‖2L2(Ω) 6 Rρeαρt‖u1(0)− u2(0)‖2H−1(Ω).

Thus, the first condition of Theorem 1.5.2 is proven and it is sufficient to provethe second condition in order to prove the existence of an exponential attractor.

Lemma 1.5.5. The semigroup S(t) is uniformly Hölder continuous on [0, T ] × B2

in the topology of H−1(Ω), where B2 =

u ∈ W, ‖u‖H2(Ω) 6 ρ

, i.e.

‖S(t1)u01 − S(t2)u02‖−1 6 Cρ,T

(

‖u01 − u02‖H−1(Ω) + |t1 − t2|1/2)

,

where u0i ∈ B2, ti 6 T, i = 1, 2, and Cρ,T is a positive constant depending on ρ andT .

26


Proof:The Lipschitz continuity with respect to the initial conditions is an immediate corol-lary of lemma 1.5.3. In order to verify the Hölder continuity with respect to t, wemultiply equation (1.4) by ∂tu and we integrate over Ω. We obtain:

|A−1/21 ∂tu|2 +

1

2

d

dt|A1/2

2 u|2 + d

dt

∫

Ω

F (u)dx = 0.

Integrating with respect to t from t1 to t2, we find:∫ t2

t1

|A−1/21 ∂tu|2ds+

1

2|A1/2

2 u(t2)|2 +∫

Ω

F (u(t2))dx

=1

2|A1/2

2 u(t1)|2 +∫

Ω

F (u(t1))dx.

Using (1.3), we have:∫ t2

t1

|A−1/21

∂u

∂t|2ds+1

2|A1/2

2 u(t2)|2 +a2p−1

4p

∫

Ω

u2p(t2)dx

6 2c3|Ω|+1

2|A1/2

2 u(t1)|2 +3a2p−1

4p

∫

Ω

u2p(t1)dx

6 C(ρ).

Consequently, we deduce:

‖u(t1)− u(t2)‖H−1(Ω) =

∥

∥

∥

∥

∫ t2

t1

∂tu(s)ds

∥

∥

∥

∥

H−1(Ω)

6

∫ t2

t1

‖∂tu(s)‖H−1(Ω)ds

6|t1 − t2|1/2(∫ t2

t1

‖∂tu(s)‖2H−1(Ω)ds

)1/2

6Cρ,T |t1 − t2|1/2,where Cρ,T is a positive constant depending on ρ and T , which yields the Hölder

continuity with respect to t.

Hence, we deduce the existence of an exponential attractor for problem (1.1).

Remark 1.5.6. We recall that an exponential attractor always contains the globalattractor. Consequently, we deduce that the global attractor has finite fractal di-mension in H−1(Ω).

Remark 1.5.7. For the problem with Neumann boundary conditions, we set g(s) =f(s)− s, thus we obtain the following reformulation:

A−11 ∂tu+ A1u+ g(u) = 0.

Noting that g satisfies the same properties as f , we proceed as above and we obtainthe same previous results.

Acknowledgments

I would like to thank Alain Miranville and Madalina Petcu, my supervisors, formany stimulating discussions and useful comments on the subject of the paper.

27


28

Chapitre 2

Well-Posedness and Long Time

Behavior of a Perturbed

Cahn-Hilliard System with Regular

Potentials

Sur le caractère bien posé et le com-portement asymptotique d’une équa-tion de type Cahn-Hilliard perturbée

Ce chapitre est constitué de l’article Well-posedness and long time behav-ior of a perturbed Cahn-Hilliard system with regular potentials, écrit encollaboration avec Alain Miranville et Madalina Petcu, article paru en 2013 dansdans Journal of Asymptotic Analysis, volume 84, pages 147-179.

Journal of Asymptotic Analysis

Volume 84, pp. 147–179, 2013

Well-posedness and long time behavior of a

perturbed Cahn-Hilliard system with regular

potentials

Haydi Israel1, Alain Miranville1 and Madalina Petcu1,2

1UMR 7348 CNRS. Laboratoire de Mathématiques et Applications - Université de Poitiers - SP2MI -

Boulevard Marie et Pierre Curie, Téléport 2


2Institute of Mathematics of the Romanian Academy, Bucharest, Romania.

Abstract. The aim of this paper is to study the well-posedness and long timebehavior, in terms of finite-dimensional attractors, of a perturbed Cahn-Hilliardequation. This equation differs from the usual Cahn-Hilliard by the presence of theterm ε(−∆u + f(u)). In particular, we prove the existence of a robust family ofexponential attractors as ε goes to zero.

2.1 Introduction

We consider the following boundary value problem in a smooth and bounded domainΩ ⊂ R

n with boundary ∂Ω:

∂tu+∆2u−∆f(u)− ε∆u+ εf(u) = 0 in Ω,u = ∆u = 0 on ∂Ω,u|t=0 = u0,

(2.1)

where f is the derivative of a nonconvex potential and the unknown u is the relativeconcentration of one phase. We assume throughout this paper that n 6 3.When ε = 0, we recover the well-known Cahn-Hilliard equation (see [38], [47], [54])and when ε > 0, equation (2.1) may be viewed as a combination of the well-knownCahn-Hilliard equation and the Allen-Cahn equation (see [30], [31], [32], [37]).We recall that the Cahn-Hilliard equation describes the behavior of two-phase sys-tems, in particular in spinodal decomposition, i.e. in the case of rapid separation ofphases when the material is cooled down sufficiently and, when ε > 0, the equationdescribes a simplified model of adsorption to and desorption from the surface.Such a model has been studied in [27], [28] and [31], for a smooth nonlinearityin [28], [31] and for a singular nonlinearity in [27] where questions such as existence

2 . Well-Posedness and Long Time Behavior of a Perturbed Cahn-Hilliard System withRegular Potentials

and uniqueness of solutions, existence of the global attractor and an exponentialattractor have been addressed.We rewrite problem (2.1) as a system of second order equations:

∂tu = ∆w − εw in Ω,w = −∆u+ f(u),u = ∆u = 0 on ∂Ω,u|t=0 = u0.

(2.2)

We denote by F the antiderivative of f vanishing at 0 and assume that

f ∈ C2(R),f(0) = 0,f ′(s) > −κ,f(s)s > pb2ps

2p − c > c(s2 + s2p)− c,|f(s)| 6 c(|s|2p−1 + 1),12b2ps

2p − c 6 F (s) 6 32b2ps

2p + c,

(2.3)

for all s ∈ R and where κ, c and b2p are positive constants and p > 2 is an integer.Typical choice for f is

f(s) = s3 − s.

This paper is organized as follows. In Section 2, we give some useful assumptions andnotation. Then, in Section 3, we derive uniform a priori estimates for approximatedsolutions which allow us to pass to the limit in the approximated problem to studythe well-posedness, namely, the existence and uniqueness of a weak solution as statedin Theorem 2.3.1. Section 4 is dedicated to the proof of some additional regularityfor the solution. It follows from the well-posedness result that the system generates acontinuous semigroup in a suitable phase space, which allows to study the existenceof the global attractor in Section 5. In Section 6, we prove that the fractal dimensionof the global attractor is finite by studying the existence of exponential attractors.Finally, Section 7 is devoted to the proof of the continuity of exponential attractorsfor the perturbed system (2.1) and to the derivation of the corresponding estimatefor the symmetric distance.

Remark 2.1.1. Neumann boundary conditions, namely, ∂nu = ∂n∆u = 0 on ∂Ω,are also relevant in the context of the Allen-Cahn and Cahn-Hilliard equations. Inthat case, the limit problem, i.e., the Cahn-Hilliard equation, is a conservation law,in the sense that the spatial average of the order parameter u is a conserved quan-tity. This brings additional difficulties in the study of the continuity of exponentialattractors and will be studied elsewhere.

2.2 Notation and assumptions

We introduce the following spaces:

H = L2(Ω), V = H10 (Ω),W = H2(Ω) ∩H1

0 (Ω).

32

2.2 Notation and assumptions

We denote by ‖ · ‖ and (·, ·) the usual norm and inner product in H.Let

Aε = −∆+ εI : D(Aε) ⊂ H → H

with D(Aε)= W . The operator Aε is a strictly positive self-adjoint linear operatorwith compact inverse A−1

ε . Then, problem (2.2) can be reformulated as follows:

A−1ε ∂tu−∆u+ f(u) = 0 in Ω,

u = 0 on ∂Ω,u|t=0 = u0.

(2.4)

For 0 6 ε 6 1 and u ∈ D(Aε), we have the following:

‖∇u‖ 6 ‖A1/2ε u‖ 6 ‖u‖H1(Ω) =⇒ ‖A1/2

ε u‖ ∼ ‖u‖H1(Ω), (2.5)

‖∆u‖ 6 ‖Aεu‖ 6 c‖u‖H2(Ω) =⇒ ‖Aεu‖ ∼ ‖u‖H2(Ω), (2.6)

‖A−1/2ε u‖ 6 c‖A1/2

ε u‖, (2.7)

‖u‖−1 ∼ ‖A−1/2ε u‖, (2.8)

‖u‖−2 ∼ ‖A−1ε u‖, (2.9)

(Aε(−∆)−1u, u) ∼ ‖u‖2, (2.10)

and((−∆)A−1

ε u, u) ∼ ‖u‖2, (2.11)

where c is independent of ε, ‖v‖2−1 = ((−∆)−1v, v) and all the equivalences areindependent of ε. Indeed, we have, if u is regular enough:

‖∇u‖2 6 ‖A1/2ε u‖2 = (Aεu, u) = ‖∇u‖2 + ε‖u‖2 6 ‖∇u‖2 + ‖u‖2 = ‖u‖2H1(Ω),

‖∆u‖2 6 ‖Aεu‖2 = (Aεu,Aεu) = ‖∆u‖2 + 2ε‖∇u‖2 + ε2‖u‖2 6 c‖u‖2H2(Ω).

We then use Poincaré’s inequality since u = 0 on ∂Ω and we deduce (2.5) and (2.6).Now, using the inclusions V ⊂ H ⊂ V ′, the scalar product:

a(u, v) = (∇u,∇v), ∀u, v ∈ V

defines a linear operator A : D(A) = W → H. The operator A is the Laplace opera-tor with Dirichlet boundary conditions and is a nonnegative self-adjoint operator; ithas an orthonormal basis of eigenvectors ejj associated to the eigenvalues λjj,with

0 < λ0 6 λ1 6 · · · 6 λj 6 · · · , λj → +∞ as j → +∞.

The family ejj may be assumed to be normalized in the norm of H, i.e.,

(ei, ej) = δij,

33


where

δij =

1 if i = j,0 otherwise.

We also note that the operator Aε has the same orthonormal basis of eigenvectorsejj associated to the eigenvalues λε,jj, with λε,j = λj + ε. We have

λj 6 λε,j 6 λj + 1, ∀j ∈ IN,

hence (2.7). We note that

Aε(−∆)−1u = (−∆+ εI)(−∆)−1u = u+ ε(−∆)−1u,

hence(−∆)−1u = A−1

ε u+ εA−1ε (−∆)−1u.

Therefore,

‖u‖2−1 = ((−∆)−1u, u) = (A−1ε u+εA−1

ε (−∆)−1u, u) = ‖A−1/2ε u‖2+ε(A−1

ε (−∆)−1u, u).

Using the fact that A−1ε and (−∆)−1 are nonnegative self-adjoint operators which

commute, we obtainε(A−1

ε (−∆)−1u, u) > 0,

which yields‖A−1/2

ε u‖2 6 ‖u‖2−1. (2.12)

We also have A−1/2ε (−∆)−1/2 = (−∆)−1/2A

−1/2ε , hence

‖u‖2−1 = ((−∆)−1u, u) =(A−1ε u+ εA−1

ε (−∆)−1u, u)

=‖A−1/2ε u‖2 + ε‖A−1/2

ε (−∆)−1/2u‖2

6‖A−1/2ε u‖2 + ‖(−∆)−1/2A−1/2

ε u‖2

6‖A−1/2ε u‖2 + c′‖A−1/2

ε u‖2,

which yields‖u‖2−1 6 c‖A−1/2

ε u‖2. (2.13)

From estimates (2.12) and (2.13), we deduce (2.8).We further have

‖u‖2−2 =((−∆)−1u, (−∆)−1u)

=(A−1ε u+ εA−1

ε (−∆)−1u,A−1ε u+ εA−1

ε (−∆)−1u)

=‖A−1ε u‖2 + ε2‖A−1

ε (−∆)−1u‖2 + 2ε(A−1ε (−∆)−1u,A−1

ε u)

=‖A−1ε u‖2 + ε2‖A−1

ε (−∆)−1u‖2 + 2ε‖(−∆)−1/2A−1ε u‖2

6‖A−1ε u‖2 + ‖(−∆)−1A−1

ε u‖2 + 2‖(−∆)−1/2A−1ε u‖2

6c‖A−1ε u‖2,

and, since ε2‖A−1ε (−∆)−1u‖2 + 2ε(A−1

ε (−∆)−1u,A−1ε u) > 0, we find

‖A−1ε u‖2 6 ‖u‖2−2.

34

2.3 Uniform a priori estimates. Existence and uniqueness of solutions.

Then, we deduce that

‖A−1ε u‖2 6 ‖u‖2−2 6 c‖A−1

ε u‖2,

where c is independent of ε.Now,

(Aε(−∆)−1u, u) = ((−∆+ εI)(−∆)−1u, u) =(u+ ε(−∆)−1u, u)

=‖u‖2 + ε‖u‖2−1

6‖u‖2 + ‖u‖2−1

6c‖u‖2

and‖u‖2 6 (Aε(−∆)−1u, u),

so that(Aε(−∆)−1u, u) ∼ ‖u‖2.

We also have

((−∆)A−1ε u, u) =((−∆)A−1

ε u,Aε(−∆)−1(−∆)A−1ε u)

6c‖(−∆)A−1ε u‖2(using (2.10))

6c‖AεA−1ε u‖2(using (2.6))

6c‖u‖2

and

‖u‖2 =‖A1/2ε (−∆)−1/2(−∆)1/2A−1/2

ε u‖2

6c‖(−∆)1/2A−1/2ε u‖2

6c((−∆)A−1ε u, u),

which yields((−∆)A−1

ε u, u) ∼ ‖u‖2.The variational formulation of problem (2.4) readsFind u : [0, T ] → V such that

(A−1ε ∂tu, q) + (∇u,∇q) + (f(u), q) = 0, ∀q ∈ V,

u(0) = u0,(2.14)

∀T > 0.In what follows, unless mentioned explicitly, the same letter Q denotes monotoneincreasing functions and the same letter c denotes positive constants independent ofε, possibly changing at different occurrences.

2.3 Uniform a priori estimates. Existence and unique-ness of solutions.

In order to prove the existence of solutions for problem (2.4), we use a Galerkinscheme. We first define an m-dimensional system which approximates the initial

35


problem. The resolution of the system proves the existence of an approximatedsolution. We then obtain a priori estimates which allow us to justify the passage tothe limit in the approximated problem and obtain a solution to the initial problem.We have the following theorem.

Theorem 2.3.1. Let us take u0 ∈ H. Then, there exists a unique solution u ofproblem (2.4) with initial datum u0 such that

u ∈ L∞([0, T ];H) ∩ L2([0, T ];W ) ∩ L2p(0, T ;L2p(Ω)).

Furthermore, if u0 ∈ V ∩ L2p(Ω), then

u ∈ L∞([0, T ];V ∩ L2p(Ω)) ∩ L2([0, T ];W ) and ∂tu ∈ L2([0, T ];H−1(Ω)).

Uniqueness of the solution: Let u and v be two solutions of (2.4) on the timeinterval [0, T ]. We set w = u− v, then w verifies the following equation:

A−1ε ∂tw +∇w + l(t)w = 0,

w(0) = u(0)− v(0),(2.15)

where l(t) =∫ 1

0

f ′ (su(t) + (1− s)v(t)) ds. Multiplying (2.15) by w and integrating

over Ω, we obtain

1

2

d

dt‖A−1/2

ε w‖2 + ‖∇w‖2 + (l(t)w,w) = 0.

Using the fact that f ′(s) > −κ, we have

(l(t)w,w) > −κ‖w‖2.

Using the following interpolation inequality:

‖w‖2L2(Ω) 6 c‖w‖−1‖∇w‖ 6 c‖A−1/2ε w‖‖∇w‖,

we obtain1

2

d

dt‖A−1/2

ε w‖2 + 1

2‖∇w‖2 6 c2κ2

2‖A−1/2

ε w‖2. (2.16)

Applying Gronwall’s lemma to (2.16), we find

‖A−1/2ε w(t)‖2 +

∫ t

0

‖∇w(s)‖2ds 6 ‖A−1/2ε (u(0)− v(0))‖2ec2κ2t. (2.17)

Relation (2.17) shows the continuous dependence of the solution on the initial datain the H−1−norm and, in particular, when u(0) = v(0), it implies the uniqueness ofthe solution.

36


Existence of a local solution: We denote by Em the space

Em = span e1, e2, . . . , em

and by Pm the orthogonal projection from V onto Em:

Pmh =m∑

j=1

(h, ej) ej,

where ejj are the eigenfunctions of the operator A.For any m ∈ N, we look for functions of the form

um (t) =m∑

j=1

umj(t) ej,

solving the approximated problem

A−1ε ∂tum −∆um + f(um) = 0,

um (0) = Pmu0.(2.18)

The variational formulation of problem (2.18) readsFind um : [0, T ] → Em such that

d

dt

(

A−1ε

m∑

j=1

umj(t) ej, ei

)

+

(

m∑

j=1

umj(t)∇ej,∇ei

)

+ (f (um) , ei) = 0, i = 1, ...,m,

um (0) = Pmu0.(2.19)

We rewrite problem (2.19) as follows:

MεdY

dt+MY +N (Y ) = 0,

where Mε =(

(A−1ε (ei), ej)

)

i,j=1,...,m, M =

(

(∇ei,∇ej))

i,j=1,...,m, Y =

um1

...umm

and N(Y ) =

(f(um), e1)...

(f(um), em)

.

We can easily check that matrix Mε is invertible. Indeed, setting X =

x1...xm

6= 0

and X ′ =m∑

i=1

xiei, we have

(MεX,X) =m∑

j=1

m∑

i=1

(

A−1ε ej, ei

)

xixj =(

A−1ε X

′

, X′

)

> 0,

37


due to the fact that A−1ε is a positive definite operator. Moreover, the matrix M

is positive definite and N(Y ) depends continuously on Y. Applying Cauchy’s theo-rem for a system of ordinary differential equations, it follows that there exists a time

tm ∈ (0, T ) and a unique solution Y for the equationdY

dt+M−1

ε MY +M−1ε N(Y ) = 0

on the time interval t ∈ [0, tm[. Based on the a priori estimates with respect to tthat will be derived below for the solution um(t), we deduce that any local solutionof (2.14) is actually a global solution defined on the whole interval [0, T ].

Now, we give the a priori estimates for the solution um(t).

A priori estimates: Multiplying (2.18) by um, we find

1

2

d

dt‖A−1/2

ε um‖2 + ‖∇um‖2 + (f(um), um) = 0. (2.20)

By (2.3), we have

(f(um), um) > c(‖um‖2 + ‖um‖2pL2p(Ω))− c|Ω|

and estimate (2.20) yields

d

dt‖A−1/2

ε um‖2 + c(‖um‖2H1(Ω) + ‖um‖2pL2p(Ω)) 6 c. (2.21)

Multiplying (2.18) by Aεum and using (2.3), we get

1

2

d

dt‖um‖2 + ‖∆um‖2 + ε‖∇um‖2 6 κ‖∇um‖2 + κε‖um‖2

6 κ(

(−∆um, um) + ‖um‖2)

6 κ‖um‖2 + 1/2‖∆um‖2 + κ2/2‖um‖2,

which yieldsd

dt‖um‖2 + ‖∆um‖2 + 2ε‖∇um‖2 6 c‖um‖2. (2.22)

Using the fact that ‖A−1/2ε um‖ 6 c‖um‖H1(Ω) and ‖um‖ 6 c‖∆um‖, estimates (2.21)

and (2.22) give, for a positive constant α independent of ε,

d

dt‖A−1/2

ε um‖2 + α‖A−1/2ε um‖2 + cα‖um‖2H1(Ω) + c‖um‖2pL2p(Ω) 6 c, (2.23)

andd

dt‖um‖2 + α‖um‖2 + cα‖∆um‖2 + 2ε‖∇um‖2 6 c‖um‖2, (2.24)

where the constant cα > 0 depends on α and is independent of ε.Applying Gronwall’s lemma, estimate (2.23) leads to

‖A−1/2ε um(t)‖2+

∫ t

0

(‖um(s)‖2H1(Ω)+‖um(s)‖2pL2p(Ω))e−α(t−s)ds 6 c‖A−1/2

ε um(0)‖2e−αt+c.

(2.25)

38


Integrating (2.21) with respect to s ∈ [t, t+1] and using estimate (2.25), we obtain∫ t+1

t

(‖um(s)‖2H1(Ω)+‖um(s)‖2pL2p(Ω))ds 6 ‖A−1/2ε um(t)‖2+c 6 c‖A−1/2

ε um(0)‖2e−αt+c.

(2.26)Estimates (2.25) and (2.26) give

‖A−1/2ε um(t)‖2 +

∫ t+1

t

(‖um(s)‖2H1(Ω) + ‖um(s)‖2pL2p(Ω))ds 6 c‖A−1/2ε um(0)‖2e−αt + c,

so thatum ∈ L2(0, T ;V ) ∩ L2p(0, T ;L2p(Ω)) ∩ L∞(0, T ;H−1(Ω)).

Applying Gronwall’s Lemma and using estimate (2.25), relation (2.22) leads to

‖um(t)‖2 6 ‖um(0)‖2e−αt + c

∫ t

0

‖um(s)‖2e−α(t−s)ds

6 c‖um(0)‖2e−αt + c,

(2.27)

which implies that um ∈ L∞(0, T ;H). Now, integrating estimate (2.22) with respectto s ∈ [t, t+ 1] and using (2.27), we obtain

‖um(t)‖2 +∫ t+1

t

(‖∆um(s)‖2 + 2ε‖∇um(s)‖2)ds 6 c‖um(0)‖2e−αt + c, (2.28)

where the positive constant c is independent of m. Integrating (2.22) with respectto t ∈ [0, T ] and using (2.27), we find∫ T

0

‖∆um(t)‖2dt 6 ‖um(0)‖2 +∫ T

0

‖um(t)‖2dt 6 ‖um(0)‖2 + c‖um(0)‖2 + cT

and we deduce that um ∈ L2(0, T ;W ).

We introduce the energy functional E : V → IR defined by

E(v) = 1

2‖∇v‖2 +

∫

Ω

F (v)dx.

Multiplying (2.18) by ∂tum, we obtain

‖A−1/2ε ∂tum‖2 +

d

dt

(

1

2‖∇um‖2 +

∫

Ω

F (um)dx

)

= 0, (2.29)

which shows that the energy E decays along the trajectories. Integrating equation(2.29) with respect to t, we get

E(um(t)) +∫ t

0

‖A−1/2ε ∂tum(s)‖2ds 6

1

2‖∇um(0)‖2 +

∫

Ω

F (um(0))

6 ‖∇u0‖2 +3

2b2p

∫

Ω

u2pm (0)dx+ c

6 ‖∇u0‖2 + c‖u0‖2pL2p(Ω) + c.

Thus, if u0 ∈ V ∩ L2p(Ω), we have a solution um which verifies

um ∈ L∞(0, T ;V ∩ L2p(Ω)) and ∂tum ∈ L2(0, T ;H−1(Ω)), ∀T > 0.

39


Passage to the limit: We now pass to the limit as m → ∞ and study the con-

vergence of the sequence (um)m. Let q satisfy1

2p+

1

q= 1, which implies q < 2.

According to the a priori estimates derived in the previous section, ‖um‖L2(0,T ;W ) isuniformly bounded and, consequently, (um)m is bounded in Lq(0, T ;Lq(Ω)). More-over, thanks to (2.3) and the uniform boundedness of (um)m in L2p(0, T ;L2p(Ω)),‖f(um)‖Lq(0,T ;Lq(Ω)) is uniformly bounded, which implies that (A−1

ε ∂tum)m is boundedin Lq(0, T ;Lq(Ω)). It follows that

‖∂tum‖Lq(0,T ;W−2,q(Ω)) 6 c. (2.30)

Starting from this point, all convergence relations will be intended to hold up to theextraction of suitable subsequences, generally not relabeled. Thus, we observe thatweak and weak star compactness results applied to the sequence (um)m entail thatthere exists a function u such that the following properties hold:

um u weakly in Lq(0, T ;W ), (2.31)

∂tum ∂tu weakly in Lq(0, T ;W−2,q(Ω)), (2.32)

as m→ ∞. It follows from (2.31), (2.32) and the Aubin-Lions compactness theoremthat

um → u strongly in Lq(0, T ;Lq(Ω)). (2.33)

Consequently um(t, x) → u(t, x) a.e. (t, x) ∈ [0, T ]× Ω.Moreover, we have

um(t, x) → u(t, x) a.e.f is a continuous function

=⇒ f(um(t, x)) → f(u(t, x)) a.e.

f(um(t, x)) → f(u(t, x)) a.e.‖f(um)‖Lq(ΩT ) 6 constant

=⇒ f(um) f(u) weakly in Lq(ΩT ),

where we have used the weak dominated convergence theorem. Finally, we deducethat A−1

ε ∂tum A−1ε ∂tu weakly in Lq(ΩT ). Thus, passing to the limit in (2.18), we

obtainA−1

ε ∂tu−∆u+ f(u) = 0, in Lq(ΩT ). (2.34)

To prove that u(0) = u0, we consider a test function ψ ∈ C1([0, T ];L2p(Ω)) suchthatψ(T ) = 0. Multiplying (2.4) by ψ and integrating over Ω× [0, T ], we obtain

∫ T

0

⟨

A−1ε ∂tu, ψ

⟩

dt−∫ T

0

〈∆u, ψ〉 dt+∫ T

0

〈f(u), ψ〉 dt = 0. (2.35)

Integrating by parts in (2.35), we have

−∫ T

0

⟨

A−1ε u, ∂tψ

⟩

dt−⟨

A−1ε u(0), ψ(0)

⟩

−∫ T

0

〈∆u, ψ〉 dt+∫ T

0

〈f(u), ψ〉 dt = 0.

(2.36)

40


Multiplying (2.18) by ψ and integrating over Ω× [0, T ], we obtain

−∫ T

0

⟨

A−1ε um, ∂tψ

⟩

dt−⟨

A−1ε Pmu(0), ψ(0)

⟩

−∫ T

0

〈∆um, ψ〉 dt+∫ T

0

〈f(um), ψ〉 dt = 0.

(2.37)

Having ψ ∈ L2p(0, T ;L2p(Ω)) and∂ψ

∂t∈ L2(0, T ;H) we deduce that

−∫ T

0

〈∆um, ψ〉 dt→ −∫ T

0

〈∆u, ψ〉 dt

and∫ T

0

⟨

A−1ε um, ∂tψ

⟩

dt→∫ T

0

⟨

A−1ε u, ∂tψ

⟩

dt.

Then, passing to the limit in (2.37) as m→ ∞, we obtain

−∫ T

0

⟨

A−1ε u, ∂tψ

⟩

dt−⟨

A−1ε u0, ψ(0)

⟩

−∫ T

0

〈∆u, ψ〉 dt+∫ T

0

〈f(u), ψ〉 dt = 0. (2.38)

We deduce from (2.36) and (2.38) that A−1ε u(0) = A−1

ε u0, which implies u(0) = u0.


In this section, we will derive some additional regularity for the solution u(t). Wehave the following theorem.

Theorem 2.4.1. Let us take u0 ∈ W . Then, there exists a unique solution u ofproblem (2.4) with initial datum u0 such that

u ∈ L∞([0, T ];H2(Ω)) ∩ L2([0, T ];H4(Ω)). (2.39)

Furthermore, we have

‖u‖H2(Ω) 6 Q(‖u0‖H2(Ω))e−αt + c, (2.40)

where the constants c, α > 0 and the monotonic function Q are independent of ε.

The proof is based on a priori estimates that we formally derive below, a rigorousjustification being based on the classical Galerkin method.

Lemma 2.4.2. Let u(t) be a solution of (2.4) with u0 ∈ W . Then, there exists atime T0 = T0(‖u0‖H2(Ω)), 0 < T0 < 1/2, and a monotonic function Q such that

‖∆u(t)‖ 6 Q(‖u0‖H2(Ω)), t 6 T0(‖u0‖H2(Ω)). (2.41)

Proof:Multiplying equation (2.4) by ∆2Aεu(t), we obtain the following inequality:

1

2

d

dt‖∆u‖2 + ‖∆2u‖2 + ε‖∇∆u‖2 =−

(

Aεf(u),∆2u)

61

2‖Aεf(u)‖2 +

1

2‖∆2u‖2

61

2c‖f(u)‖2H2(Ω) +

1

2‖∆2u‖2,

(2.42)

41


where c is independent of ε. Taking into account that f ∈ C2(Ω) and that H2(Ω) ⊂C(Ω), we can prove that there exists a monotonic functions Q (depending on f)such that

‖f(u)‖2H2(Ω) 6 Q(

‖u‖2H2(Ω)

)

6 Q(

‖∆u‖2)

. (2.43)

Indeed, we have for n 6 3 (where n is the space dimension)

‖f ′(u)‖2L∞(Ω) 6 Qf ′(‖u‖2L∞(Ω)) 6 Qf ′(‖u‖2H2(Ω)), since f ′ ∈ C(Ω),

‖f ′′(u)‖2L∞(Ω) 6 Qf ′′(‖u‖2L∞(Ω)) 6 Qf ′′(‖u‖2H2(Ω)), since f ′′ ∈ C(Ω),

and

‖f(u)‖2H2(Ω) 6c‖∆f(u)‖2

6c(‖f ′(u)∆u‖2 + ‖f ′′(u)∇u · ∇u‖2)6c(‖f ′(u)‖2L∞(Ω)‖∆u‖2 + ‖f ′′(u)‖2L∞(Ω)‖∇u‖4L4(Ω))

6c(‖f ′(u)‖2L∞(Ω)‖∆u‖2 + ‖f ′′(u)‖2L∞(Ω)‖∆u‖4)6Q(‖u‖2H2(Ω)),

(2.44)

for some monotonic increasing functions Qf ′ and Qf ′′ .Thus, replacing inequality (2.43) in (2.42), the function y(t) := ‖∆u(t)‖2 satisfiesthe inequality

y′(t) 6 Q (y(t)) .

Let z(t) be a solution of the following equation:

z′(t) = Q (z(t)) , z(0) = y(0) = ‖∆u(0)‖2.Due to the comparison principle, there exists a time T0(‖u0‖H2(Ω)) ∈ (0, 1/2) suchthat we have

y(t) 6 z(t), ∀t 6 T0(‖u0‖H2(Ω)). (2.45)

Then, the lemma is an immediate consequence of (2.45).

Lemma 2.4.3. Let the above assumptions hold and let T0 be the same as in Lemma2.4.2. Then, the following estimate holds:

t‖A−1/2ε ∂tu(t)‖2 6 Q(‖u0‖H2(Ω)), t ∈ (0, T0], (2.46)

for some monotonic function Q.

Proof:Multiplying (2.4) by ∂tu(t) and integrating over Ω, we obtain

‖A−1/2ε ∂tu‖2 +

1

2

d

dt‖∇u‖2 6 |(f(u), ∂tu)| 6 c‖f(u)‖2H1(Ω) + 1/2‖A−1/2

ε ∂tu‖2. (2.47)

Integrating (2.47) over [0, T0] and taking into account (2.41) and the fact thatH2(Ω) ⊂ C(Ω), we have

∫ T0

0

‖A−1/2ε ∂tu(t)‖2dt 6c

∫ T0

0

‖f(u(t))‖2H1(Ω)dt+ ‖∇u(0)‖2

6Q(‖u0‖H2(Ω)) + ‖∇u(0)‖26Q(‖u0‖H2(Ω)),

(2.48)

42


where we have used the fact that

‖f(u(t))‖2H1(Ω) 6 Q(‖u(t)‖H2(Ω)) 6 Q(‖u0‖H2(Ω)), ∀t ∈ [0, T0],

for some monotonic function Q. Differentiating (2.4) with respect to t and settingθ(t) = ∂tu(t), we find

A−1ε ∂tθ −∆θ + f ′(u(t))θ = 0,

θ|∂Ω = 0.(2.49)

Multiplying (2.49) by tθ(t), integrating over Ω and using the fact that f ′(u) > −κ,we obtain

d

dt(t‖A−1/2

ε θ‖2) + 2t‖∇θ‖2 62κt‖θ‖2 + ‖A−1/2ε θ‖2

6ct‖A−1/2ε θ‖‖∇θ‖+ ‖A−1/2

ε θ‖2

6t‖∇θ‖2 + c(t+ 1)‖A−1/2ε θ‖2.

Hence, we haved

dt(t‖A−1/2

ε θ‖2) 6 c(t+ 1)‖A−1/2ε θ‖2. (2.50)

Applying Gronwall’s lemma to estimate (2.50) over (0, t) for t 6 T0 6 1 and using(2.48), we deduce that

t‖A−1/2ε θ(t)‖2 6 c

∫ t

0

‖A−1/2ε θ(s)‖2ds 6 Q(‖u0‖H2(Ω)) (2.51)

and Lemma 2.4.3 is proven.

Lemma 2.4.4. Let u(t) be a solution of equation (2.4) and let t > T0, where T0is the same as in Lemma 2.4.2. Then, the following estimate holds uniformly withrespect to ε:

‖A−1/2ε ∂tu(t)‖2 + ‖u(t)‖2H2(Ω) +

∫ t+1

t

‖∂tu(s)‖2H1(Ω)ds 6 eK1tQ(‖u0‖H2(Ω)), ∀t > T0,

where K1 is a positive constant and Q is some monotonic function.

Proof:Multiplying (2.49) by θ(t), integrating over Ω and using the fact that f ′(u) > −κ,we have

d

dt‖A−1/2

ε θ‖2 + 2‖∇θ‖2 62κ‖θ‖2

62cκ‖A−1/2ε θ‖‖∇θ‖

6‖∇θ‖2 + c2κ2‖A−1/2ε θ‖2.

(2.52)

Applying Gronwall’s lemma to estimate (2.52) over (T0, t), we obtain

‖A−1/2ε θ(t)‖2 +

∫ t+1

t

‖∇θ(s)‖2ds 6 eK1t‖A−1/2ε θ(T0)‖2, t > T0, (2.53)

43


for some positive constant K1. Using Lemma 2.4.3, estimate (2.53) gives

‖A−1/2ε ∂tu(t)‖2 +

∫ t+1

t

‖∂tu(s)‖2H1(Ω)ds 6 eK1tQ(‖u0‖H2(Ω)), t > T0. (2.54)

Interpreting the parabolic equation (2.4) as an elliptic boundary value problem,

∆u(t)− f(u(t)) = h(t) := A−1ε ∂tu(t), u(t)|∂Ω = 0, (2.55)

for every fixed t > T0, estimate (2.54) implies that

‖h(t)‖2 = ‖A−1/2ε ∂tu(t)‖2 6 eK1tQ(‖u0‖H2(Ω)). (2.56)

Multiplying (2.55) by u(t), using the fact that f(u)u > −c and integrating over Ω,we obtain

‖∇u(t)‖2 6 c(1 + ‖h(t)‖2). (2.57)

Multiplying (2.55) by ∆u(t), using the fact that f ′(u) > −κ and integrating over Ω,we find

‖∆u(t)‖2 6 c(‖∇u(t)‖2 + ‖h(t)‖2)). (2.58)

We deduce from estimates (2.56), (2.57) and (2.58) that

‖u(t)‖2H2(Ω) 6 c‖∆u(t)‖2 6 ceK1tQ(‖u0‖H2(Ω)) (2.59)


Corollary 2.4.5. Let the above assumptions hold and let u(t) be a solution of prob-lem (2.4). Then, the following estimate holds:

‖u(t)‖H2(Ω) 6 Q(‖u(0)‖H2(Ω))eKt, (2.60)

for all t > 0 and where the positive constant K and the monotonic function Q areindependent of ε.

Proof:We have proved that estimate (2.60) is true for t 6 T0(‖u0‖H2(Ω)) in Lemma 2.4.2and for t > T0(‖u0‖H2(Ω)) in Lemma 2.4.4, which completes the proof.

Lemma 2.4.6. Let u(t) be a solution of problem (2.4). Then, the following estimateis valid, uniformly with respect to ε:

‖u(1)‖H2(Ω) 6 Q(‖u0‖2), (2.61)

for some monotonic function Q that is independent of ε.

Proof:Estimate (2.28) yields

∫ 1

0

‖u(s)‖H2(Ω)ds 6 c‖u(0)‖2 + c.

44


It follows that there exists a time T ∈ [0, 1] such that

‖u(T )‖H2(Ω) 6 c‖u(0)‖2 + c. (2.62)

Applying estimate (2.60) starting with the time t = T instead of t = 0 and using(2.62), we obtain the desired inequality.To complete the proof of Theorem 2.4.1, we proceed as follows. We have fromestimate (2.28) that for every t > 1, there exists a time t∗ ∈ [t− 1, t] such that:

‖∆u(t∗)‖2 6 c‖u(0)‖2e−αt + c.

Consequently, for t′ ∈ [0, 1] such that t = t∗ + t′, estimate (2.60) yields:

‖∆u(t)‖2 = ‖∆u(t∗ + t′)‖2 6 Q(‖∆u(t∗)‖2)eKt′

6 cQ(‖∆u(t∗)‖2)e−αt′

6 cQ(c‖u(0)‖2e−αt + c)

6 Q(‖u(0)‖2H2(Ω))e−αt + c,

for some monotonic function Q. Theorem 2.4.1 is thus proven.

Remark 2.4.7. Theorem 2.4.1 is also true for every function f such that f ∈C2(R), f(0) = 0, f ′(s) > −κ and f(s)s > −c, s ∈ R and κ, c > 0.

The next two Lemmata give additional regularity for the time derivative of u.

Lemma 2.4.8. Let the assumptions of Theorem 2.4.1 hold. Then, the solution u(t)of problem (2.4) satisfies

‖∂tu(t)‖2 +∫ t+1

t

‖∂tu(s)‖2H2(Ω)ds 6 Q(‖u0‖H2(Ω))e−αt + c, t > 1, (2.63)

where the positive constants c, α and the function Q are independent of ε.

Proof:Multiplying (2.49) by (t − T0)Aεθ(t), where T0 is the same as in Lemma 2.4.3 andθ = ∂tu, and integrating over Ω, we find

d

dt

(

(t− T0)‖θ(t)‖2

2

)

+α′(t− T0)‖θ(t)‖2

2+ cα′(t− T0)‖∆θ(t)‖2 + ε(t− T0)‖∇θ(t)‖2

6 (1 + κε(t− T0))‖θ(t)‖2 + c(t− T0)‖f ′(u(t))θ(t)‖26 c(t− T0)(‖θ(t)‖2 + ‖f ′(u(t))θ(t)‖2) := c(t− T0)hu(t),

(2.64)

for appropriate positive constants α′ and c′α. Applying Gronwall’s lemma to (2.64)over (T0, t), we obtain

(t− T0)‖θ(t)‖2eα′t6 c

∫ t

T0

(s− T0)hu(s)eα′sds

6 c(t− T0)

∫ t

T0

hu(s)eα′sds,

45


so that

‖θ(t)‖2 6 c

∫ t

T0

hu(s)e−α′teα

′sds = c

∫ t

T0

hu(s)e−α′(t−s)ds. (2.65)

To estimate∫ t

T0

hu(s)e−α′(t−s)ds, we proceed as follows.

Using Theorem 2.4.1 and estimate (2.54), we find∫ t+1

t

‖∂tu(s)‖2H1(Ω)ds 6 Q(‖u(t− T0)‖H2(Ω))eK1T0

6 eK1T0(Q(‖u0‖H2(Ω))e−α(t−T0) + c)

6 Q(‖u0‖H2(Ω))e−αt + c,

(2.66)

and using the fact that W ⊂ C(Ω), we obtain∫ t+1

t

hu(s)ds 6 Q( sups∈[t,t+1]

‖u(s)‖L∞(Ω))

∫ t+1

t

‖∂tu(s)‖2H1(Ω)ds 6 Q(‖u0‖H2(Ω))e−αt+c,

(2.67)for t > T0 and for an appropriate function Q and constants c, α > 0 which areindependent of ε.We have∫ t

T0

hu(s)e−α′(t−s)ds = e−α′t

∫ t

T0

hu(s)eα′sds

6e−α′t

(

∫ 1

T0

hu(s)eα′sds+

∫ 2

1hu(s)e

α′sds+ · · ·+∫ [t]

[t]−1hu(s)e

α′sds+

∫ [t]+1

[t]hu(s)e

α′sds

)

6e−α′t

(

eα′

∫ 1

T0

hu(s)ds+ e2α′

∫ 2

1hu(s)ds+ · · ·+ eα

′[t]

∫ [t]

[t]−1hu(s)ds+ eα

′([t]+1)

∫ [t]+1

[t]hu(s)ds

)

6e−α′t(eα′

(Q(‖u0‖H2(Ω))e−αT0 + c) + e2α

′

(Q(‖u0‖H2(Ω))e−α + c) + · · ·

+ eα′[t](Q(‖u0‖H2(Ω))e

−α([t]−1) + c) + eα′([t]+1)(Q(‖u0‖H2(Ω))e

−α[t] + c))

6e−α′t(eα′

(Q(‖u0‖H2(Ω)) + c) + e2α′

(Q(‖u0‖H2(Ω))e−α + c) + · · ·

+ eα′[t](Q(‖u0‖H2(Ω))e

−α([t]−1) + c) + eα′([t]+1)(Q(‖u0‖H2(Ω))e

−α[t] + c))

6e−α′teα′

Q(‖u0‖H2(Ω))

(

e(α′−α)([t]+1) − 1

e(α′−α) − 1

)

+ ce−α′teα′

(

eα′([t]+1) − 1

eα′ − 1

)

6Q(‖u0‖H2(Ω))e−αt + c,

(2.68)

where α 6 α′.From estimates (2.65) and (2.68), we deduce the lemma.

Lemma 2.4.9. Let the assumptions of Theorem 2.4.1 hold. Then, we have thefollowing:

‖∂tu(t)‖2H1(Ω) +

∫ t+1

t

‖∂2t u(s)‖2H−1(Ω)ds 6 Q(‖u(0)‖H2(Ω))e−αt + c,

where t > 2 and the constants c, α > 0 and the function Q are independent of ε.

46


Proof:Multiplying (2.49) by (t − T∗)∂tθ(t), where θ = ∂tu and t > T∗ > 1 is arbitrary,using (2.8) and integrating over Ω, we obtain

1

2

d

dt[(t− T∗)‖∇θ(t)‖2] + (t− T∗)‖∂tθ(t)‖2H−1(Ω)

=1

2‖∇θ(t)‖2 − (t− T∗)(f

′(u(t))θ(t), ∂tθ(t)) := Hu(t).

(2.69)

We have

(f ′(u(t))θ(t), ∂tθ(t)) 6 ‖f ′(u(t))θ(t)‖H1(Ω)‖∂tθ(t)‖H−1(Ω)

6(‖f ′(u(t))‖L∞(Ω)‖∇θ(t)‖+ ‖∇f ′(u(t))‖L4(Ω)‖θ(t)‖L4(Ω))‖∂tθ(t)‖H−1(Ω)

6(‖f ′(u(t))‖L∞(Ω) + ‖∇f ′(u(t))‖L4(Ω))‖θ(t)‖H1(Ω)‖∂tθ(t)‖H−1(Ω)

6(‖f ′(u(t))‖L∞(Ω) + ‖f ′′(u(t))‖L∞(Ω)‖u(t)‖H2(Ω))‖θ(t)‖H1(Ω)‖∂tθ(t)‖H−1(Ω).

(2.70)

It follows from Theorem 2.4.1 and estimates (2.63), (2.70) and from the fact thatW ⊂ C(Ω) and H1(Ω) ⊂ L6(Ω) ⊂ L4(Ω) (n 6 3) that

∫ T∗+s

T∗

Hu(t)dt 6 (1+s)(Q(‖u(0)‖H2(Ω))e−αT∗+c)+

1

2

∫ T∗+s

T∗

(t−T∗)‖∂tθ(t)‖2H−1(Ω)dt,

(2.71)where s ∈ [0, 2] and Q is an appropriate function independent of ε.Integrating (2.69) with respect to t ∈ [T∗, T∗ + s] and using (2.71), we find

s‖θ(T∗ + s)‖2H1(Ω) +

∫ T∗+s

T∗

(t− T∗)‖∂tθ(t)‖2H−1(Ω)dt 6 Q(‖u(0)‖H2(Ω))e−αT∗ + c,

for T∗ > 1 and s ∈ [0, 2]. Thus, for T∗ := t− 1 > 1 and s = 1, we deduce that

‖θ(t)‖2H1(Ω) +

∫ t

t−1

‖∂tθ(s)‖2H−1(Ω)ds 6 Q(‖u(0)‖H2(Ω))e−α(t−1) + c

6 cQ(‖u(0)‖H2(Ω))e−αt + c.

(2.72)

We thus deduce the lemma from estimates (2.72).


In this section, we will study the long time behavior of the solutions of (2.4). Wehave the following.

Lemma 2.5.1. Problem (2.4) generates the following semigroup on the phase spaceH:

Sε(t) : H −→ H

u0 7−→ Sε(t)u0 = u(t), t > 0,

47


where u(t) is the unique solution of problem (2.4), at time t, with initial datum u0.Furthermore, this semigroup is Lipschitz continuous in the H−1(Ω)−topology,

‖A−1/2ε (Sε(t)u1−Sε(t)u2)‖2+

∫ t+1

t

‖Sε(s)u1−Sε(s)u2‖2H1(Ω)ds 6 ceCt‖A−1/2ε (u1−u2)‖2,

(2.73)for any u1, u2 ∈ H, where c and C are positive constants independent of t and ε.Thus, Sε(t) can be uniquely extended, by continuity, to a semigroup, still denoted bySε(t), acting on H−1(Ω).

To prove the existence of the global attractor, we apply the following theorem.

Theorem 2.5.2. (See [50]) If Sε(t) is dissipative in H−1(Ω) and B is a compactabsorbing set in H−1(Ω), then Sε(t) has the global attractor Aε = ω(B).

We have the following theorem.

Theorem 2.5.3. The semigroup Sε(t) possesses the global attractor Aε in H−1(Ω)which is bounded in W.

We devide the proof of this theorem into several propositions.

Proposition 2.5.4. Problem (2.4) has an absorbing set in H−1(Ω). More precisely,there exists a positive constant ρ0 and a time t0 := t0(‖u0‖−1) such that

‖A−1/2ε u(t)‖ 6 ρ0, for all t > t0.

Proof:We multiply (2.4) by u, integrate over Ω and obtain

1

2

d

dt‖A−1/2

ε u‖2 + ‖∇u‖2 + (f(u), u) = 0. (2.74)

Using (2.3) with α = p− 3

2, we have

0 6

∫

Ω

|f(u)|dx 6 pb2p

∫

Ω

u2pdx− 3

2b2p

∫

Ω

u2pdx+ c|Ω|

6 (f(u), u)− 3

2b2p

∫

Ω

u2pdx+ c.

Thus, using (2.3), we get

‖∇u‖2 + (f(u), u) > ‖∇u‖2 + 3

2b2p

∫

Ω

u2pdx− c

>1

2‖∇u‖2 +

∫

Ω

F (u)dx− c

= E(u)− c.

Therefore, (2.74) yields

1

2

d

dt‖A−1/2

ε u‖2 + E(u) 6 c. (2.75)

48


Using (2.3) and the fact that

‖A−1/2ε u‖2 6 c‖u‖2 6 c‖∇u‖2, ∀u ∈ V,

we have the following inequality:

1

2

d

dt‖A−1/2

ε u‖2 + 1

2c‖A−1/2

ε u‖2 + 1

2b2p

∫

Ω

u2pdx 6 c.

By Gronwall’s lemma, we find

‖A−1/2ε u(t)‖2 6‖A−1/2

ε u(0)‖2e−t/c + c(1− e−t/c)

6c‖(−∆)−1/2u(0)‖2e−t/c + c(1− e−t/c).

It follows that if t > t0(‖(−∆)−1/2u(0)‖2) ≡ c ln(

c‖(−∆)−1/2u(0)‖2)

, then

‖A−1/2ε u(t)‖2 6 ρ20.

Proposition 2.5.5. For an arbitrary constant r > 0, there exists a positive constantρV depending on r and a time t1 := t1(‖u0‖−1, r) such that

‖∇u(t)‖ 6 ρV (r) and ‖u(t)‖pL2p(Ω) 6ρV (r)√

b2p(2.76)

for all t > t0 + r := t1.

Proof:Fixing a constant r > 0 and integrating (2.75) with respect to t, we obtain

∫ t+r

t

E(u(s))ds 6 1

2ρ20 + rc, ∀t > t0, (2.77)

where t0 is as in Proposition 2.5.4. Since E decays along the trajectories, we concludefrom (2.77) that

E(u(t+ r)) 61

2rρ20 + c, ∀t > t0.

By the definition of E and (2.3), we have

‖∇u(t)‖2 + b2p

∫

Ω

u2p(x, t)dx 6 2c|Ω|+ 1

rρ20 + 2c := ρ2V , (2.78)

for all t > t0 + r := t1. The proposition is thus proven.

Proposition 2.5.6. For an arbitrary constant r > 0, there exists a positive constantρ depending on r and a time t2 := t2(‖u0‖−1, r) such that

‖∆u(t)‖ 6 ρ(r) for all t > t1 + r := t2.

49


Proof:Multiplying (2.4) by ∂tu, we obtain

1

2

d

dt‖∇u‖2 + d

dt

∫

Ω

F (u)dx+ ‖A−1/2ε ∂tu‖2 = 0. (2.79)

Using (2.76), we find∫ t+r

t

‖A−1/2ε ∂tu(s)‖2ds+

1

2‖∇u(t+ r)‖2 + b2p

2

∫

Ω

u2p(t+ r)dx

61

2‖∇u(t)‖2 + b2p

2

∫

Ω

u2p(t)dx+ c

63

2ρ2V + c, ∀t > t1.

(2.80)

Multiplying (2.49) by ∂tu and using the fact that f ′(u) > −κ, we obtain

1

2

d

dt‖A−1/2

ε ∂tu‖2 + ‖∇∂tu‖2 6 κ‖∂tu‖2 6 c‖A−1/2ε ∂tu‖‖∇∂tu‖,

which yieldsd

dt‖A−1/2

ε ∂tu‖2 6 c‖A−1/2ε ∂tu‖2, (2.81)

for some positive constant c. Using (2.80), (2.81) and the uniform Gronwall lemma,we deduce that

‖A−1/2ε ∂tu(t)‖2 6 c(r), ∀t > t1 + r. (2.82)

We rewrite (2.4) in the following form:

−∆u+ f(u) = −A−1ε ∂tu. (2.83)

Multiplying (2.83) by −∆u(t) and using the fact that f ′(u) > −κ, we obtain

‖∆u(t)‖2 6κ‖∇u(t)‖2 + (A−1ε ∂tu(t),∆u(t))

6κ‖∇u(t)‖2 + ‖A−1ε ∂tu(t)‖‖∆u(t)‖

6κ‖∇u(t)‖2 + c‖A−1/2ε ∂tu(t)‖‖∆u(t)‖,

which yields‖∆u(t)‖2 6 2κ‖∇u(t)‖2 + c‖A−1/2

ε ∂tu(t)‖2. (2.84)

Using (2.78) and (2.82), estimate (2.84) gives

‖∆u(t)‖ 6 ρ(r), ∀t > t2 := t1 + r,

for some positive constant ρ(r) which depend on r and Proposition 2.5.6 is proven.

LetB :=

u ∈ W, ‖∆u‖2 6 ρ

, (2.85)

where ρ is the same as in Proposition 2.5.6. Then, B is a compact absorbing setin H−1(Ω) since it is an absorbing set in W and W is compactly embedded intoH−1(Ω). Applying Theorem 2.5.2, we deduce the existence of the global attractorAε = ω(B) and Theorem 2.5.3 is thus proven.

50



To prove the existence of an exponential attractor, we apply the following theorem.

Theorem 2.6.1. (See [44]) Let E and E1 be two Hilbert spaces such that E1 iscompactly embedded into E and S(t) : X → X be a semigroup acting on a closedsubset X of E. We assume that

1. ∀x1, x2 ∈ X, ∀t > 0,

‖S(t)x1 − S(t)x2‖E16 h(t)‖x1 − x2‖E,

where the function h is continuous;

2. (t, x) 7→ S(t)x is uniformly Hölder continuous in the topology of E on [0, T ]×B,∀T > 0, ∀B ⊂ X bounded.

Then, S(t) possesses an exponential attractor on X.

In order to apply this result to the semigroup Sε(t) associated with problem (2.4),we set E = H−1(Ω), E1 = H2(Ω) and we consider the set

X =⋃

t> t2

S(t)BH−1(Ω)

,

where B is defined in (2.85). X is thus compact in H−1(Ω), bounded in H2(Ω) andpositively invariant by Sε(t).

Let u1 and u2 be two solutions of (2.4) with initial data u01, u02 ∈ X, respectively.We set w = u1 − u2. Then, w verifies

A−1ε ∂tw −∆w + l(t)w = 0,

w(0) = u1(0)− u2(0),(2.86)

where l(t) =∫ 1

0

f ′(su1(t) + (1− s)u2(t))ds.

The next lemma gives the H−1(Ω) → H-smoothing property for the difference oftwo solutions.

Lemma 2.6.2. Let u1(t) and u2(t) be two solutions of (2.4) such that ‖∆ui(0)‖ 6

ρ, i = 1, 2. Then, the following estimate is valid:

t‖u1(t)− u2(t)‖2 +∫ t+1

t

‖∆u1(s)−∆u2(s)‖2ds 6 Rρeαt‖u1(0)− u2(0)‖2−1, t > 0,

(2.87)where the constants Rρ, α > 0 are independent of ε and Rρ depends on ρ.

51


Proof:Multiplying (2.86) by −t∆w and integrating over Ω, we find

1

2td

dt(−∆A−1

ε w,w) + t‖∆w‖2 − t (l(t)w,∆w) = 0. (2.88)

Setting (−∆A−1ε w,w) = ‖w‖2∗, where ‖.‖∗ ∼ ‖.‖L2(Ω), equation (2.88) yields

1

2

d

dt

(

t‖w‖2∗)

+ t‖∆w‖2 61

2‖w‖2∗ + t (l(t)w,∆w)

61

2‖w‖2∗ + t‖l(t)‖L∞(Ω)‖w‖‖∆w‖

61

2‖w‖2∗ + tcρ‖w‖‖∆w‖

61

2‖w‖2∗ +

t

2c2ρ‖w‖2 +

t

2‖∆w‖2

61

2‖w‖2∗ +

t

2c2ρ‖∇w‖2 +

t

2‖∆w‖2,

(2.89)

where ‖l(t)‖L∞(Ω) 6 C(‖u1(0)‖H2(Ω), ‖u2(0)‖H2(Ω)) 6 cρ and cρ is a positive constantdepending on ρ. Integrating with respect to s ∈ [0, t], we obtain

t‖w(t)‖2∗ +∫ t

0

s‖∆w(s)‖2ds 6∫ t

0

‖w(s)‖2∗ds+ c2ρ

∫ t

0

s‖∇w(s)‖2ds

6c

∫ t

0

‖∇w(s)‖2ds+ c2ρ

∫ t

0

s‖∇w(s)‖2ds

6Rρeαt‖u1(0)− u2(0)‖2−1,

(2.90)

where we have used the fact that ‖w‖∗ 6 c‖∇w‖ and estimates (2.8) and (2.17).Now, multiplying (2.86) by −∆w and integrating over Ω, we find

1

2

d

dt

(

‖w‖2∗)

+ ‖∆w‖2 6 (l(t)w,∆w)

6c2ρ2‖∇w‖2 + 1

2‖∆w‖2.

(2.91)

Integrating (2.91) over (t, t+ 1), t > 0, we obtain

∫ t+1

t

‖∆w(s)‖2ds 6 ‖u1(0)− u2(0)‖2−1Rρeαt. (2.92)

Thus, the lemma is an immediate consequence of estimates (2.90) and (2.92).

Lemma 2.6.3. Let the assumptions of Lemma 2.6.2 hold. Then, the followingestimate holds:

‖u1(t)−u2(t)‖2H1(Ω)+

∫ t+1

t

‖∂tu1(s)−∂tu2(s)‖2−1ds 6 Rρeαρt‖u1(0)−u2(0)‖2−1, t > 2,

(2.93)where Rρ and αρ are positive constants depending on ρ and independent of ε.

52


Proof:We have

‖∇(l(t)w)‖ 6 ‖l(t)‖L∞(Ω)‖∇w‖+ c‖∇l(t)‖L4(Ω)‖w‖L4(Ω)

6 c(‖l(t)‖L∞(Ω) + ‖∇l(t)‖L4(Ω))‖∇w‖ (n 6 3).

Using the embeddings W ⊂ C(Ω) and W ⊂ W 1,4(Ω), we find

‖∇l(t)‖L4(Ω) 6 Q(maxi=1,2

‖ui(t)‖L∞(Ω))maxi=1,2

‖ui(t)‖W 1,4(Ω) 6 Cρ,

where Q is an appropriate function and Cρ is a positive constant depending on ρwhich are independent of ε. Consequently, we obtain

‖∇(l(t)w)‖ 6 cρ‖∇w‖,

where cρ > 0 depend on ρ and independent of ε.Now, multiplying (2.86) by (t − T∗)∂tw, where T∗ > 1 is arbitrary and such thatt− T∗ > 0 , we have

1

2

d

dt((t− T∗)‖∇w‖2) + c(t− T∗)‖∂tw‖2−1 =

1

2‖∇w‖2 − (t− T∗)(l(t)w, ∂tw)

61

2‖∇w‖2 + (t− T∗)‖∇(l(t)w)‖‖∂tw‖−1

61

2‖∇w‖2 + (t− T∗)cρ‖∇w‖‖∂tw‖−1

6 (1 + cρ(t− T∗))‖∇w‖2

2+c

2(t− T∗)‖∂tw‖2−1.

(2.94)

Integrating (2.94) with respect to t ∈ [T∗, T∗+s], for T∗ > 1 and s ∈ [0, 2], and usingestimate (2.17), we find

s‖∇w(T∗ + s)‖2 + c

∫ T∗+s

T∗

(t− T∗)‖∂tw(t)‖2−1dt 6 ‖u1(0)− u2(0)‖2−1RρeαρT∗ .

(2.95)

Setting T∗ := t− 1 > 1 and s = 1, estimate (2.95) yields

‖∇w(t)‖2 + c

∫ t

t−1

‖∂tw(s)‖2−1ds 6 ‖u1(0)− u2(0)‖2−1Rρeαρ(t−1)

6 ‖u1(0)− u2(0)‖2−1Rρeαρt,

(2.96)

and the lemma is thus proven.We now have a H−1(Ω) → H2(Ω)−smoothing property.

Lemma 2.6.4. Let u1(t) and u2(t) be two solutions of (2.4) such that ‖∆ui(0)‖ 6


‖u1(t)−u2(t)‖2H2(Ω)+‖∂tu1(t)−∂tu2(t)‖2−1 6 Rρeαρt‖u1(0)−u2(0)‖2−1, t > 3, (2.97)

where Rρ and αρ are positive constants depending on ρ and independent of ε.

53


Proof:We differentiate equation (2.86) with respect to t and set θ(t) = ∂tw(t). This functionsatisfies the equation

A−1ε ∂tθ −∆θ + l(t)θ + ∂tl(t)w = 0, θ|∂Ω = 0.

Multiplying this equation by (t− 2)θ(t), integrating over Ω and noting that

‖∂tl(t)w(t)‖2 6 maxi=1,2

(‖ui(t)‖L∞(Ω))(‖∂tu1(t)‖2H1(Ω) + ‖∂tu2(t)‖2H1(Ω))‖∇w(t)‖2

6cρ‖∇w(t)‖2 (using Lemma 2.4.9),

we obtain the following estimate:

d

dt((t− 2)‖A−1/2

ε θ‖2) + (t− 2)‖∇θ‖2 6 cρ(t− 1)‖A−1/2ε θ‖2 + cρ(t− 2)‖∇w‖2, t > 3.

Applying Gronwall’s Lemma, using the bound for ‖∇w‖2 and estimate (2.93), weprove that

‖θ(t)‖2−1 +

∫ t+1

t

‖∇θ(s)‖2ds 6 Rρeαρt‖u1(0)− u2(0)‖2−1, ∀t > 3.

Now, interpreting equation (2.86) as an elliptic equation,

∆w − l(t)w = A−1ε ∂tw,

we have

‖∆w(t)‖2 = (A−1ε ∂tw(t),∆w(t)) + (l(t)w(t),∆w(t))

6 c‖∂tw(t)‖2−1 +1

2‖∆w(t)‖2 + ‖∇(l(t)w(t))‖‖∇w(t)‖

6 c‖∂tw(t)‖2−1 +1

2‖∆w(t)‖2 + cρ‖∇w(t)‖2.

Using the above estimates and (2.93), we find

‖∆w(t)‖2 6 Rρeαρt‖u1(0)− u2(0)‖2−1, ∀t > 3.

Theorem 2.6.5. Let u1(t) and u2(t) be two solutions of (2.4) such that ‖∆ui(0)‖ 6


‖u1(t2)− u2(t2)‖2H2(Ω) 6 Rρ‖u1(0)− u2(0)‖2−1, (2.98)

where Rρ is a positive constant depending on ρ and independent of ε.

Proof:Having estimate (2.97) for every t > 3, we rescale the time (t→ αt) and we deducethat estimate (2.98) is valid for every t > T ′, T ′ > 0 arbitrary. In particular, thisholds for t = t2.

54

2.7 Construction of a robust family of exponential attractors

Lemma 2.6.6. The semigroup Sε(t) is uniformly Hölder continuous on [0, T ]×B,having endowed B with the H−1−topology, i.e.,

‖A−1/2ε (Sε(t1)u01 − Sε(t2)u02)‖ 6 Rρ,T

(

‖A−1/2ε (u01 − u02)‖+ |t1 − t2|1/2

)

,

where u0i ∈ B, ti 6 T, i = 1, 2, and Rρ,T is a positive constant depending on ρ andT and independent of ε.

Proof:The Lipschitz continuity with respect to the initial conditions is an immediate con-sequence of estimate (2.17). In order to verify the Hölder continuity with respect tot, we integrate (2.79) from t1 to t2 and find

∫ t2

t1

‖A−1/2ε ∂tu(s)‖2ds+

1

2‖∇u(t2)‖2 +

∫

Ω

F (u(t2))dx

=1

2‖∇u(t1)‖2 +

∫

Ω

F (u(t1))dx.

Using (2.3), we have∫ t2

t1

‖A−1/2ε ∂tu(s)‖2ds+

1

2‖∇u(t2)‖2 +

b2p2

∫

Ω

u2p(t2)dx

6 2c|Ω|+ 1

2‖∇u(t1)‖2 +

3

2b2p

∫

Ω

u2p(t1)dx

6 cρ (using (2.80)).

Consequently, we deduce that

‖A−1/2ε (u(t1)− u(t2))‖ 6 c‖u(t1)− u(t2)‖−1 =c

∥

∥

∥

∥

∫ t2

t1

∂tu(s)ds

∥

∥

∥

∥

H−1(Ω)

6c

∫ t2

t1

‖∂tu(s)‖H−1(Ω)ds

6c|t1 − t2|1/2(∫ t2

t1

‖∂tu(s)‖2H−1(Ω)ds

)1/2

6c|t1 − t2|1/2(∫ t2

t1

‖A−1/2ε ∂tu(s)‖2ds

)1/2

6cρ|t1 − t2|1/2,where cρ is a positive constant depending on ρ. We have thus proved the Höldercontinuity with respect to t.

Hence, we deduce the existence of an exponential attractor for problem (2.4).

2.7 Construction of a robust family of exponentialattractors

In this section, we will apply the following theorem to the study of the long timebehavior of the solutions of (2.4).

55


Theorem 2.7.1. (See [11]) Let E1 and E be two Banach spaces such that theinclusion E1 ⊂ E is compact and let B be a bounded set of E. We assume thatthere exists a family of operators Sε : B → B, ε ∈ [0, 1], which satisfies the followingassumptions:

1. For every x1, x2 ∈ B, the following estimate is valid:

‖Sεx1 − Sεx2‖E16 L‖x1 − x2‖E, (2.99)

where the constant L is independent of ε.

2. For every ε ∈ [0, 1], for i ∈ IN and for x ∈ B, we have the estimate

‖Siεx− Si

0x‖E 6 Kiε. (2.100)

Then, for every ε ∈ [0, ε0], there exists an exponential attractor Mε for the map Sε

in B. Moreover, the exponential attractor Mε can be chosen such that the followingestimate is valid:

distsym(Mε,M0) 6 C1εν ,

where the constant C1 and the exponent 0 < ν < 1 can be calculated explicitly anddistsym denotes the symmetric Hausdorff distance between two sets in E. Finally, thefractal dimension of the exponential attractors considered above is uniformly boundedwith respect to ε ∈ [0, ε0],

dimF (Mε, E) 6 C = C(L),

where the constant C is independent of ε and can be calculated explicitly.

In order to apply the theorem above, let u0(t) and uε(t) be two solutions of (2.1)with respectively zero (ε = 0) and nonzero (0 < ε 6 1) parameters,

∂tuε + A2uε + Af(uε) + εAuε + εf(uε) = 0

and

∂tu0 + A2u0 + Af(u0) = 0,

where A = −∆.We have the following result on the difference of the two solutions.

Theorem 2.7.2. Let ‖u0(0)‖H2(Ω) and ‖uε(0)‖H2(Ω) 6 ρ, with ρ a positive constant.Then, the following estimate holds:

‖u0(t)− uε(t)‖2−1 6 Rρeαt(‖u0(0)− uε(0)‖2−1 + ε2), (2.101)

where t > 0, α is a positive constant independent of ε and Rρ is a positive constantdepending on ρ and independent of ε.

56


Proof:Let wε(t) = uε(t)− u0(t). Then, wε satisfies the following equation:

A−1∂twε + Awε + lε(t)w

ε + εuε + εA−1f(uε) = 0, (2.102)

where lε(t) :=∫ 1

0f ′(suε(t)+(1−s)u0(t))ds. Multiplying (2.102) by wε(t), we obtain

1

2

d

dt‖wε‖2−1 + ‖∇wε‖2 + (lε(t)w

ε, wε) + ε(uε, wε) + ε(A−1f(uε), wε) = 0. (2.103)

We have

(lε(t)wε, wε) > −κ‖wε‖2

> −1

4‖∇wε‖2 − c‖wε‖2−1,

ε(uε, wε) 6 ‖wε‖2 + ε2

4‖uε‖2

61

4‖∇wε‖2 + c‖wε‖2−1 +

ε2

4‖uε‖2

and

ε(A−1f(uε), wε) 6 ε‖f(uε)‖−1‖wε‖−1

6 ‖wε‖2−1 +ε2

4‖f(uε)‖2−1

6 ‖wε‖2−1 + ε2Q(‖uε‖H2(Ω)) (using (2.43))

6 ‖wε‖2−1 + cρε2 (using (2.40)),

where cρ depends on ρ and is independent of ε. Then, (2.103) gives

d

dt‖wε‖2−1 + ‖∇wε‖2 6 c‖wε‖2−1 + c′ε2hε(t), (2.104)

where hε(t) := ‖uε(t)‖2+ cρ. Applying Gronwall’s lemma and using estimate (2.40),inequality (2.104) gives

‖wε(t)‖2−1 +

∫ t+1

t

‖∇wε(s)‖2ds 6 Rρeαt(‖wε(0)‖2−1 + ε2). (2.105)

Theorem 2.7.3. Let ‖u0(0)‖H2(Ω) and ‖uε(0)‖H2(Ω) 6 ρ. We have the following:

t‖uε(t)− u0(t)‖2 6 Rρeαρt(‖uε(0)− u0(0)‖2−1 + ε2), t > 0, (2.106)


57


Proof:Multiplying (2.102) by tAwε and integrating over Ω, where wε = uε − u0, we obtain

t

2

d

dt‖wε‖2 + t‖Awε‖2 + t(lε(t)w

ε, Awε) + εt(uε, Awε) + εt(f(uε), wε) = 0.

Thus,

1

2

d

dt(t‖wε‖2) + t‖∆wε‖2 =t(lε(t)wε,∆wε) + εt(uε,∆wε)− εt(f(uε), wε) +

1

2‖wε‖2

6t‖lε(t)‖∞‖wε‖‖∆wε‖+ εt‖uε‖‖∆wε‖+ εt‖f(uε)‖‖wε‖+ 1

2‖wε‖2

6tc2ρ‖wε‖2 + t

4‖∆wε‖2 + ε2t‖uε‖2 + t

4‖∆wε‖2 + ε2t‖f(uε)‖2

+ t‖wε‖2 + 1

2‖wε‖2,

where ‖lε(t)‖L∞(Ω) 6 Q(max(‖uε(t)‖L∞(Ω), ‖u0(t)‖L∞(Ω))) 6 cρ. Therefore, we have

d

dt(t‖wε‖2) + t‖∆wε‖2 6(c2ρ + 1)t‖wε‖2 + ε2t(‖uε‖2 + ‖f(uε)‖2) + ‖wε‖2

6(c2ρ + 1)t‖wε‖2 + ε2t(‖uε‖2 + cρ) + c‖∇wε‖2,

where we have used (2.44). Applying Gronwall’s lemma and using (2.40) and esti-mate (2.105), we deduce that

t‖uε(t)− u0(t)‖2 +∫ t+1

t

s‖∆wε(s)‖2ds 6 Rρeαρt(‖uε(0)− u0(0)‖2−1 + ε2). (2.107)

Theorem 2.7.4. Let the assumptions of Theorem 2.7.2 hold. Then, the followingestimate is valid:

t‖uε(t)−u0(t)‖2H1(Ω)+

∫ t+1

t

s‖∂tuε(s)−∂tu0(s)‖2−1ds 6 Rρeαρt(‖uε(0)−u0(0)‖2−1+ε

2), t > 0,

(2.108)where Rρ and αρ are positive constants depending on ρ and independent of ε.

Proof:Multiplying (2.102) by t∂twε, we find

1

2

d

dt(t‖∇wε‖2) + t‖∂twε‖2−1 =

1

2‖∇wε‖2 − t(lε(t)w

ε, ∂twε)− εt(uε, ∂tw

ε)− εt(A−1f(uε), ∂twε)

61

2‖∇wε‖2 + t‖∇(lε(t)w

ε)‖‖∂twε‖−1 + εt‖∇uε‖‖∂twε‖−1

+ εt‖f(uε)‖−1‖∂twε‖−1

61

2‖∇wε‖2 + cρt‖∇wε‖2 + t

2‖∂twε‖2−1 + cρε

2t(‖∇uε‖2 + 1).

Applying Gronwall’s lemma, using estimates (2.105), (2.107) and Theorem 2.4.1, wededuce estimate (2.108).

58


Theorem 2.7.5. Let ‖u0(0)‖H2(Ω) and ‖uε(0)‖H2(Ω) 6 ρ. We have the following:

t‖uε(t)−u0(t)‖2H2(Ω)+t‖∂tuε(t)−∂tu0(t)‖2−1 6 Rρeαρt(‖uε(0)−u0(0)‖2−1+ε

2), t > 0,(2.109)


Proof:Differentiating (2.102) with respect to t and setting θε = ∂tw

ε, we obtain

A−1∂tθε + Aθε + lε(t)θ

ε + ∂tlε(t)wε + ε∂tu

ε + εA−1f ′(uε(t))∂tuε = 0. (2.110)

Multiplying (2.110) by tθε, using the fact that lε(t) > −κ, ‖∂tlε(t)wε‖ 6 cρ‖∇wε‖and‖f ′(uε)‖L∞(Ω) 6 cρ, we find

1

2

d

dt(t‖θε‖2−1) + t‖∇θε‖2 61

2‖θε‖2−1 + κt‖θε‖2 − t(∂tlε(t)w

ε, θε)

− εt(∂tuε, θε)− εt(A−1f ′(uε)∂tu

ε, θε)

61

2‖θε‖2−1 + ct‖θε‖−1‖∇θε‖+ t‖∂tlε(t))wε‖‖θε‖+ εt‖∂tuε‖−1‖∇θε‖+ εt‖f ′(uε)∂tu

ε‖−1‖θε‖−1

6cρ(t+ 1)‖θε‖2−1 + cρt(‖∇wε‖2 + ε2‖∂tuε‖2−1) +t

2‖∇θε‖2.

Using estimates (2.48) and (2.108), we deduce that

t‖∂tuε(t)− ∂tu0(t)‖2−1 6 Rρe

αρt(‖uε(0)− u0(0)‖2−1 + ε2), t > 0. (2.111)

Interpreting equation (2.102) as an elliptic equation,

Awε(t) + lε(t)wε(t) = −A−1∂tw

ε(t)− εuε(t)− εA−1f(uε(t)) (2.112)

and multiplying (2.112) by Awε(t), we obtain

‖Awε(t)‖2 =− (lε(t)wε(t), Awε(t))− (A−1∂tw

ε(t), Awε(t))− (εuε(t), Awε(t))

− (εA−1f(uε(t)), Awε(t))

6cρ‖∇wε(t)‖‖Awε(t)‖+ ‖∂twε(t)‖−1‖∇wε(t)‖+ ε‖uε(t)‖‖Awε(t)‖+ ε‖f(uε(t))‖−1‖∇wε(t)‖,

hence‖Awε(t)‖2 6 cρ(‖∂twε(t)‖2−1 + ‖∇wε(t)‖2 + ε2). (2.113)

Using (2.111) and (2.113), we have the desired inequality.

Theorem 2.7.6. Let the above assumptions hold. Then, for every ε > 0, the semi-group Sε(t) generated by equation (2.4) possesses an exponential attractor Mε ⊂ W.Moreover, these exponential attractors can be chosen such that

dimF (Mε,W ) 6 C, (2.114)

59


distsym,W (Mε,M0) 6 C ′εη, (2.115)

where the constants C,C ′ > 0 and 0 < η < 1 are independent of ε and can becalculated explicitly and where the rate of convergence to these attractors is alsouniform with respect to ε, i.e., there exists a constant α > 0 such that, for everybounded subset B0 ⊂ W, there exists a constant C ′′ = C ′′(B0) such that

distW (Sε(t)B0,Mε) 6 C ′′e−αt,

where C ′′ and α are also independent of ε

Proof:We recall that the set

B = u0 ∈ W, ‖∆u0‖ 6 ρis a uniformly absorbing set for the semigroup Sε(t), ε ∈ [0, 1], i.e., for every boundedset B0 ⊂ H2(Ω), there exists a time T which is independent of ε, such that

Sε(t)B0 ⊂ B, ∀t > T, ∀ε ∈ [0, 1].

Therefore, it is sufficient to construct exponential attractors Mε on B only. Wedefine a family of maps Sε := Sε(t2) : B → B, for ε ∈ [0, 1] and then constructexponential attractors Md

ε for the discrete semigroups generated by these maps.Moreover, it is convenient to construct these attractors on B endowed first with themetric of the space H−1(Ω). Then, we apply Theorem 2.7.1 with E := H−1(Ω) andE1 := H2(Ω). Estimate (2.99) is an immediate consequence of Theorem 2.6.5 andestimate (2.100) is a corollary of Theorem 2.7.2. Thus, all assumptions of Theorem2.7.1 are satisfied and, consequently, we have

dimF (Mdε, H

−1(Ω)) 6 C,

distsym,H−1(Ω)(Mdε,Md

0) 6 C ′εν ,

distH−1(Ω)(SnεB,Md

ε) 6 C ′′e−αn,

for appropriate constants C,C ′, C ′′, α and ν which are independent of ε. Finally, weset

Mcε := ∪t∈[0,t2]Sε(t)Md

ε.

Since, (t, x) → Sε(t)x is uniformly Hölder continuous (Lemma 2.6.6) on [0, t2]× B,the exponential attractors Mc

ε, ε ∈ [0, 1], satisfy

1.dimF (Mc

ε, H−1(Ω)) 6 C + 2, (2.116)

2.distH−1(Ω)(Sε(t)B,Md

ε) 6 C1e−αt/t2 , (2.117)

3.

distsym,H−1(Ω)(Mcε,Mc

0) 6 C(

distsym,H−1(Ω)(Mdε,Md

0) + ε)

6 C2εν . (2.118)

60


Thus, we have constructed exponential attractors Mcε for the metric of H−1(Ω). We

note that it follows from its definition that the set Sε(1)Mcε is also an exponential

attractor. We finally setMε := Sε(1)Mc

ε.

Thus, we deduce estimates (2.114) and (2.115) from estimates (2.116), (2.118) andfrom Lemma 2.6.2 and Theorem 2.7.5 for t = 1.Acknowledgments: The authors wish to thank the referees for their careful readingof the paper and useful comments.

61


62

Chapitre 3

A Cahn-Hilliard Type Equation

With Dynamic Boundary Conditions

and Regular Potentials

Equation de type Cahn-Hilliard avecdes conditions dynamiques sur le bordet des potentiels réguliers

3.1 Introduction

Given a bounded and smooth domain Ω ⊂ R3 with boundary Γ := ∂Ω, we consider

the following problem:

∂tu = ∆w − w = −(−∆+ I)w = −Aw, ∂nw|Γ = 0,w = −∆u+ f(u), u|t=0 = u0,∂tv = ∆Γv − g(v)− λv − ∂nu, x ∈ Γ, v|t=0 = v0u|Γ = v

(3.1)

where ∆Γ is the Laplace-Beltrami operator on the boundary Γ, n is the outward nor-mal direction to the boundary and λ is some given positive constant. The nonlinearfunctions f and g represent the derivative of a typically nonconvex configurationpotential and of a boundary potential, respectively.The boundary condition will be interpreted as an additional second-order parabolicequation on the boundary Γ.We note that equation (3.1) may be viewed as a combination of the well-knownCahn-Hilliard equation

∂tu = −∆(∆u− f(u)), u(0, x) = u0(x)

3 . A Cahn-Hilliard Type Equation With Dynamic Boundary Conditions and RegularPotentials


∂tu = ∆u− f(u), u(0, x) = u0(x).

Problem (3.1) is introduced by [30] as a simplification of a mesoscopic model formultiple microscopic mechanisms in surface processes such that surface diffusionand adsorption-desorption; u is the order parameter representing typically the nor-malized concentration of one component and w is the chemical potential.This problem was derived and studied in [28], [27], [29], [31] and [32] where questionssuch as existence and uniqueness of solutions, existence of the global attractor andof an exponential attractor have been answered under various assumptions on thenonlinearities.We have the following free energy for our problem:

E(u) =∫

Ω

(

1

2|∇u|2 + F (u)

)

dx+

∫

Γ

(

1

2|∇Γu|2 +

λ

2|u|2 +G(u)

)

dσ

where F (s) :=∫ s

0

f(t)dt and G(s) :=∫ s

0

g(t)dt. The first integral is the bulk energy

and the second one is the surface energy. If u is a regular solution of problem (3.1),then, u dissipates E since

d

dtE(u(t)) = −

∫

Ω

(|∇w|2 + |w|2)dx−∫

Γ

|ut|2dσ.

In what follows, unless mentioned explicitly, the same letter Q denotes monotoneincreasing functions and the same letter c denotes positive constants independent oft, possibly changing at different occurrences.

3.1.1 Assumptions and notations

We assume that the nonlinearities f and g belong to C2(R,R) and satisfy the fol-lowing standard dissipativity assumptions

lim inf|s|→+∞

f ′(s) > 0, lim inf|s|→+∞

g′(s) > 0. (3.2)

We note that assumptions (3.2) imply that (see [42])

1

2|f(u)|(1 + |u|) 6 f(u)u+ c

1

2|g(v)|(1 + |v|) 6 g(v)v + c

(3.3)

and

|F (s)| 6 |f(s)|(1 + |s|)− c

|G(s)| 6 |g(s)|(1 + |s|)− c.(3.4)

Consequently, estimates (3.3) and (3.4) yield

(f(u), u)Ω > c(‖f(u)‖L1(Ω) + ‖F (u)‖L1(Ω))− c

(g(v), v)Γ > c(‖g(v)‖L1(Γ) + ‖G(v)‖L1(Γ))− c,(3.5)

64

3.2 Uniform a priori estimates

where (·, ·)Ω and (·, ·)Γ denote the standard scalar products in L2(Ω) and L2(Γ),respectively.Now, we introduce the following natural phase space for problem (3.1)

W := (u, v) ∈ H2(Ω)×H2(Γ), w = −∆u+ f(u) ∈ H1(Ω), u|Γ = v, ∂nw|Γ = 0endowed with the obvious norm

‖(u, v)‖2W:= ‖u‖2H2(Ω) + ‖w‖2H1(Ω) + ‖v‖2H2(Γ).

We also need to introduce the following spaces

H := L2(Ω)× L2(Γ)

V := (u, v) ∈ H1(Ω)×H1(Γ), u|Γ = vwhere the scalar products are given by

(U,U ′)H= (u, u′)Ω + (v, v′)Γ , ∀U = (u, v), U ′ = (u′, v′) ∈ H,

(U,U ′)V= (∇u,∇u′)Ω + λ (v, v′)Γ + (∇Γv,∇Γv

′)Γ , ∀U = (u, v), U ′ = (u′, v′) ∈ V,

respectively. We further introduce the space L = H−1(Ω) × L2(Γ) defined by thefollowing norm

‖(u, v)‖2L:= ‖u‖2H−1(Ω) + ‖v‖2L2(Γ)

where H−1(Ω) = [H1(Ω)]′. We endow H−1(Ω) with the norm

‖µ‖2H−1(Ω) := ‖A−1/2µ‖2L2(Ω) =(

A−1µ, µ)

Ω

where A−1 : L2(Ω) → L2(Ω) denotes the inverse of the operator (−∆ + I) withNeumann boundary conditions.


In this section, the main task is to derive several estimates for the solutions ofproblem (3.1) which are necessary for the study of the asymptotic behavior.

Definition 3.2.1. A solution of problem (3.1) is a pair (u(t), v(t)) of functionsbelonging to the space L∞([0, T ],W) with (∂tu, ∂tv) ∈ L2([0, T ],V) and which satis-fies the equations (3.1) in the sense of equalities in the spaces L2([0, T ], L2(Ω)) andL2([0, T ], L2(Γ)).

We have the following:

Proposition 3.2.2. Let the nonlinearities f and g satisfy (3.2) and let (u(t), v(t))be a solution of problem (3.1). Then, the following estimate holds

‖(u(t), v(t))‖2L+c

∫ t+1

t

(‖u(s)‖2H1(Ω) + ‖v(s)‖2H1(Γ) + ‖f(u(s))‖L1(Ω) + ‖g(v(s))‖L1(Γ))ds

+ c

∫ t+1

t

(‖F (u(s))‖L1(Ω) + ‖G(v(s))‖L1(Γ))ds

6‖(u(0), v(0))‖2Le−kt + c,

(3.6)

where the positive constant k is independent of t.

65


Proof:Multiplying the first equation of (3.1) by A−1u(t), the second one by u(t) and thethird one by v(t) and taking the sum of the resulting equations that we obtain, wehave

1

2

d

dt

(

‖u(t)‖2H−1(Ω) + ‖v(t)‖2L2(Γ)

)

+‖∇u(t)‖2L2(Ω) + ‖∇Γv(t)‖2L2(Γ) + λ‖v‖2L2(Γ)

+ (f(u(t)), u(t))Ω + (g(v(t)), v(t))Γ = 0.

(3.7)

Using (3.5), we obtain

1

2

d

dt

(

‖u(t)‖2H−1(Ω) + ‖v(t)‖2L2(Γ)

)

+ ‖∇u(t)‖2L2(Ω) + ‖∇Γv(t)‖2L2(Γ) + λ‖v‖2L2(Γ)

+ c(‖f(u(t))‖L1(Ω) + ‖g(v(t))‖L1(Γ) + ‖F (u(t))‖L1(Ω) + ‖G(v(t))‖L1(Γ)) 6 c.

(3.8)

Using the fact that

‖u‖2H1(Ω) 6 cΩ(‖∇u‖2L2(Ω) + ‖v‖2H1(Γ)) (3.9)

equation (3.8) gives

d

dt(‖u(t)‖2H−1(Ω) + ‖v(t)‖2L2(Γ))

+k(‖u(t)‖2H−1(Ω) + ‖v(t)‖2L2(Γ)) + ck(‖u(t)‖2H1(Ω) + ‖v(t)‖2H1(Γ))

+c(‖f(u(t))‖L1(Ω) + ‖g(v(t))‖L1(Γ) + ‖F (u(t))‖L1(Ω) + ‖G(v(t))‖L1(Γ)) 6 c,

(3.10)

where k > 0 is small enough and ck > 0 depends on k and is independent of t.Consequently, applying Gronwall’s inequality to (3.10) followed by an integrationover the time interval (t, t+ 1), we obtain estimate (3.6).

Proposition 3.2.3. Let the assumptions of Proposition 3.2.2 hold. Then, we havethe following estimate:

‖u(t)‖2H1(Ω) + ‖v(t)‖2H1(Γ) + (F (u(t)), 1)Ω + (G(u(t)), 1)Γ

+

∫ t

0

(

‖∂tu(s)‖2H−1(Ω) + ‖∂tv(s)‖2L2(Γ)

)

e−τ(t−s)ds

+

∫ t+1

t

(

‖∂tu(s)‖2H−1(Ω) + ‖∂tv(s)‖2L2(Γ)

)

ds

6c(

‖u(0)‖2H1(Ω) + ‖v(0)‖2H1(Γ) + (F (u(0)), 1)Ω + (G(u(0)), 1)Γ

)

e−τt + c,

(3.11)

where the positive constant τ is independent of t.

Proof:Multiplying the first equation of (3.1) by A−1∂tu(t), the second one by ∂tu(t) and

66


the third one by ∂tv(t) and taking the sum of the resulting equations, we have thefollowing identity

1

2

d

dt(‖∇u(t)‖2L2(Ω) + ‖∇Γv(t)‖2L2(Γ) + λ‖v(t)‖2L2(Γ) + 2 (F (u(t)), 1)Ω + 2 (G(v(t)), 1)Γ)

+ ‖∂tu(t)‖2H−1(Ω) + ‖∂tv(t)‖2L2(Γ) = 0.

(3.12)

Summing up (3.8) and (3.12), we obtain

d

dtE(t) + 2‖∂tu(t)‖2H−1(Ω) + 2‖∂tv(t)‖2L2(Γ)

+ 2(‖∇u(t)‖2L2(Ω) + ‖∇Γv(t)‖2L2(Γ) + λ‖v‖2L2(Γ) + c‖F (u(t))‖L1(Ω) + c‖G(v(t))‖L1(Γ))

6c,

where

E(t) =‖u(t)‖2H−1(Ω) + ‖v(t)‖2L2(Γ) + ‖∇u(t)‖2L2(Ω) + ‖∇Γv(t)‖2L2(Γ) + λ‖v(t)‖2L2(Γ)

+ 2 (F (u(t)), 1)Ω + 2 (G(v(t)), 1)Γ .

Thus, for τ > 0 small enough, we have

d

dtE(t) + τE(t) + ‖∂tu(t)‖2H−1(Ω) + ‖∂tv(t)‖2L2(Γ) 6 c.

Applying the Gronwall lemma, we obtain (3.11).

Proposition 3.2.4. Let the assumptions of Proposition 3.2.2 hold. Then, we havethe following estimate:

‖∂tu(t)‖2H−1(Ω) + ‖∂tv(t)‖2L2(Γ)+

∫ t

0

(‖∂tu(s)‖2H1(Ω) + ‖∂tv(s)‖2H1(Γ))e−ρ(t−s)ds

6 c‖(u(0), v(0))‖2We−ρt + c,

(3.13)

where the constant ρ is independent of t.

Proof:We differentiate (3.1) with respect to t and set (φ(t), µ(t), ψ(t)) := ∂t(u(t), w(t), v(t)).Then, we have

∂tφ = ∆µ− µ = −(−∆+ I)µ = −Aµ, ∂nµ|Γ = 0,µ = −∆φ+ f ′(u)φ,∂tψ = ∆Γψ − g′(v)ψ − λψ − ∂nψ, x ∈ Γ,φ|Γ = ψ

(3.14)

Multiplying the first, second and third equations of (3.14) by A−1φ(t), φ(t) andψ(t), respectively, then integrate over Ω, we obtain

1

2

d

dt(‖φ(t)‖2H−1(Ω) + ‖ψ(t)‖2L2(Γ))+‖∇φ(t)‖2L2(Ω) + ‖∇Γψ(t)‖2L2(Γ) + λ‖ψ(t)‖2L2(Γ)

+ (f ′(u)φ(t), φ(t))Ω + (g′(v)ψ(t), ψ(t))Γ = 0.

(3.15)

67


We define h(t) := − (f ′(u)φ(t), φ(t))Ω− (g′(v)ψ(t), ψ(t))Γ . Due to assumption (3.1),we have f ′(s) > −c and g′(s) > −c, for all s ∈ IR. Thus, using the followinginterpolation inequality ‖φ‖2L2(Ω) 6 c‖φ‖H−1(Ω)‖φ‖H1(Ω), h(t) is estimated as follows

h(t) 6 c‖φ(t)‖2L2(Ω) + c‖ψ(t)‖2L2(Γ)

6 η‖φ(t)‖2H1(Ω) + cη‖φ(t)‖2H−1(Ω) + c‖ψ(t)‖2L2(Γ),

for a sufficiently small η > 0 and a large constant cη > 0. Choosing η < 1 small andusing estimate (3.11) as well as the embeddings H2(Ω) ⊂ L∞(Ω), H2(Γ) ⊂ L∞(Γ),we obtain

supt>0

∫ t+1

t

h(s)ds 6 Q(‖(u(0), v(0))‖2W)e−τt + c. (3.16)

Thus, for a ρ small enough, (3.15) gives:

d

dt(‖φ(t)‖2H−1(Ω)+‖ψ(t)‖2L2(Γ)) + ρ(‖φ(t)‖2H−1(Ω) + ‖ψ(t)‖2L2(Γ))

+ ‖φ(t)‖2H1(Ω) + ‖ψ(t)‖2H1(Γ) 6 cη(‖φ(t)‖2H−1(Ω) + ‖ψ(t)‖2L2(Γ)).

(3.17)

Applying the Gronwall lemma and using (3.11), (3.16), we find

‖φ(t)‖2H−1(Ω) + ‖ψ(t)‖2L2(Γ) +

∫ t

0

(‖φ(s)‖2H1(Ω) + ‖ψ(s)‖2H1(Γ))e−ρ(t−s)ds

6c(‖φ(0)‖2H−1(Ω) + ‖ψ(0)‖2L2(Γ) + ‖(u(0), v(0))‖2V+ (F (u(0)), 1)Ω + (G(u(0)), 1)Γ)e

−ρt

+ c.

Moreover, from (3.1), we have

φ(0) = ∆w(0)− w(0) = −Aw(0),

ψ(0) = ∆Γv(0)− ∂nu(0)− λv(0)− g(v(0)).

Therefore, we deduce from H2(Ω) ⊂ L∞(Ω) and H2(Γ) ⊂ L∞(Γ) that

‖φ(0)‖2H−1(Ω) + ‖ψ(0)‖2L2(Γ) =‖Aw(0)‖2H−1(Ω) + ‖ψ(0)‖2L2(Γ)

=‖A1/2w(0)‖2L2(Ω) + ‖ψ(0)‖2L2(Γ)

6c(

‖w(0)‖2H1(Ω) + ‖(u(0), v(0))‖2H2(Ω)×H2(Γ)

)

=c‖(u(0), v(0))‖2W.

Using (3.3) and the above estimates, we deduce (3.13).

Theorem 3.2.5. Let the assumptions of Proposition 3.2.2 hold. Then, we have thefollowing estimate:

‖(u(t), v(t))‖W 6 Q(‖(u(0), v(0))‖W)e−αt + c, (3.18)

where the positive constant α is independent of t.

68


Proof:We recall that φ = ∂tu = ∆w − w = −Aw. Therefore, (3.13) can be also rewrittenas

‖A1/2w(t)‖2L2(Ω) + ‖ψ(t)‖2L2(Γ)+

∫ t

0

(‖φ(s)‖2H1(Ω) + ‖ψ(s)‖2H1(Γ))e−ρ(t−s)ds

6 Q(‖u(0), v(0)‖2W)e−ρt + c.

(3.19)

In order to obtain the required estimates of u and v in the H2(Ω)−norms, we canrewrite (for every fixed t) problem (3.1) as a second order nonlinear elliptic boundaryvalue problem. We have

−∆u(t) + f(u(t)) = h1(t) := w(t), u|Γ = v−∆Γv(t) + λv(t) + g(v(t)) + ∂nu(t) = h2(t) := −∂tv(t) (3.20)

According to (3.19), we find

‖h1(t)‖2L2(Ω) + ‖h2(t)‖2L2(Γ) 6 e−ρtQ(‖(u(0), v(0))‖2W) + c. (3.21)

Applying the maximal principle (see, Lemma A.2, [42]) to this problem, we obtain

‖u(t)‖2L∞(Ω) + ‖v(t)‖2L∞(Γ) 6 e−ρtQ(‖(u(0), v(0))‖2W) + c.

Finally, applying the above estimate combined with a H2−regularity theorem (see,Lemma A.1, [42]) to the elliptic boundary value problem (3.20), we find

‖u(t)‖2H2(Ω) + ‖v(t)‖2H2(Γ) 6 Q(‖(u(0), v(0))‖2W)e−ρt + c,

where Q is a monotonic function independent of t.In the sequel, we shall also need the uniform bounds on the solutions in H3(Ω) ×H3(Γ) which are formulated in the next theorem.

Theorem 3.2.6. Let the assumptions of Theorem (3.2.5) hold and let (u(t), v(t))be a solution of problem (3.1). Then (u(t), v(t)) ∈ H3(Ω) × H3(Γ) for every t > 0and the following estimate holds:

‖u(t)‖2H3(Ω) + ‖v(t)‖2H3(Γ) 61

tQ(‖(u(0), v(0))‖2

W), (3.22)

for every t ∈]0, 1].Proof:We differentiate (3.1) and set (φ(t), µ(t), ψ(t)) := ∂t(u(t), w(t), v(t)). Then, thesefunctions satisfy (3.14). We multiply scalarly the first, second and third equationsof (3.14) respectively by tA−1∂tφ(t), t∂tφ(t) and t∂tψ(t) and take the sum of theequations that we obtain. Then, we have

1

2

d

dt(t‖∇φ(t)‖2L2(Ω) + t‖∇Γψ(t)‖2L2(Γ) + tλ‖ψ(t)‖2L2(Γ))

+ t‖∂tφ(t)‖2H−1(Ω) + t‖∂tψ(t)‖2L2(Γ)

=1

2(‖∇φ(t)‖2L2(Ω) + ‖∇Γψ(t)‖2L2(Γ) + λ‖ψ(t)‖2L2(Γ))

− (f ′(u(t))φ(t), t∂tφ(t))Ω − (g′(v(t))ψ(t), t∂tψ(t))Γ .

(3.23)

69


We estimate the last two terms on the right-hand side of (3.23) as follows

(f ′(u(t))φ(t), t∂tφ(t))Ω 6 ct‖f ′(u(t))φ(t)‖H1(Ω)‖∂tφ(t)‖H−1(Ω) (see Chapter 2)

6 Q(‖u(t)‖H2(Ω))t‖φ(t)‖2H1(Ω) + t/2‖∂tφ(t)‖2H−1(Ω)

6 Q(‖u(t)‖H2(Ω))‖φ(t)‖2H1(Ω) + t/2‖∂tφ(t)‖2H−1(Ω),

(g′(v(t))ψ(t), t∂tψ(t))Γ 6 ct‖g′(v(t))ψ(t)‖L2(Γ)‖∂tψ(t)‖L2(Γ)

6 Q(‖v(t)‖H2(Γ))t‖ψ(t)‖2L2(Γ) + t/2‖∂tψ(t)‖2L2(Γ)

6 Q(‖v(t)‖H2(Γ))‖ψ(t)‖2L2(Γ) + t/2‖∂tψ(t)‖2L2(Γ).

Then, estimate (3.23) gives

d

dt(t‖∇φ(t)‖2L2(Ω) + t‖∇Γψ(t)‖2L2(Γ) + tλ‖ψ(t)‖2L2(Γ))

+ t‖∂tφ(t)‖2H−1(Ω) + t‖∂tψ(t)‖2L2(Γ)

6‖∇φ(t)‖2L2(Ω) + ‖∇Γψ(t)‖2L2(Γ) + λ‖ψ(t)‖2L2(Γ)

+Q(‖(u(0), v(0))‖2W)(‖∂tu(t)‖2H1(Ω) + ‖∂tv(t)‖2L2(Γ))

6c(‖φ(t)‖2H1(Ω) + ‖ψ(t)‖2H1(Γ)) +Q(‖(u(0), v(0))‖2W)(‖∂tu(t)‖2H1(Ω) + ‖∂tv(t)‖2L2(Γ)).

(3.24)

Integrating (3.24) with respect to t and using Theorem 3.2.5 and (3.19), we deduce

t(‖∂tu(t)‖2H1(Ω) + ‖∂tv(t)‖2H1(Γ)) +

∫ t

0

(s‖∂ttu(s)‖2H−1(Ω) + s‖∂ttv(s)‖2L2(Γ))ds

6(c+Q(‖(u(0), v(0))‖2W))

∫ t

0

(‖φ(s)‖2H1(Ω) + ‖ψ(s)‖2H1(Γ))ds

6Q(‖(u(0), v(0))‖2W),

(3.25)

t ∈ [0, 1], for some monotonic Q which is independent of t. Having obtained estimate(3.25), we can rewrite problem (3.1) as a linear elliptic boundary value problem

−∆u(t) = h3(t) := w(t)− f(u(t)), u|Γ = v−∆Γv(t) + λv(t) + ∂nu(t) = h4(t) := −∂tv(t)− g(v(t)).

(3.26)

According to Theorem 3.2.5, estimates (3.19) and (3.25), we have

‖h3(t)‖2H1(Ω) + ‖h4(t)‖2H1(Γ) 61

tQ(‖(u(0), v(0))‖2

W), t ∈ (0, 1],

for some monotonic function Q. Applying the H1 − H3 regularity theorem to thelinear elliptic problem (3.26), we obtain

‖u(t)‖2H3(Ω) + ‖v(t)‖2H3(Γ) 6 c(‖h3(t)‖2H1(Ω) + ‖h4(t)‖2H1(Γ))

61

tQ(‖(u(0), v(0))‖2

W), t ∈ (0, 1].

We will also verify the uniqueness of the solution of (3.1) and the Lipschitz continuitywith respect to the initial data.

70


Proposition 3.2.7. Let the assumptions of Proposition 3.2.2 hold and let the func-tions (u1(t), v1(t)) and (u2(t), v2(t)) be two solutions of (3.1). Then, we have

‖u1(t)− u2(t)‖2H−1(Ω)+‖v1(t)− v2(t)‖2L2(Γ)

+

∫ t+1

t

(‖u1(s)− u2(s)‖2H1(Ω) + ‖v1(s)− v2(s)‖2H1(Γ))ds

6cect(

‖u1(0)− u2(0)‖2H−1(Ω) + ‖v1(0)− v2(0)‖2L2(Γ)

)

,

(3.27)

where the positive constant c is independent of the initial data.

Proof:Set (u, w, v) = (u1 − u2, w1 − w2, v1 − v2). These functions satisfy the followingproblem:

∂tu = ∆w − w = −(−∆+ I)w = −Aw, ∂nw|Γ = 0,w = −∆u+ (f(u1)− f(u2)) = −∆u+ l1(t)u,∂tv = ∆Γv − (g(u1)− g(v2))− λv − ∂nu = ∆Γv − λv − ∂nu− l2(t)v, x ∈ Γ,u|t=0 = u1(0)− u2(0), v|t=0 = v1(0)− v2(0),u|Γ = v

(3.28)

where l1(t) :=∫ 1

0

f ′(su1(t)+(1−s)u2(t))ds and l2(t) :=∫ 1

0

g′(sv1(t)+(1−s)v2(t))ds.We multiply the first equation of (3.28) scalarly by A−1u, the second by u and thethird by v and take the sum of the equations that we obtain. Then, we have

1

2

d

dt(‖u(t)‖2H−1(Ω) + ‖v(t)‖2L2(Γ)) + ‖∇u(t)‖2L2(Ω) + ‖∇Γv(t)‖2L2(Γ) + λ‖v‖2L2(Γ)

= − (l1(t)u(t), u(t))Ω − (l2(t)v(t), v(t))Γ . (3.29)

Due to assumptions (3.2), f ′(v) > −c and g′(v) > −c, and thus estimate (3.29)implies

d

dt(‖u(t)‖2H−1(Ω)+‖v(t)‖2L2(Γ)) + 2‖∇u(t)‖2L2(Ω) + 2‖∇Γv(t)‖2L2(Γ) + 2λ‖v‖2L2(Γ)

6c(‖u(t)‖2L2(Ω) + ‖v(t)‖2L2(Γ)).

(3.30)

Using now the interpolation inequality ‖µ‖2L2(Ω) 6 c‖µ‖H1(Ω)‖µ‖H−1(Ω) in order toestimate the first term in the right-hand side of (3.30), we obtain

d

dt(‖u(t)‖2H−1(Ω)+‖v(t)‖2L2(Γ)) + 2‖u(t)‖2H1(Ω) + 2‖v(t)‖2H1(Γ)

6c(‖u(t)‖2L2(Ω) + ‖v(t)‖2L2(Γ))

6‖u(t)‖2H1(Ω) + c(‖u(t)‖2H−1(Ω) + ‖v(t)‖2L2(Γ)).

(3.31)

71


Applying Gronwall’s inequality, we find

‖u(t)‖2H−1(Ω)+‖v(t)‖2L2(Γ) +

∫ t

0

(‖u(s)‖2H1(Ω) + ‖v(s)‖2H1(Γ))ds

6(‖u(0)‖2H−1(Ω) + ‖v(0)‖2L2(Γ))ect.

(3.32)

Integrating (3.31) with respect to t ∈ [t, t+ 1] and using (3.32), we obtain

‖u(t+ 1)‖2H−1(Ω)+‖v(t+ 1)‖2L2(Γ) +

∫ t+1

t

(‖u(s)‖2H1(Ω) + ‖v(s)‖2H1(Γ))ds

6c

∫ t+1

t

(‖u(s)‖2H−1(Ω) + ‖v(s)‖2L2(Γ))ds+ ‖u(t)‖2H−1(Ω) + ‖v(t)‖2L2(Γ)

6cect(‖u(0)‖2H−1(Ω) + ‖v(0)‖2L2(Γ)).

(3.33)

Finally, estimates (3.32) and (3.33) give (3.27).

Proposition 3.2.8. Let the assumptions of Theorem 3.2.2 hold. Then, we have:

‖u1(t)− u2(t)‖2H1(Ω)+‖v1(t)− v2(t)‖2H1(Γ)

+

∫ t+1

t

(‖∂tu1(s)− ∂tu2(s)‖2H−1(Ω) + ‖∂tv1(s)− ∂tv2(s)‖2L2(Γ))ds

6ceC0t(

‖u1(0)− u2(0)‖2H1(Ω) + ‖v1(0)− v2(0)‖2H1(Γ)

)

,

(3.34)

where the positive constant C0 depends on the initial data.

Proof:Let (u, w, v) = (u1−u2, w1−w2, v1−v2). Then, these functions satisfy (3.28). Thus,multiplying scalarly the first, second and third equations of (3.28) by respectivelyA−1∂tu(t), ∂tu(t) and ∂tv(t), then integrating by parts, we obtain

1

2

d

dt(‖∇u(t)‖2L2(Ω)+‖∇Γv(t)‖2L2(Γ) + λ‖v(t)‖2L2(Γ)) + ‖∂tu(t)‖2H−1(Ω) + ‖∂tv(t)‖2L2(Γ)

= − (l1(t)u(t), ∂tu(t))Ω − (l2(t)v(t), ∂tv(t))Γ .

(3.35)

It follows from Theorem 3.2.5, the assumptions (3.2) on the nonlinearities f and gand the embeddings H2(Ω) ⊂ L∞(Ω), H2(Γ) ⊂ L∞(Γ) that

‖∂tl1(t)‖L2(Ω) + ‖∂tl2(t)‖L2(Γ)+‖l1(t)‖H2(Ω) + ‖l2(t)‖H2(Γ)

6 Q (‖(u1(t), v1(t))‖W + ‖(u2(t), v2(t))‖W)6 C0 := Q (‖(u1(0), v1(0))‖W + ‖(u2(0), v2(0))‖W)

(3.36)

consequently

(l1(t)u(t), ∂tu(t))Ω + (l2(t)v(t), ∂tv(t))Γ6C0‖u(t)‖H1(Ω)‖∂tu(t)‖H−1(Ω) + C0‖v(t)‖L2(Γ)‖∂tv(t)‖L2(Γ)

6C2

0

2(‖u(t)‖2H1(Ω) + ‖v(t)‖2L2(Γ)) +

1

2(‖∂tu(t)‖2H−1(Ω) + ‖∂tv(t)‖2L2(Γ)).

72

3.3 Existence of solutions.

Inserting the above estimate into the right-hand of (3.35), we obtain

d

dt(‖∇u(t)‖2L2(Ω) + ‖∇Γv(t)‖2L2(Γ) + λ‖v(t)‖2L2(Γ)) + ‖∂tu(t)‖2H−1(Ω) + ‖∂tv(t)‖2L2(Γ)

6C20

(

‖u(t)‖2H1(Ω) + ‖v(t)‖2H1(Γ)

)

.

(3.37)

Applying Gronwall’s inequality to relation (3.37), we obtain our desired estimate.

3.3 Existence of solutions.

In this section, we establish the existence of a solution for problem (3.1). The proofis based on the priori estimates derived in the first section and a Faedo-Galerkinapproximation scheme. To do this, let us consider the operator A : D(A) ⊂ V → V

′

defined by⟨

A(

uv

)

,

(

u′

v′

)⟩

V′,V

= (∇u,∇u′)Ω + λ(v, v′)Γ + (∇Γv,∇Γv′)Γ,

for all

(

uv

)

,

(

u′

v′

)

∈ V. We have that A is linear, selfadjoint, continuous,

positive and invertible operator with compact inverse on V and that

A(

uv

)

=

(

−∆u−∆Γv + λv + ∂nu

)

, ∀(

uv

)

∈ D(A).

Consequently, let Φ1 =

(

ϕm

χm

)

, · · · ,Φm =

(

ϕm

χm

)

be an orthonormal family in

H associated with the eigenvalues 0 < λ1 6 λ2 6 · · · 6 λm of A. We consider theapproximate problem:

Find Um =

(

umvm

)

=m∑

i=1

ci(t)Φi =

m∑

i=1

ci(t)ϕi

m∑

i=1

ci(t)χi

such that

d

dt(A−1um, ϕj)Ω = −(∇um,∇ϕj)Ω + (∂nu, χj)Γ − (f(um), ϕj)Ω,

d

dt(vm, χj)Γ = −(∇Γv,∇Γχj)Γ − λ(v, χj)Γ − (g(vm), χj)Γ − (∂nu, χj)Γ,

(3.38)

for all j = 1, · · · ,m, that is to say, setting

F(

uv

)

=

(

f(u)g(v)

)

and N(

uv

)

=

(

A−1uv

)

,

we search for Um such that:

d

dt(NUm,Φj)H + (AUm,Φj)V + (F(Um),Φj)H = 0, ∀j = 1, · · · ,m. (3.39)

73


We define the following matrices

G0 =

((

N(

ϕi

χi

)

,

(

ϕj

χj

))

H

)

16i,j6m

G1 =

((

λi

(

ϕi

χi

)

,

(

ϕj

χj

))

H

)

16i,j6m

F =

f(m∑

i=1

ci(t)ϕi)

g(m∑

i=1

ci(t)χi)

,

(

ϕj

χj

)

H

16i,j6m

and the column vector

c(t) =

c1(t)...

cm(t)

.

Consequently, (3.39) can be rewriten in the following form

G0c′(t) +G1c(t) + F(c(t)) = 0. (3.40)

We have that G0 is symmetric and positive-definite, hence non-singular. G0 andG1 are uniformly continuous with respect to t. Applying Cauchy’s theorem forODE’s, we find that there exists a small time tm ∈ (0, T ) such that the equationc′(t) = −G

−10 G1c(t)−G

−10 F(c(t)) holds for all t ∈ [0, tm]. Now, based on the uniform

a priori estimates with respect to t, derived for the solution (u, v) of (3.1), we ob-tain, in particular, that any local solution of (3.40) is actually a global solution thatis defined on the whole interval [0, T ]. It remains then to pass to the limit as m→ ∞.

According to the a priori estimates derived in the Section 2, we have

‖um‖L∞([0,T ];H2(Ω)) + ‖vm‖L∞([0,T ];H2(Γ)) 6 c

‖∂tum‖L2([0,T ];H1(Ω)) + ‖∂tvm‖L2([0,T ];H1(Γ)) 6 c.

The approximate chemical potential wm = −∆um + f(um), n ∈ IN, satisfies

‖wm‖L∞([0,T ];H1(Ω)) 6 c,

where c depends on Ω, Γ, u0, v0 and T but is independent of n and t. From thispoint on, all convergence relations will be intended to hold up to the extractionof suitable subsequences, generally not relabeled. Thus, we observe that weak andweak star compactness results applied to the sequences Um = (um, vm) entail thatthere exists the function U = (u, v) such that, as m → ∞, the following propertieshold

Um → U weakly star in L∞([0, T ];H2(Ω)×H2(Γ)), (3.41)

∂tUm → ∂tU weakly in L2([0, T ];H1(Ω)×H1(Γ)), (3.42)

74

3.4 Global and exponential attractor

wm → ζ weakly star in L∞([0, T ];H1(Ω))(we prove later that ζ = −∆u+f(u) = w).

It follows from (3.41), (3.42) and the compactness theorem of J.P Aubin-J.L Lionsthat

Um → U strongly in C([0, T ];H2−δ(Ω)×H2−δ(Γ)) (3.43)

with δ > 0 sufficiently small. Moreover, due to the embedding H2−δ(Ω) ⊂ C(Ω), forδ ∈ (0, 1/2), since Ω ⊂ R

n with n 6 3, the convergence (3.43) yields

Um → U strongly in C([0, T ];C(Ω)× C(Γ)).

We have

‖f(um)− f(u)‖2L2(Ω) 6 c‖um − u‖2L2(Ω)

‖g(vm)− g(v)‖2L2(Γ) 6 c‖vm − v‖2L2(Γ).

Consequently, it follows that

f(um) → f(u) strongly in C([0, T ];L2(Ω)) (3.44)

andg(vm) → g(v) strongly in C([0, T ];L2(Γ)). (3.45)

Thus, passing to the limit in (3.39) and using the above convergence properties, weimmediately have that (u, v) satisfies (3.1). We also deduce from (3.44) that ζ = wand wm → w weakly star in L∞([0, T ];H1(Ω)).

Corollary 3.3.1. Let (u0, v0) ∈ W and suppose that the nonlinearities f and gsatisfy assumptions (3.2). Then, problem (3.1) has a unique solution (u(t), v(t))that belongs to the space C([0, T ];W). Moreover this problem defines a semigroupS(t) in the phase space W by

S(t) : W → W, (3.46)

such thatS(t)(u0, v0) = (u(t), v(t)). (3.47)

Proof:The uniqueness of a solution has been proved in Proposition 3.2.7 and the existenceof solutions in the space W has been proved above using a Faedo-Galerkin approx-imation scheme. Thus, problem (3.1) generates a semigroup S(t) on W given by(3.46), where (u(t), v(t)) solves (3.1) with initial data in W.


We introduce the following ball B with sufficiently large radius R in the spaceH3(Ω)×H3(Γ):

BR :=

(u, v) ∈ H3(Ω)×H3(Γ), ‖(u, v)‖H3(Ω)×H3(Γ) 6 R

.

75


Then, obviously, BR ⊂ W and

‖BR‖W 6 cR, (3.48)

where the constant cR depends on R. Moreover, due to the dissipative estimate(3.18) and the smoothing property (3.22), there exists a sufficiently large R and atime t∗ which are independent of t such that BR := B is a bounded absorbing setfor the semigroup S(t) acting on W and

S(t)B ⊂ B, t > t∗.

We deduce from above the existence of a connected compact global attractor A ⊂W ∩ (H3(Ω)×H3(Γ)) for the semigroup S(t) associated to the system (3.1).

Lemma 3.4.1. For, every (u1(0), v1(0)) and (u2(0), v2(0)) belonging to B, the cor-responding solutions of problem (3.1) satisfy the following estimate

‖u1(t)− u2(t)‖2H1(Ω)+‖v1(t)− v2(t)‖2H1(Γ)

6 cRect t+ 1

t‖(u1(0)− u2(0), v1(0)− v2(0))‖2L, t > 0,

(3.49)

where the positive constant cR depends only on R and is independent of t.

Proof:Let (u, w, v) := (u1 − u2, w1 − w2, v1 − v2). Then, these functions satisfy equations(3.28). From Proposition 3.2.7, we have

‖(u(t), v(t))‖2L+

∫ t

0

(‖u(s)‖2H1(Ω) + ‖v(s)‖2H1(Γ))ds 6 cect‖(u(0), v(0))‖2L. (3.50)

To verify estimate (3.49), we multiply scalarly the first equation of (3.28) scalarlyby tA−1∂tu, the second by t∂tu and the third by t∂tv. Arguing as in the derivationof (3.23), we have

1

2

d

dt(t‖∇u(t)‖2L2(Ω) + t‖v(t)‖2L2(Γ) + λt‖v(t)‖2L2(Γ)) + t‖∂tu(t)‖2H−1(Ω) + t‖v(t)‖2L2(Γ)

=1

2

(

‖∇u(t)‖2L2(Ω) + ‖v(t)‖2L2(Γ) + λ‖v(t)‖2L2(Γ)

)

− (f(u1(t))− f(u2(t)), t∂tu(t))Ω

− (g(v1(t))− g(v2(t)), t∂tv(t))Γ .

Due to estimates (3.18) and (3.48) and to the embedding H2(Ω) ⊂ C(Ω), we have

‖ui(t)‖L∞(Ω) + ‖vi(t)‖L∞(Γ) 6 c(‖ui(t)‖H2(Ω) + ‖vi(t)‖H2(Γ)) 6 cR, i = 1, 2, t > 0.(3.51)

Estimating the last two terms in the right-hand side of the equation above as follows

(f(u1(t))− f(u2(t)), t∂tu(t))Ω + (g(v1(t))− g(v2(t)), t∂tv(t))Γ=(l1(t)u(t), t∂tu(t))Ω + (l2(t)v(t), t∂tv(t))Γ6t‖l1(t)‖L∞(Ω)‖u(t)‖H1(Ω)‖∂tu(t)‖H−1(Ω) + t‖l2(t)‖L∞(Γ)‖v(t)‖L2(Γ)‖∂tv(t)‖L2(Γ)

61/2(

t‖∂tu(t)‖2H−1(Ω) + t‖∂tv(t)‖2L2(Γ)

)

+ cRt(

‖u(t)‖2H1(Ω) + ‖v(t)‖2L2(Γ)

)

,

(3.52)

76


where we have used estimate (3.51). We obtain

d

dt(t‖∇u(t)‖2L2(Ω) + t‖v(t)‖2L2(Γ) + λt‖v(t)‖2L2(Γ)) + t‖∂tu(t)‖2H−1(Ω) + t‖v(t)‖2L2(Γ)

6

(

‖∇u(t)‖2L2(Ω) + ‖v(t)‖2L2(Γ) + λ‖v(t)‖2L2(Γ)

)

+ cRt(

‖u(t)‖2H1(Ω) + ‖v(t)‖2L2(Γ)

)

6cR(t+ 1)(

‖u(t)‖2H1(Ω) + ‖v(t)‖2H1(Γ)

)

.

(3.53)

Integrating with respect to t and using (3.50), we infer

‖u(t)‖2H1(Ω) + ‖v(t)‖2H1(Γ) 6 cRect t+ 1

t‖(u(0), v(0))‖2

L

6cRect t+ 1

t‖(u(0), v(0))‖2L2(Ω)×L2(Γ),

(3.54)


Lemma 3.4.2. Let the assumptions of Lemma 3.4.1 hold. Then, for every t > 0,we have the following smoothing estimate

‖u1(t)− u2(t)‖2H2(Ω) + ‖v1(t)− v2(t)‖2H2(Γ)

6cRect t+ 1

t(‖(u1(0)− u2(0), v1(0)− v2(0))‖2L),

(3.55)

where the positive constant cR depends only on R and independent of t.

Proof:Recall that the function u and v satisfy (3.28). Differentiating now all equations of(3.28) with respect of t, multiplying scalarly the first, second and third equationsof (3.28) by A−1t∂tu, t∂tu and t∂tv, respectively, and summing up the relationsobtained, we have

td

dt(‖∂tu(t)‖2H−1(Ω) + ‖∂tv(t)‖2L2(Γ))

+ 2t‖∇∂tu‖2L2(Ω) + 2t‖∇Γ∂tv(t)‖2L2(Γ) + 2tλ‖∂tv(t)‖2L2(Γ)

=− 2t (l1(t)∂tu(t), ∂tu(t))Ω − 2t (∂tl1(t)u(t), ∂tu(t))Ω − 2t (l2(t)∂tv(t), ∂tv(t))Γ− 2t (∂tl2(t)v(t), ∂tv(t))Γ .

Using (3.36) and the embeddings H1(Ω) ⊂ L4(Ω), H1(Γ) ⊂ L4(Γ), we have

(l1(t)∂tu(t), ∂tu(t))Ω + (∂tl1(t)u(t), ∂tu(t))Ω + (l2(t)∂tv(t), ∂tv(t))Γ+ (∂tl2(t)v(t), ∂tv(t))Γ

6‖l1(t)‖L∞(Ω)‖∂tu(t)‖2L2(Ω) + ‖l2(t)‖L∞(Γ)‖∂tv(t)‖2L2(Γ)

+ ‖∂tl1(t)‖L2(Ω)‖u(t)‖L4(Ω)‖∂tu(t)‖L4(Ω) + ‖∂tl2(t)‖L2(Γ)‖v(t)‖L4(Γ)‖∂tv(t)‖L4(Γ)

6c‖l1(t)‖L∞(Ω)‖∂tu(t)‖H−1(Ω)‖∂tu(t)‖H1(Ω) + ‖l2(t)‖L∞(Γ)‖∂tv(t)‖2L2(Γ)

+ ‖∂tl1(t)‖L2(Ω)‖u(t)‖L4(Ω)‖∂tu(t)‖L4(Ω) + ‖∂tl2(t)‖L2(Γ)‖v(t)‖L4(Γ)‖∂tv(t)‖L4(Γ)

61

2cR(‖∂tu(t)‖2H−1(Ω) + ‖∂tv(t)‖2L2(Γ) + ‖u(t)‖2H1(Ω) + ‖v(t)‖2H1(Γ))

+1

2(‖∂tu(t)‖2H1(Ω) + ‖∂tv(t)‖2H1(Γ)),

77


which yields

d

dt[t(‖∂tu(t)‖2H−1(Ω) + ‖∂tv(t)‖2L2(Γ))] + t(‖∂tu(t)‖2H1(Ω) + ‖∂tv(t)‖2H1(Γ))

6cRt(‖∂tu(t)‖2H−1(Ω) + ‖∂tv(t)‖2L2(Γ) + ‖u(t)‖2H1(Ω) + ‖v(t)‖2H1(Γ))

+ ‖∂tu(t)‖2H−1(Ω) + ‖∂tv(t)‖2L2(Γ)

6(cRt+ 1)(‖∂tu(t)‖2H−1(Ω) + ‖∂tv(t)‖2L2(Γ)) + cRt(‖u(t)‖2H1(Ω) + ‖v(t)‖2H1(Γ))

(3.56)

where cR depends on the norm of the initial data in B but independent of t. Inte-grating (3.56) over (0, t) and using Proposition 3.2.8 and estimate (3.50), we obtainthat

t(‖∂tu(t)‖2H−1(Ω) + ‖∂tv(t)‖2L2(Γ)) 6 cRect(t+ 1)(‖u(0)‖2H−1(Ω) + ‖v(0)‖2L2(Γ)).(3.57)

The estimate of Lemma 3.4.2 is then a consequence of (3.57), standard parabolicarguments and maximum principle as in the proof of Theorem 3.2.5.

Lemma 3.4.3. Let (u, v) be the solution of (3.1) with initial data (u(0), v(0)) in B.Then, (u, v) is Hölder continuous with respect to t, that is, for every t > t∗ > 0 and0 6 s 6 1, we have

‖u(t+ s)− u(t)‖H−1(Ω) + ‖v(t+ s)− v(t)‖L2(Γ) 6 Q(‖(u(0), v(0))‖2W)s1/2. (3.58)

Proof:From estimate (3.11), we have that for every t ∈ [t∗, T ], with T fixed, the following

∫ t+1

t

(‖∂tu(s)‖2H−1(Ω) + ‖∂tv(s)‖2L2(Γ))ds 6 Q(‖(u(0), v(0))‖2W) 6 cR,

for some positive constant cR which depends on R. Moreover, for every t > t∗ > 0and 0 6 s 6 1, we have

‖u(t+ s)− u(t)‖H−1(Ω)+‖v(t+ s)− v(t)‖L2(Γ)

= ‖∫ t+s

t

∂tu(σ)dσ‖H−1(Ω) + ‖∫ t+s

t

∂tv(σ)dσ‖L2(Γ)

6

∫ t+s

t

(‖∂tu(σ)‖H−1(Ω) + ‖∂tv(σ)‖L2(Γ))dσ

6 cRs1/2.

(3.59)

Thus, the Lemma is proved.In order to prove the existence of an exponential attractor, we will apply the follow-ing theorem (see [10]),

Theorem 3.4.4. Let E and E1 be two Banach spaces such that E1 is compactlyembedded into E. Let X be a bounded subset of E1 and consider a nonlinear mapS : X → X satisfying the smoothing property

‖Sx1 − Sx2‖E16 C‖x1 − x2‖E, (3.60)

78


for all x1, x2 ∈ X, where C > 0 depends on X. Then the discrete dynamical system(X, (S∗)n) possesses a discrete exponential attractor Md ⊂ E, that is, a compact setwith finite fractal dimension such that

S(Md) ⊂ Md, (3.61)

distE(Sn(X),Md) 6 CXe

−ρn, n ∈ IN, (3.62)

where CX and ρ are positive constants independent of n.

We will apply this theorem with X = B, E = L, E1 = H2(Ω) × H2(Γ) andS = S(t∗). Thus, the mapping S : X → X satisfies the smoothing property (3.60).Therefore, Theorem 3.4.4 applies to S and there exists a compact set Md ⊂ B offinite fractal dimension (with respect to the metric in L) that satisfies (3.61) and(3.62). In order to extend this result to continuous times, we use the standardformula

Mc = ∪t∈[t∗,2t∗]S(t)Md.

Then, since the semigroup S(t) is uniformly Lipschitz continuous on [t∗, 2t∗]× B inthe norm of L (Lemma 3.4.3), we have that Mc is a compact set with finite fractaldimension such that

dimF (Mc,L) 6 dimF (Md,L) + 2. (3.63)

79


80

Chapitre 4

Numerical Analysis of a

Cahn-Hilliard Type Equation With

Dynamic Boundary Conditions

Analyse numérique d’une équation detype Cahn-Hilliard avec des conditionsdynamiques sur le bord

4.1 Introduction

We consider the following problem in a smooth and bounded domain Ω ⊂ Rn with

boundary ∂Ω = Γ:

ut = ∆w − w, t > 0, x ∈ Ω, (4.1)

w = f(u)−∆u, t > 0, x ∈ Ω, (4.2)

ut = ∆Γu− λu− g(u)− ∂nu, t > 0, x ∈ Γ, (4.3)

∂nw = 0, t > 0, x ∈ Γ, (4.4)

where ∆Γ is the Laplace-Beltrami operator on the boundary Γ, f and g are givennonlinear interaction functions and λ is some given positive constant. The boundarycondition (4.3) will be interpreted as an additional second-order parabolic equationon the boundary Γ.Problem (4.1)-(4.2) was introduced by Karali and Katsoulakis in [30] as a simplifi-cation of a mesoscopic model for multiple microscopic mechanisms in surface pro-cesses such that surface diffusion and adsorption-desorption and studied in [28], [31]and [32]; questions related to the well-posedness and to the asymptotic behavior,such as the existence of the global attractor and an exponential attractor have been

4 . Numerical Analysis of a Cahn-Hilliard Type Equation With Dynamic BoundaryConditions

answered under various assumptions on the nonlinearities and when the system isendowed with Dirichlet and Neumann boundary conditions; Here u is the orderparameter and corresponds to a rescaled density of atoms and w is the chemicalpotential. Now, the question of how the phase separation process is influenced bythe presence of walls has recently gained much attention and was mainly studiedfor polymer mixtures when the boundary conditions are given by (4.3)-(4.4). Thewell-posedness and the long time behavior of problem (4.1)-(4.4) were studied in [28]and [27], respectively with regular and singular potentials.The chapter is organized as follows. We first introduce in Section 2 some notationand assumptions and, in Section 3, we study a space discretization of (4.1)-(4.4) bya Galerkin method. In Section 4, we prove optimal error estimates for the differencebetween the approximate and the exact solution uh−u in energy norms and weakernorms as the mesh step h tends to 0, where uh is the solution of the space semi-discrete scheme and u is the solution of the continuous problem. In Section 5, westudy the numerical stability of the fully discrete problem obtained by applying theEuler implicit method to the space semi-discrete problem. In particular, we showthat this fully discrete problem is unconditionally stable and the solution convergesto equilibrium as h→ 0. Finally, numerical simulations in two space dimensions arepresented in Section 6.

4.2 Assumptions and notation

In what follows, we consider Ω to be a 2d or 3d slab, i.e.

Ω = Πd−1i=1 (IR/(LiZ))× (0, Ld), Li > 0, i = 1, ..., d, d = 2 or 3,

with smooth boundary

Γ = ∂Ω = Πd−1i=1 (IR/(LiZ))× 0, Ld .

More precisely, when d = 2, Ω is the rectangle (0, L1)× (0, L2) and u, w are periodicin the x1−direction while the boundary conditions in problem (4.1) are valid forx2 = 0 and x2 = L2; when d = 3, Ω is a parallelepiped (0, L1) × (0, L2) × (0, L3),u and w are periodic in the x1 and x2−directions and the boundary conditions inproblem (4.1) are valid for x3 = 0 and x3 = L3.We assume that the nonlinearities f and g belong to C2(R,R) and satisfy the fol-lowing standard dissipativity assumptions

lim inf|v|→∞

f ′(v) > 0, lim inf|v|→∞

g′(v) > 0. (4.5)

Typical choices are

f(v) = v3 − v and g(v) = kv − h (v ∈ IR), (4.6)

where k > 0 and h ∈ IR are constants. The evolution boundary value problem(4.1)-(4.4) is completed by the initial condition u(0) = u0.We introduce the space

V =

v ∈ H1p (Ω) and v(·, 0), v(·, Ld) ∈ H1

per

(

Πd−1i=1 ]0, Li[

)

,

82

4.3 The semi-discrete scheme

where H1per is the classical space of periodic functions and

H1p (Ω) =

v ∈ H1(Ω), v is periodic in the x1, · · · , xd−1 − directions

.

Then, V is a Hilbert space for the Hilbertian norm

‖v‖V =(

‖v‖2H1(Ω) + ‖v‖2H1(Γ)

)1/2

.

The space V can also be seen as the closure for the ‖ · ‖V−norm of C1(Ω) withthe periodicity condition; V is continuously and densely embedded in H1(Ω) and isisometric to the closed subspace V of H1

p (Ω)×H1per(Γ) defined by:

V =

(u, ϕ) ∈ H1p (Ω)×H1

per(Γ), u|Γ = ϕ in the sense of traces

.

We denote by (·, ·)Ω the L2(Ω)−scalar product and by | · |0 the L2(Ω)−norm. Sim-ilarly, (·, ·)Γ denotes the L2(Γ)−scalar product and | · |0,Γ the associated norm. Asa shortcut, we will denote by ‖ · ‖k the Hilbertian norm in Hk(Ω) and | · |k theassociated seminorm, i.e.

∀v ∈ Hk(Ω), ‖v‖2k =∑

|α|6k

|∂αv|20 and |v|2k =∑

|α|=k

|∂αv|20.

Similarly, ‖ · ‖k,Γ denotes the Hk(Γ)−norm and | · |k,Γ the associated seminorm.

We note that from (4.5), we can deduce that (see [7])

F (v) > c1v2 − c2, G(v) > c1v

2 − c2, ∀v ∈ IR, (4.7)

for some constants c1 > 0 and c2 > 0 and where F is an antiderivative of f and Gis an antiderivative of g.


The variational formulation of (4.1)-(4.4) reads

(ut, ϕ)Ω = −(∇w,∇ϕ)Ω − (w,ϕ)Ω,(w, χ)Ω = (f(u), χ)Ω + (∇u,∇χ)Ω + (∇Γu,∇Γχ)Γ + λ(u, χ)Γ + (g(u), χ)Γ

+(ut, χ)Γ,(4.8)

for all ϕ ∈ H1p (Ω) and for all χ ∈ V.

We introduce the functional E : V → IR defined by

E(u) =∫

Ω

(

1

2|∇u|2 + F (u)

)

dx+

∫

Γ

(

1

2|∇Γu|2 +

λ

2|u|2 +G(u)

)

dσ. (4.9)

If u is a regular solution of (4.1)-(4.4), then u dissipates E . Indeed, choosing ϕ = wand χ = ut in (4.8) and subtracting the two equations, we obtain

d

dtE(u(t)) = −

∫

Ω

|∇w|2dx−∫

Ω

|w|2dx−∫

Γ

|ut|2dσ, ∀t > 0. (4.10)

83


For the space discretization, we consider a quasiuniform family of decompositions

Ωh

hof Πd

i=1[0, Li] into d−simplices which take into account the periodic boundaryconditions on Ω, so that

Ωh

his also a triangulation of Ω. The triangulation Ωh of

Ω induces a triangulation Γh of Γ into d− 1 simplices in a natural way. For a giventriangulation Ωh = ∪T∈ΩhT , we define V h as the usual P 1 conforming finite elementspace

V h =

vh ∈ C0(Ω), vh|T is affine ∀T ∈ Ωh

.

For u ∈ C0(Ω), let Ihu denote the P 1 interpolate of u on Ωh, i.e. Ihu is the uniquefunction in V h which takes the same values as u on the nodes of the triangulation.We have the following standard approximation results, where C > 0 denotes aconstant which depends only on

Ωh

h

∀u ∈ H2p (Ω), |u− Ihu|0 + h|u− Ihu|1 6 Ch2|u|2 (4.11)

and∀ϕ ∈ H2

per(Γ), |ϕ− Ihϕ|0,Γ + h|ϕ− Ihϕ|1,Γ 6 Ch2|ϕ|2,Γ. (4.12)

Moreover, we have the following inverse estimate (see [15])

∀vh ∈ V h, ‖vh‖C0(Ω) 6 Ch−d/2|vh|0, (4.13)

where d is the space dimension.

The space semi-discrete version of (4.8) reads:Find (uh, wh) : [0, T ] → V h × V h such that

(uht , ϕ)Ω = −(∇wh,∇ϕ)Ω − (wh, ϕ)Ω,(wh, χ)Ω = (f(uh), χ)Ω + (∇uh,∇χ)Ω + (∇Γu

h,∇Γχ)Γ + λ(uh, χ)Γ + (g(uh), χ)Γ+(uht , χ)Γ,

(4.14)for all ϕ, χ ∈ V h.We define the operator Gh : L2(Ω) → V h, v → Ghv, where Ghv is the uniquesolution of the problem

(∇Ghv,∇χ)Ω + (Ghv, χ)Ω = (v, χ)Ω, ∀χ ∈ V h. (4.15)

We also define the discrete norm

|v|−1,h = (Ghv, v)1/2Ω =

(

|∇Ghv|20 + |Ghv|20)

1

2 , ∀v ∈ L2(Ω).

The norm |·|−1,h is a discrete version of theH−1-norm. We note that Gh is selfadjointand positive definite on L2(Ω). Indeed, for χ = Ghv in (4.15), we have

(Ghv, v)Ω = |∇Ghv|20 + |Ghv|20 > 0

and(v,Ghv′)Ω = (∇Ghv,∇Ghv′)Ω + (Ghv,Ghv′)Ω = (v′, Ghv)Ω,

for all v, v′ ∈ L2(Ω). Moreover, the following interpolation inequalities hold

|vh|20 6 |vh|−1,h‖vh‖1, ∀vh ∈ V h, (4.16)

84


and|v|−1,h 6 |v|0, ∀v ∈ L2(Ω). (4.17)

In order to prove (4.16), we write

|vh|20 = (∇Ghvh,∇vh)Ω + (Ghvh, vh)Ω

6 |∇Ghvh|0|∇vh|0 + |Ghvh|0|vh|06 (|∇Ghvh|20 + |Ghvh|20)

1

2 (|∇vh|20 + |vh|20)1

2

= |vh|−1,h‖vh‖1.

To prove (4.17), we write

|v|2−1,h = (Ghv, v)Ω 6 |Ghv|0|v|0

and|Ghv|20 6 ‖Ghv‖21 = |∇Ghv|20 + |Ghv|20 = |v|2−1,h.

Proposition 4.3.1. For every uh0 ∈ V h, problem (4.14) has a unique solution

(uh, wh) ∈ C1([0,+∞);V h × V h),

such that uh(0) = uh0 . Moreover,

E(uh(t)) +∫ t

0

(|wh|21 + |wh|20 + |uht |20,Γ)ds 6 E(uh(0)), ∀t > 0, (4.18)

where E is defined by (4.9).

Proof:Let (ϕ1, . . . , ϕm) be an orthonormal basis of V h for the L2(Ω)-scalar product. Weseek for uh(t) =

∑mi=1 ui(t)ϕi and wh(t) =

∑mi=1wi(t)ϕi. We define the matrices

Aij = (∇ϕi,∇ϕj)Ω, (MΓ)ij = (ϕi, ϕj)Γ and (AΓ)ij = (∇Γϕi,∇Γϕj)Γ,

for 1 6 i, j 6 m, the vectors U =

u1...um

, W =

w1...wm

, and the functions

F h(U) =

(f(uh), ϕ1)Ω...

(f(uh), ϕm)Ω

, Gh

Γ(U) =

(λuh + g(uh), ϕ1)Γ...

(λuh + g(uh), ϕm)Γ

.

Then (4.14) can be written as

(

(A+ I) I−I MΓ

)(

WU ′

)

= −(

0AU + F h(U) + AΓU +Gh

Γ(U)

)

. (4.19)

85


Let B denote the square matrix of size 2M in the left-hand side of (4.19). We claimthat B is invertible. Indeed, let X, Y ∈ IRM . We have

(X t, Y t)B

(

XY

)

= X t(A+ I)X +X tY − Y tX + Y tMΓY

= (∇xh,∇xh)Ω + (xh, xh)Ω + (yh, yh)Γ

> 0,

(4.20)

where xh =∑m

i=1 xiϕi and yh =∑m

i=1 yiϕi. This shows that B is positive semidefi-nite. Now, if X, Y ∈ IRM satisfy

B

(

XY

)

=

(

00

)

⇐⇒

(A+ I)X +BY = 0−X +MΓY = 0,

then by multiplying this equality on the left by (X t, Y t), we find that (∇xh,∇xh)Ω+(xh, xh)Ω + (yh, yh)Γ = 0. This implies X = Y = 0 which yields that B is invert-ible, as claimed. Thus, problem (4.14) has a unique maximal solution (uh, wh) ∈C1([0, T+);V h × V h) such that uh(0) = uh0 . Choosing ϕ = wh and χ = uht in (4.14),we find

d

dtE(uh(t)) + |wh|21 + |wh|20 + |uht |20,Γ = 0. (4.21)

Integrating with respect to t, we deduce (4.18). Using (4.7), equation (4.18) leadsto the a priori bound ‖uh‖1 6 C(R) provided that uh0 ∈ V h with ‖uh0‖1 6 R.Since uh ∈ L∞(0, T+;H1(Ω)), i.e. the bound is independent of time, we find thatthe solution is global, i.e. T+ = +∞, and the proof is complete.Exclusively for the next Theorem 4.3.2, we assume that f has a subcritical growth.In other words, we assume that there exists a positive constant c3 such that

|f(s)| 6 c3(1 + |s|p−1), ∀s ∈ IR, (4.22)

with p ∈ [2, 6] when d = 3 and p > 2 arbitrary when d = 2. When d = 3, we alsoassume that there exists a positive constant c4 such that

|g(s)| 6 c4(1 + |s|q−1), ∀s ∈ IR, (4.23)

where q > 2 is arbitrary. The typical choices (4.6) satisfy these assumptions withp = 4 and q = 2. We have the following theorem:

Theorem 4.3.2. Assume that f, g ∈ C1(IR) satisfy (4.5), (4.22) and (4.23). Letu0 ∈ V and let uh0 ∈ V h be such that uh0 → u0 in V as h → 0. Then, for all T > 0,we have

uh → u weak * in L∞(0, T ;H1p (Ω)) and strongly in C0([0, T ];L2(Ω)),

(uh)|Γ → u|Γ weak * in L∞(0, T ;H1per(Γ)) and strongly in C0([0, T ];L2(Γ)),

wh → w weakly in L2(0, T ;H1p (Ω)),

where (u, w) is the unique solution of (4.8) such that u(0) = u0 and

u ∈ L∞(0, T ;V ), u|Γ ∈ W 1,2(0, T ;L2(Γ)) and w ∈ L2(0, T ;H1p (Ω)). (4.24)

86

4.4 Error estimates for the space semi-discrete scheme

Proof:By (4.22) and (4.23), we have that

|F (σ)| 6 c5|σ|p + c6 and |G(σ)| 6 c7|σ|q + c8, ∀σ ∈ IR, (4.25)

where c5, c6, c7 and c8 are positive constants. Since uh0 → u0 in V, using (4.25) and theSobolev embeddings H1

p (Ω) ⊂ Lp(Ω) and H1per(Γ) ⊂ Lq(Γ) (with q = +∞ when d =

2), we know that E(uh0) is bounded by a constant independent of h. The discreteenergy estimate (4.18) implies that (uh)h is bounded in L∞(0, T ;V ), (∇wh)h, (w

h)hare bounded in L2(0, T ;L2(Ω)) and that ((uh|Γ)t)h is bounded in L2(0, T ;L2(Γ)).

Thus, we obtain that, up to a subsequence, uh → u weak * in L∞(0, T ;V ) andwh → w weakly in L2(0, T ;H1(Ω)).If ϕ ∈ H1

p (Ω) and χ ∈ V, then choosing sequences ϕh ∈ V h and χh ∈ V h such thatϕh → ϕ strongly in H1(Ω) and χh → χ strongly in V . Using standard compactnessresults, we can pass to the limit in (4.14) and we obtain that (u, w) satisfies (4.1)-(4.4) and (4.24). By uniqueness, the whole sequence (uh, wh) converges to (u, w).For the strong convergence of (uh|Γ) to u|Γ, we use the fact that the space

v ∈ L∞(0, T ;H1per(Γ)), vt ∈ L2(0, T ;L2(Γ))

is compactly embedded into C0([0, T ];L2(Γ)). Finally, for the strong convergence of(uh)h, we use the fact that for all 0 6 s 6 t 6 T, we have

|uh(t)− uh(s)|20

=2

∫ t

s

(

uht (σ), uh(σ)− uh(s)

)

Ωdσ

6− 2

∫ t

s

(

∇wh(σ),∇(uh(σ)− uh(s)))

Ωdσ − 2

∫ t

s

(

wh(σ), uh(σ)− uh(s))

Ωdσ

64c‖uh‖L∞(0,T ;H1p(Ω))

(

‖∇wh‖L2(0,T ;L2(Ω)) + ‖wh‖L2(0,T ;L2(Ω))

)

|t− s|1/2.(4.26)

Thus, the sequence (uh)h is uniformly equicontinuous in C0([0, T ];L2(Ω)). Since (uh)is bounded in C0([0, T ];H1

p (Ω)) with H1p (Ω) compactly embedded into L2(Ω), the

Ascoli theorem implies that uh → u strongly in C0([0, T ];L2(Ω)).


In order to estimate the errors uh − u and wh − w in appropriate norms, we followa standard approach (see [7], [12] and [55]) and we write

uh(t)− u(t) = θu(t) + ρu(t), with θu = uh − uh, ρu = uh − u,

wh(t)− w(t) = θw(t) + ρw(t), with θw = wh − wh, ρw = wh − w,

for all t ∈ [0, T ], where uh = uh(t) and wh = wh(t) are the elliptic projections ofu = u(t) and w = w(t), defined by

(∇wh,∇χ)Ω + (wh, χ)Ω = (∇w,∇χ)Ω + (w, χ)Ω, ∀χ ∈ V h, (4.27)

87


(∇uh,∇χ)Ω + (∇Γuh,∇Γχ)Γ + λ(uh, χ)Γ = (∇u,∇χ)Ω + (∇Γu,∇Γχ)Γ

+ λ(u, χ)Γ, ∀χ ∈ V h.(4.28)

For a given w ∈ H1(Ω), equation (4.27) defines a unique wh ∈ V h. Indeed, thebilinear form defined by

a(ϕ, χ) = (∇ϕ,∇χ)Ω + (ϕ, χ)Ω (4.29)

is the scalar product on H1(Ω). Thus, applying the Lax-Milgram theorem, we obtainthe result. Similarly, for a given u ∈ V, equation (4.28) defines a unique uh ∈ V h.Indeed, the norm v 7→ |∇v|20 + |v|20,Γ is equivalent to the H1−norm, so that thebilinear form defined by

a(ϕ, χ) = (∇ϕ,∇χ)Ω + (∇Γϕ,∇Γχ)Γ + λ(ϕ, χ)Γ (4.30)

is coercive on V, i.e. there exists c0 > 0 such that

a(ϕ, ϕ) = (∇ϕ,∇ϕ)Ω + (∇Γϕ,∇Γϕ)Γ + λ(ϕ, ϕ)Γ > c0‖ϕ‖2V , ∀ϕ ∈ V.

The bilinear continuous form a(·, ·) is a fortiori coercive on V h ⊂ V and the Lax-Milgram theorem applies.

Lemma 4.4.1. For all w ∈ H2(Ω), the function wh ∈ V h defined by

(∇wh,∇χ)Ω + (wh, χ)Ω = (∇w,∇χ)Ω + (w, χ)Ω, ∀χ ∈ V h,

satisfies|wh − w|0 + h|wh − w|1 6 Ch2|w|2, (4.31)

where C is a positive constant, independent of h.

Proof:By definition, we have

a(wh, χ) = a(w, χ), ∀χ ∈ V h, (4.32)

where a(·, ·) is defined by (4.29). Since wh−Ihw ∈ V h, we have that a(wh−w, wh−Ihw) = 0, which yields

a(wh − w, wh − w) = a(wh − w, wh − Ihw) + a(wh − w, Ihw − w),

= a(wh − w, Ihw − w),

implying

‖wh − w‖21 = a(wh − w, wh − w) 6 ‖wh − w‖1‖Ihw − w‖1.

By (4.11), we have

‖wh − w‖1 6 ‖Ihw − w‖1 6 Ch|w|2, (4.33)

88


which gives the H1−estimate. In order to have the L2−estimate, we set ϕ ∈ H1(Ω)to be the unique solution of

a(ϕ, χ) = (z, χ)Ω, ∀χ ∈ V, (4.34)

for a given function z ∈ L2(Ω). Then, ϕ ∈ H2p (Ω) and thanks to the elliptic regularity,

we have|ϕ|2 6 C|z|0, (4.35)

for some constant C > 0 independent of z.Choosing χ = wh −w in (4.34) and using the fact that a(wh −w, Ihϕ) = 0, we find

(z, wh − w)Ω =a(ϕ, wh − w)

=a(ϕ− Ihϕ, wh − w)

6‖ϕ− Ihϕ‖1‖wh − w‖1.(4.36)

Choosing z = wh − w, using (4.11), (4.33) and (4.35), we obtain

|wh − w|20 6‖ϕ− Ihϕ‖1‖wh − w‖16Ch|ϕ|2Ch|w|26Ch2|wh − w|0|w|2,

(4.37)

which gives|wh − w|0 6 Ch2|w|2. (4.38)

By (4.33) and (4.38), we can conclude and the proof is complete.

Lemma 4.4.2. For all u ∈ H2p (Ω) with u|Γ ∈ H2

per(Γ) the function uh ∈ V h definedby (4.28) satisfies

|uh − u|0 + |uh − u|0,Γ + h|uh − u|1 + h|uh − u|1,Γ 6 Ch2(|u|2 + |u|2,Γ), (4.39)

where the positive constant C is independent of h.

Proof:Arguing as above, we have

a(uh, χ) = a(u, χ), ∀χ ∈ V h, (4.40)

where a(·, ·) is defined by (4.30). Since uh−Ihu ∈ V h, we have a(uh−u, uh−Ihu) = 0,which yields

a(uh − u, uh − u) = a(uh − u, uh − Ihu) + a(uh − u, Ihu− u),

= a(uh − u, Ihu− u).(4.41)

From (4.41) and using the coercivity of a, we obtain

c0‖uh − u‖2V 6 a(uh − u, uh − u) 6 c‖uh − u‖V ‖Ihu− u‖V .

By (4.12), we have

‖uh − u‖V 6 c‖Ihu− u‖V 6 ch(|u|2 + |u|2,Γ), (4.42)

89


which gives the H1−estimates. In order to have the L2−estimates, we set ϕ ∈ V tobe the unique solution of

a(ϕ, χ) = (z, χ)Ω + (ψ, χ)Γ, ∀χ ∈ V, (4.43)

for some given functions (z, ψ) ∈ L2(Ω)×L2(Γ). Then, thanks to an elliptic regularityresult (see [42]), we have ϕ ∈ H2

p (Ω), ϕ|Γ ∈ H2per(Γ) and

|ϕ|2 + |ϕ|2,Γ 6 C(|z|0 + |ψ|0,Γ), (4.44)

for some constant C > 0 independent of z and ψ.Choosing χ = uh − u in (4.43) and using the fact that a(uh − u, Ihϕ) = 0, we find

(z, uh − u)Ω + (ψ, uh − u)Γ =a(ϕ, uh − u)

=a(ϕ− Ihϕ, uh − u)

6c‖ϕ− Ihϕ‖V ‖uh − u‖V .(4.45)

Choosing z = uh − u and ψ = (uh − u)Γ, using (4.12), (4.42) and (4.44), we obtain

|uh − u|20 + |uh − u|20,Γ 6c‖ϕ− Ihϕ‖1‖uh − u‖16Ch(|ϕ|2 + |ϕ|2,Γ)Ch(|u|2 + |u|2,Γ)6Ch2(|uh − u|20 + |uh − u|20,Γ)1/2(|u|2 + |u|2,Γ).

(4.46)

Estimate (4.46) leads to

(|uh − u|20 + |uh − u|20,Γ)1/2 6 Ch2(|u|2 + |u|2,Γ). (4.47)

By (4.42) and (4.47), we can conclude and the proof is complete.

Lemma 4.4.3. Let (u, w) the a solution of (4.8) with the initial condition u(0) =u0 ∈ V and (uh, wh) be the solution of (4.14) with the initial condition uh(0) = uh0 ∈V h. Assume that

supt∈[0,T ]

‖u(t)‖C0(Ω) < R, supt∈[0,T ]

‖ut(t)‖C0(Ω) < R, ‖uh(0)‖C0(Ω) < R,

for some constant R < +∞, and let T h ∈ (0, T ] be the maximal time such that‖uh(t)‖L∞(Ω) 6 R for all t ∈ [0, T h]. Then, the following estimate holds:

N (t) +

∫ t

0

(|θut |21 + ‖θw‖21 + ‖θut ‖21,Γ)ds

6CN (0) + C ′

∫ t

0

(|ρut |20 + |ρu|20 + |ρu|20,Γ + |ρut |20,Γ + |ρw|20 + |ρutt|20 + |ρutt|20,Γ + |ρwt |20)ds,(4.48)

for some positive constants C and C ′ which are independent of u, uh and h, where

N (t) = |θu|21 + λ|θu|20,Γ + |θu|21,Γ + |θut |2−1,h + |θut |20,Γ.

90


Proof:Subtracting the first equation of (4.8) from the first equation of (4.14), we obtain

(uht − ut, ϕ)Ω + (∇(wh − w),∇ϕ)Ω + (wh − w,ϕ)Ω = 0, ∀ϕ ∈ V h. (4.49)

Using the definitions of θu and θw as well as (4.27), we find

(θut , ϕ)Ω + (∇θw,∇ϕ)Ω + (θw, ϕ)Ω = −(ρut , ϕ)Ω, ∀ϕ ∈ V h. (4.50)

Choosing ϕ = θw in (4.50), we obtain

(θut , θw)Ω + |θw|21 + |θw|20 = −(ρut , θ

w)Ω. (4.51)

Now, subtracting the second equation of (4.8) from the second equation of (4.14)and using (4.28), we find

−(θw, χ)Ω + (∇θu,∇χ)Ω + (∇Γθu,∇Γχ)Γ + λ(θu, χ)Γ + (θut , χ)Γ

= (ρw, χ)Ω − (f(uh)− f(u), χ)Ω − (g(uh)− g(u), χ)Γ − (ρut , χ)Γ,(4.52)

for all χ ∈ V h.Choosing χ = θut , estimate (4.52) gives

−(θw, θut )Ω + |θut |20,Γ +1

2

d

dt(|θu|21 + λ|θu|20,Γ + |θu|21,Γ)

= (ρw, θut )Ω − (f(uh)− f(u), θut )Ω − (g(uh)− g(u), θut )Γ − (ρut , θut )Γ.

(4.53)

Summing (4.51) and (4.53), we have

1

2

d

dt(|θu|21 + λ|θu|20,Γ + |θu|21,Γ) + |θw|21 + |θw|20 + |θut |20,Γ

= −(ρut , θw)Ω + (ρw, θut )Ω − (f(uh)− f(u), θut )Ω − (g(uh)− g(u), θut )Γ − (ρut , θ

ut )Γ.

(4.54)

We have:

|f(uh)− f(u)|0 6 Lf |uh − u|0,|g(uh)− g(u)|0,Γ 6 Lg|uh − u|0,Γ,

(4.55)

on [0, T h], where Lf and Lg are respectively the Lipschitz constants of f and g on[−R,R]. Thus, using (4.55) and the Hölder inequality, estimate (4.54) yields

1

2

d

dt(|θu|21 + λ|θu|20,Γ + |θu|21,Γ) + |θw|21 + |θw|20 + |θut |20,Γ

6|ρut |0|θw|0 + |ρw|0|θut |0 + Lf (|θu|0 + |ρu|0)|θut |0+ Lg(|θu|0,Γ + |ρu|0,Γ)|θut |0,Γ + |ρut |0,Γ|θut |0,Γ.

(4.56)

Using the inequality

ab 6 ǫa2 + 1/(4ǫ)b2, ∀a, b > 0, ∀ǫ > 0,

91


with ε conveniently chosen, estimate (4.56) gives

d

dt(|θu|21 + λ|θu|20,Γ + |∇Γθ

u|20,Γ) + |θw|21 + |θw|20 + |θut |20,Γ6 C1(|ρut |20 + |ρu|20 + |ρu|20,Γ + |ρut |20,Γ + |ρw|20) + C2(|θu|20,Γ + |θu|20 + |θut |20),

(4.57)

for some positive constants C1 and C2 which depend on |Ω|, |Γ|, Lf and Lg.To estimate θut , we differentiate (4.50) and (4.52) with respect to t. We obtain

(θutt, ϕ)Ω + (∇θwt ,∇ϕ)Ω + (θwt , ϕ)Ω = −(ρutt, ϕ)Ω, ∀ϕ ∈ V h (4.58)

and

−(θwt , χ)Ω + (∇θut ,∇χ)Ω + (∇Γθut ,∇Γχ)Γ + λ(θut , χ)Γ + (θutt, χ)Γ

= (ρwt , χ)Ω − ([f(uh)− f(u)]t, χ)Ω − ([g(uh)− g(u)]t, χ)Γ − (ρutt, χ)Γ, ∀χ ∈ V h.

(4.59)

Choosing ϕ = Ghθut in (4.58) and χ = θut in (4.59), adding the resulting equationsand using the fact that

(∇θwt ,∇Ghθut )Ω + (θwt , Ghθut )Ω = (θwt , θ

ut )Ω,

we find1

2

d

dt(|θut |2−1,h + |θut |20,Γ) + |θut |21 + |θut |21,Γ + λ|θut |20,Γ

=− (ρutt, Ghθut )Ω + (ρwt , θ

ut )Ω − ([f(uh)− f(u)]t, θ

ut )Ω − ([g(uh)− g(u)]t, θ

ut )Γ

− (ρutt, θut )Γ.

(4.60)

Employing[f(uh)− f(u)]t = f ′(uh)[uht − ut] + [f ′(uh)− f ′(u)]ut

and[g(uh)− g(u)]t = g′(uh)[uht − ut] + [g′(uh)− g′(u)]ut,

we find

([f(uh)− f(u)]t, θut )Ω =(f ′(uh)[uht − ut], θ

ut )Ω + ([f ′(uh)− f ′(u)]ut, θ

ut )Ω

6 sup[−R,R]

|f ′|(|θut |0 + |ρut |0)|θut |0 +RLf ′(|θu|0 + |ρu|0)|θut |0

and

([g(uh)− g(u)]t, θut )Γ =(g′(uh)[uht − ut], θ

ut )Γ + ([g′(uh)− g′(u)]ut, θ

ut )Γ

6 sup[−R,R]

|g′|(|θut |0,Γ + |ρut |0,Γ)|θut |0,Γ +RLg′(|θu|0,Γ + |ρu|0,Γ)|θut |0,Γ,

where Lf ′ and Lg′ are respectively the Lipschitz constants of f ′ and g′ on [−R,R].Thus, (4.60) implies

1

2

d

dt(|θut |2−1,h + |θut |20,Γ) + |θut |21 + |θut |21,Γ + λ|θut |20,Γ

6|ρutt|−1,h|θut |−1,h + |ρutt|0,Γ|θut |0,Γ + |ρwt |0|θut |0+ sup

[−R,R]

|f ′|(|θut |0 + |ρut |0)|θut |0 +RLf ′(|θu|0 + |ρu|0)|θut |0

+ sup[−R,R]

|g′|(|θut |0,Γ + |ρut |0,Γ)|θut |0,Γ +RLg′(|θu|0,Γ + |ρu|0,Γ)|θut |0,Γ.

(4.61)

92


Using (4.17), the fact that the norm v 7→ |∇v|20+ |v|20,Γ is equivalent to the H1−normand the interpolation property (4.16) applied to θut , (4.61) gives

d

dt(|θut |2−1,h + |θut |20,Γ) + |θut |21 + |θut |21,Γ + λ|θut |20,Γ6 C3(|θut |2−1,h + |θut |20,Γ + |θu|20 + |θu|20,Γ)+ C4(|ρutt|20 + |ρutt|20,Γ + |ρwt |20 + |ρut |20 + |ρu|20 + |ρut |20,Γ + |ρu|20,Γ), on (0, T h),

(4.62)

for some positive constants C3 and C4 which depend on λ,R, cp, Lf ′ , Lg′ , sup[−R,R]

|f ′|

and sup[−R,R]

|f ′|. Adding (4.57) and (4.62), we find

d

dt(|θu|21 + λ|θu|20,Γ + |θu|21,Γ + |θut |2−1,h + |θut |20,Γ) + |θw|21 + |θw|20 + |θut |20,Γ + |θut |21 + |θut |21,Γ

6 C5(|ρut |20 + |ρu|20 + |ρu|20,Γ + |ρut |20,Γ + |ρw|20 + |ρutt|20 + |ρutt|20,Γ + |ρwt |20)+ C6(|θu|21 + λ|θu|20,Γ + |θu|21,Γ + |θut |2−1,h + |θut |20,Γ + |θut |20),

(4.63)

where we use the fact that |θu|20 6 c(|θu|21 + λ|θu|20,Γ) and for θut we use again theinterpolation property (4.16). Applying Gronwall’s lemma, we find estimate (4.48)with C = eC6T and C ′ = C5e

C6T .

Theorem 4.4.4. Let (u, w) be the solution of problem (4.1)-(4.4) with the initialcondition u(0) = u0 such that

u, ut, utt, w, wt ∈ L2(0, T ;H2p (Ω)) (4.64)

andu|Γ, (ut)|Γ, (utt)|Γ, w|Γ, (wt)|Γ ∈ L2(0, T ;H2

per(Γ)) (4.65)

and let (uh, wh) be the solution of problem (4.14) with initial condition uh(0) = uh0 .If

θu(0) = 0 and θw(0) = 0, (4.66)

then, the following estimates hold:

sup[0,T ]

(

|uh − u|0 + |uh − u|0,Γ + |uht − ut|−1,h + |uht − ut|0,Γ)

6 Ch2,

∫ T

0

|wh − w|20ds 6 Ch4,

sup[0,T ]

(

|uh − u|1 + |uh − u|1,Γ)

6 Ch,

∫ T

0

(

|wh − w|21 + |uht − ut|21 + |uht − ut|21,Γ)

ds 6 Ch2.

Proof:If we differentiate equations (4.27) and (4.28) with respect to t, we obtain that theelliptic projections of ut and wt are respectively (u)t and (w)t. A similar statement

93


holds for utt. Therefore, Lemma 4.4.1 applies with w replaced by wt and Lemma4.4.2 applies with u replaced by ut, utt, i.e.

|ρut |0 + |ρut |0,Γ + h|ρut |1 + h|ρut |1,Γ 6 Ch2(|ut|2 + |ut|2,Γ),|ρutt|0 + |ρutt|0,Γ + h|ρutt|1 + h|ρutt|1,Γ 6 Ch2(|utt|2 + |utt|2,Γ),

|ρwt |0 + h|ρwt |1 6 Ch2|wt|2.(4.67)

The regularity required on u, ut implies that u ∈ C1([0, T ];H2p (Ω)) and by the

Sobolev continuous injection H2p (Ω) ⊂ C0(Ω), we see that u and ut belong to

C0([0, T ];C0(Ω)). Thus,

supt∈[0,T ]

‖u(t)‖C0(Ω) < R and supt∈[0,T ]

‖ut(t)‖C0(Ω) < R,

for some R > 0. We also have

‖uh0 − u0‖C0(Ω) 6 ‖uh0 − Ihu0‖C0(Ω) + ‖Ihu0 − u0‖C0(Ω)

6 Ch−d/2|uh0 − Ihu0|0 + ‖Ihu0 − u0‖C0(Ω) (using (4.13))

6 Ch−d/2(|uh0 − u0|0 + |u0 − Ihu0|0) + ‖Ihu0 − u0‖C0(Ω).

(4.68)

Using the embedding that H2p (Ω) ⊂ C0,γ(Ω), where γ ∈ (0, 1), we find

‖Ihu0 − u0‖C0(Ω) 6 C ′hγ|u0|2. (4.69)

Due to estimates (4.11), (4.39) and assumptions (4.66), we have

|uh0 − u0|0 + |u0 − Ihu0|0 6 Ch2(|u0|2 + |u0|2,Γ). (4.70)

Thus by (4.69) and (4.70), we deduce that

‖uh0 − u0‖C0(Ω) 6 (Ch2−d/2 + C ′hγ)|u0|2 + Ch2−d/2|u0|2,Γ. (4.71)

For h small enough, we obtain

‖uh0‖C0(Ω) < R, (4.72)

and we may apply Lemma (4.4.3).We claim that N (0) 6 Ch4, with N (t) = |θu|21+λ|θu|20,Γ+|∇Γθ

u|20,Γ+|θut |2−1,h+|θut |20,Γ.Indeed, by assumptions (4.66), we have

N (0) = |θut (0)|2−1,h + |θut (0)|20,Γ. (4.73)

Using the fact that θw(0) = 0 and estimate (4.50) at t = 0, we see that uh(0) satisfies

(θut (0), ϕ)Ω = −(ρut (0), ϕ)Ω, ∀ϕ ∈ V h. (4.74)

Choosing ϕ = Ghθut (0) in (4.74) and using (4.17) and (4.67), we obtain

|θut (0)|2−1,h =− (ρut (0), Ghθut (0))Ω

6|ρut (0)|−1,h|θut (0)|−1,h

6c|ρut (0)|0|θut (0)|−1,h

6Ch2(|ut(0)|2 + |ut(0)|2,Γ)|θut (0)|−1,h.

(4.75)

94


For ϕ = θut (0) in (4.74), we find

|θut (0)|0 6 |ρut (0)|0 6 Ch2(|ut(0)|2 + |ut(0)|2,Γ). (4.76)

Using assumptions (4.66) and choosing χ = θut (0) in (4.52) considered at t = 0, weobtain

|θut (0)|20,Γ =(ρw(0), θut (0))Ω − (f(uh0)− f(u0), θut (0))Ω

− (g(uh0)− g(u0), θut (0))Γ − (ρut (0), θ

ut (0))Γ.

(4.77)

Using the fact that ‖u0‖C0(Ω) < R and the fact that uh0 − u0 = ρu(0), (4.77) gives

|θut (0)|20,Γ 6(|ρw(0)|0 + Lf |ρu(0)|0)|θut (0)|0+ (Lg|ρu(0)|0,Γ + |ρut (0)|0,Γ)|θut (0)|0,Γ,

(4.78)

where Lf and Lg are respectively the Lipschitz constants of f and g on [−R,R]. By(4.31), (4.39), (4.67) and (4.76), estimate (4.78) yields

|θut (0)|20,Γ 6Ch2(|w(0)|2 + |u0|2 + |u0|2,Γ)Ch2(|ut(0)|2 + |ut(0)|2,Γ)+ (LgCh

2(|u0|2 + |u0|2,Γ) + Ch2(|ut(0)|2 + |ut(0)|2,Γ))|θut (0)|0,Γ,(4.79)

and|θut (0)|20,Γ 6 Ch4 + Ch2|θut (0)|0,Γ. (4.80)

In particular, (4.80) implies |θut (0)|0,Γ 6 Ch2. Thus,

N (0) 6 Ch4. (4.81)

The regularity assumptions on u and w and estimates (4.31), (4.39) and (4.67),imply that

∫ t

0

|ρut |20 + |ρw|20 + |ρu|20 + |ρutt|20ds 6 Ch4. (4.82)

Using estimates (4.81) and (4.82) we deduce from (4.48) that

N (t) 6 Ch4, ∀t ∈ [0, T h]. (4.83)

Estimate (4.83) implies in particular that

|θu(t)|20 6 Ch2, ∀t ∈ [0, T h]. (4.84)

Arguing as in (4.68), we deduce that

supt∈[0,Th]

‖uh(t)− u(t)‖C0(Ω) → 0 as h→ 0. (4.85)

We conclude by noticing that for h small enough, T h = T.

Remark 4.4.5. We remark here that the regularity required in (4.64) and (4.65) isa strong one, this is due to the fact that we need strong regularity results in orderto estimate the term θut .

95


4.5 Stability of the backward Euler scheme

In what follows, we denote by δt = T/N the time step with N ∈ IN∗. We study thefollowing backward in time Euler scheme:Let u0h ∈ V h and for n = 1, 2, ..., find (unh, w

nh) ∈ V h × V h such that

(∂unh, ϕ)Ω = −(∇wnh ,∇ϕ)Ω − (wn

h , ϕ)Ω,

(wnh , χ)Ω = (∇unh,∇χ)Ω + (f(unh), χ)Ω + (∇Γu

nh,∇Γχ)Γ + (g(unh), χ)Γ + (∂unh, χ)Γ,

(4.86)for all ϕ, χ ∈ V h, where we denote by ∂ the operator which to a sequence (vn)n>0

associates the sequence defined by

∂vn =vn − vn−1

δt, n > 1, (4.87)

and the function g is given by

g(σ) = λσ + g(σ), ∀σ ∈ IR.

Note that the dissipativity assumptions (4.5) imply that

f ′(v) > −Cf and g′(v) > −Cg, ∀v ∈ IR, (4.88)

where Cf and Cg are positive constants.

Theorem 4.5.1. For every u0h ∈ V h, there exists a sequence (unh, wnh)n>1 generated

by (4.86) and which satisfies

E(unh) +1

2δt|unh − un−1

h |2−1,h +1

2δt|unh − un−1

h |20,Γ 6 E(un−1h ), ∀n > 1. (4.89)

Furthermore, if δt < δt∗, where δt∗ = min

4

C2f + C2

fC,1

Cg

, then this sequence is

uniquely defined.

Proof:Consider the variational problem:

Jh(u) = infv∈V h

Jh(v), (4.90)

where

Jh(v) = E(v) + 1

2δt|v − un−1

h |2−1,h +1

2δt|v − un−1

h |20,Γ. (4.91)

Using (4.7), we have

Jh(v) >1

2(|v|21 + |v|21,Γ + λ|v|20,Γ)− c2(|Ω|+ |Γ|), ∀v ∈ V h. (4.92)

Since Jh is continuous, there exists a solution to (4.90). Such a solution u satisfies

0 =(∇u,∇χ)Ω + (f(u), χ)Ω + (∇Γu,∇Γχ)Γ + (g(u), χ)Γ

+1

δt(Gh(u− un−1

h ), χ)Ω +1

δt(u− un−1

h , χ)Γ, ∀χ ∈ V h.(4.93)

96


Setting unh = u, wnh = − 1

δtGh(u−un−1

h ), we find that (unh, wnh) solves problem (4.86).

Thanks to (4.90), Jh(unh) 6 Jh(un−1h ), which implies (4.89).

In order to prove uniqueness, set ξ = (unh)1 − (unh)

2 and η = (wnh)

1 − (wnh)

2 to bethe difference of two possible solutions ((unh)

i, (wnh)

i) (i = 1, 2) of (4.86) for a givenun−1h . Then, (ξ, η) satisfies

(ξ, ϕ)Ω =− δt(∇η,∇ϕ)Ω − δt(η, ϕ)Ω,

(η, χ)Ω =(∇ξ,∇χ)Ω + (f((unh)1)− f((unh)

2), χ)Ω + (∇Γξ,∇Γχ)Γ

+(

g((unh)1)− g((unh)

2), χ)

Γ+ (ξ/δt, χ)Γ,

(4.94)

for all ϕ, χ ∈ V h. Taking ϕ = η in the first equation of (4.94) and χ = ξ in thesecond equation of (4.94) and subtracting the resulting equations, we obtain

δt|η|21 + δt|η|20 + |ξ|21 + |ξ|21,Γ +1

δt|ξ|20,Γ 6 Cf |ξ|20 + Cg|ξ|20,Γ, (4.95)

where we have used the inequalities

(f((unh)1)− f((unh)

2), ξ)Ω > −Cf |ξ|20,

(g((unh)1)− g((unh)

2), ξ)Γ > −Cg|ξ|20,Γ.By choosing χ = ξ in the first equation of (4.94), we find

Cf |ξ|20 =− Cfδt(∇η,∇ξ)Ω − Cfδt(η, ξ)Ω

6δt|η|21 +C2

fδt

4|ξ|21 + δt|η|20 +

C2fδt

4|ξ|20,

6δt|η|21 +C2

fδt

4|ξ|21 + δt|η|20 +

C2fCδt

4|ξ|21,

(4.96)

which leads to(

1− δt(C2

f

4+C2

fC

4)

)

|ξ|21 + |ξ|21,Γ +(

1

δt− Cg

)

|ξ|20,Γ 6 0. (4.97)

The smallness assumption on δt implies ξ = 0 and by (4.94), we deduce that η = 0.

Corollary 4.5.2. If f and g are analytic, then, for all u0h ∈ V h, any sequence(unh, w

nh)n>1 generated by (4.86) and which satisfies the energy estimate (4.89) con-

verges to a steady state (uh, wh) as n→ +∞.

Proof:Let u0h ∈ V h. By (4.89), the sequence (E(unh))n is non-increasing and since it isbounded from below by 0, we have E(unh) → E∗. We assume without loss of generalitythat E∗ = 0. By (4.9), E(v) → +∞ as ‖v‖V → +∞ and (unh)n is bounded: thereexist u∞h ∈ V h and a subsequence (unk

h )k such that unk

h → u∞h in V h as k → +∞.Using the same matrix notation as introduced in the proof of Proposition 4.3.1,problem (4.86) reads(

(A+ I) I−I MΓ

)(

W n

(Un − Un−1)/δt

)

= −(

0AUn + F h(Un) + AΓU

n +GhΓ(U

n)

)

,

97


where Un (resp. W n) is the vector of the coordinates of unh (resp. wnh). The matrix

A+ I is invertible. Thus, eliminating W n, we obtain

((A+ I)−1+MΓ)Un − Un−1

δt= −(AUn+F h(Un)+AΓU

n+GhΓ(U

n)) = −∇Eh(Un),

(4.98)where

Eh(V ) = E(M∑

i=1

viϕi), ∀V = (v1, · · · , vM) ∈ IRM .

If we take the Euclidean norm of (4.98), we see that

λ1‖Un − Un−1‖

δt6 ‖∇Eh(Un)‖ 6 λM

‖Un − Un−1‖δt

, (4.99)

where 0 6 λ1 < λM < +∞ are respectively the smallest and the largest eigenvaluesof ((A+ I)−1 +MΓ) since ((A+ I)−1 +MΓ) is a symmetric positive definite matrix.On the other hand, since f and g are real analytic, the function Eh is real analyticon IRM and it satisfies the Lojasiewicz inequality; more precisely, there exist σ, γ > 0and ν ∈ (0, 1/2] such that

∀V ∈ IRM , ‖V − U∞‖ < σ ⇒ |Eh(V )|1−ν6 γ‖∇Eh(V )‖, (4.100)

where we have used the fact that Eh(U∞) = E(u∞h ) = E∗ = 0 and where ‖·‖ denotesthe Euclidean norm in IRM . Now let n be such that ‖Un −U∞‖ 6 σ. We recall thatEh satisfies the following inequality:

Eh(Un) +1

2δt|Un − Un−1|2−1,h +

1

2δt|Un − Un−1|20,Γ 6 Eh(Un−1), ∀n > 1. (4.101)

We consider the following two cases:Case 1: Eh(Un) > Eh(Un−1)/2. In what follows, we will use the fact that all normsare equivalent on V h. Since x 7→ xν−1 is non-increasing, we have

2Eh(Un) > Eh(Un−1) ⇒ Eh(Un−1)ν−1 > 2ν−1Eh(Un)ν−1. (4.102)

We also know that

|vh|2−1,h + |vh|20,Γ > ch|vh|20, ∀vh ∈ V h (4.103)

for some positive constant ch > 0 since all norms are equivalent on V h. Then

Eh(Un−1)ν − Eh(Un)ν =

∫ Eh(Un−1)

Eh(Un)

νxν−1dx

>

∫ Eh(Un−1)

Eh(Un)

ν(Eh(Un−1))ν−1dx (using (4.102))

> 2ν−1νEh(Un)ν−1[Eh(Un−1)− Eh(Un)] (using (4.101))

> 2ν−1νEh(Un)ν−1 1

2δt(|Un − Un−1|2−1,h + |Un − Un−1|20,Γ)

= 2ν−2νEh(Un)ν−1 1

δt(|Un − Un−1|2−1,h + |Un − Un−1|20,Γ)

> 2ν−2νch‖Un − Un−1‖2δtEh(Un)1−ν

.

(4.104)

98


Using (4.100) and (4.99), we obtain

Eh(Un−1)ν − Eh(Un)ν >2ν−2νchγδt

‖Un − Un−1‖2‖∇Eh(Un)‖

>2ν−2νchλMγ

‖Un − Un−1‖.(4.105)

Case 2: Eh(Un) 6 Eh(Un−1)/2. We have

Eh(Un) 6Eh(Un−1)

2

⇐⇒(Eh(Un))1/2 6(Eh(Un−1))1/2√

2

⇐⇒(Eh(Un−1))1/2 − (Eh(Un−1))1/2√2

6 (Eh(Un−1))1/2 − (Eh(Un))1/2

⇐⇒(

1− 1√2

)

(Eh(Un−1))1/2 6 (Eh(Un−1))1/2 − (Eh(Un))1/2

⇐⇒(Eh(Un−1))1/2 6

(

1− 1√2

)−1(

(Eh(Un−1))1/2 − (Eh(Un))1/2)

.

(4.106)

Using (4.101), (4.103) and (4.106), we obtain

‖Un − Un−1‖ 61√ch(|Un − Un−1|−1,h + |Un − Un−1|0,Γ)

6 2

(

δt

ch

)1/2(

Eh(Un−1)− Eh(Un))1/2

6 2

(

δt

ch

)1/2

Eh(Un−1)1/2

6 2

(

1− 1√2

)−1(δt

ch

)1/2(

Eh(Un−1)1/2 − Eh(Un)1/2)

.

(4.107)

Thus, in both cases, we have

‖Un − Un−1‖ 622−νλMγ

νch

(

Eh(Un−1)ν − Eh(Un)ν)

+ 2

(

1− 1√2

)−1(δt

ch

)1/2(

Eh(Un−1)1/2 − Eh(Un)1/2)

622−νλMγ

νch

(

Eh(Un−1)ν − Eh(Un)ν)

+

(

δt

ch

)1/2(

Eh(Un−1)1/2 − Eh(Un)1/2)

.

(4.108)

Now, let E > 0 be small enough so that

22−νλMγ

νchEν +

(

δt

ch

)1/2

E1/26 σ/3. (4.109)

99


We choose n large enough such that

‖Un − U∞‖ 61√ch(|Un − U∞|−1,h + |Un − U∞|0,Γ) < σ/3 and Eh(Un) 6 E. Let

N − 1 > n be the largest integer (including +∞) such that

‖Un − U∞‖ 61√ch(|Un − U∞|−1,h + |Un − U∞|0,Γ) < 2σ/3,

for all n with n 6 n 6 N − 1. Assume by contradiction that N is finite. We deducefrom (4.101) that

‖UN − U∞‖ 61√ch(|UN − U∞|−1,h + |UN − U∞|0,Γ)

61√ch

(

|UN − UN−1|−1,h + |UN − UN−1|0,Γ)

+1√ch

(

|UN−1 − U∞|−1,h + |UN−1 − U∞|0,Γ)

61√ch

(

√

2δtEh(UN−1) + |UN−1 − U∞|−1,h + |UN−1 − U∞|0,Γ)

6σ/3 + 2σ/3 = σ.

(4.110)

So we may apply (4.108) to every n 6 n 6 N − 1 and since (Eh(Un))n is non-increasing, we obtain

N∑

n=n

‖Un − Un−1‖ 622−νλMγ

νchEh(UN−1)ν + 5

(

δt

ch

)1/2

Eh(UN−1)1/2 6 σ/3. (4.111)

Thus,

‖UN − U∞‖ 6 ‖UN − Un‖+ ‖Un − U∞‖

6

N∑

n=n

‖Un − Un−1‖+ ‖Un − U∞‖

6 σ/3 + σ/3 = 2σ/3,

(4.112)

which is in contradiction with the definition of N − 1. So N = +∞ and the wholesequence (Un) converges to U∞. Since wn

h , defined by (4.86), is a continuous functionof unh, w

nh also has a limit w∞

h as n → +∞. We see that (u∞h , w∞h ) is necessarily a

steady state by passing to the limit in (4.86).

4.6 Numerical simulations

In this section, we illustrate some numerical simulations in two space dimensions.The fully discrete scheme (4.86) requires at each time step the resolution of a nonlin-ear system and for the numerical computation of solutions of the space semi-discretescheme (4.14), we propose instead of (4.86) a semi-implicit time discretizations which

100

4.6 Numerical simulations

is the semi-implicit Euler (SIE) scheme, i.e. (4.14) but the implicit nonlinear termsf(unh) and g(unh) are respectively replaced by the explicit terms f(un−1

h ) and g(un−1h ).

Using the same arguments as in the proof of Proposition 4.3.1 we see that the matrixof the SIE scheme is positive semidefinite and invertible, so that the SIE scheme iswell-posed.

Figure 4.1: t = 5

Figure 4.2: t = 10

Figure 4.3: t = 25

Figure 4.4: t = 50

In Figures 4.1-4.4, we see the result of the SIE scheme on the slab Lx×Ly = 80×10.The triangulation Ωh was obtained by dividing the slab into 256 × 50 rectanglesand by dividing every rectangle along the same diagonal into two triangles. Thenonlinearities are

f(v) =1

2(v3 − v) and g(v) = (λ+ k)v − h, v ∈ R, (4.113)

with λ + k = 1 (λ = 0.5, for instance), h = 0 and δt = 0.1. In each picture, themaximum and minimum values of u are colored in white and black and values of u inbetween correspond to different shades of grey. In these numerical simulations, wechose the same parameters as in [7] and [33]. Since h = 0, none of the componentsis preferably attracted by the walls, which is visible on the fact that both white andblack zones appear at the boundary.

101


Figure 4.5: t = 2 (h = 0)

Figure 4.6: t = 2 (h = 0.7)

In Figures 4.5 and 4.6, we consider the nonlinearity f(v) = v3− v

2, v ∈ R, and the

same nonlinearity g(v) = (λ+k)v−h, v ∈ R. This time, δt = 0.01 and the geometryis different; the domain Ω is a disk of radius 80 centered at (0, 0) from which wehave cut off a disk of radius 40 and centered at (20, 0). The exterior boundary isdivided into 600 intervals and the internal boundary into 400 intervals, yielding atriangulation Ωh of Ω with 59 048 triangles and 30 024 vertices. In these figures, wesee the difference between the case h = 0, where no phase is preferentially attractedby the walls, and the case h = 0.7, where one of the components is preferentiallyattracted by the walls. We also remark that away from the boundary, Figures 4.5and 4.6 present the same patterns.

102

Chapitre 5

Long Time Behavior of an

Allen-Cahn Type Equation With a

Singular Potential and Dynamic

Boundary Conditions

Comportement asymptotique d’une équa-tion de type Allen-Cahn avec un po-tentiel régulier et des conditions dy-namiques sur le bord

Ce chapitre est constitué de l’article Long time behavior of an Allen-Cahn typeequation with a singular potential and dynamic boundary conditions, paruen 2012 dans dans Journal of Applied Analysis and Computation, volume 1, numero3, pages 29-56.

Journal of Applied Analysis and Computation

Volume 1, Number 3, pp. 29–56, August 2012

Long time behavior of an Allen-Cahn type equation

with a singular potential and dynamic boundary

conditions

Haydi ISRAEL

UMR 7348 CNRS. Laboratoire de Mathématiques et Applications - Université de Poitiers

SP2MI - Boulevard Marie et Pierre Curie - Téléport 2


Abstract. The aim of this paper is to study the well-posedness and the long timebehavior of solutions for an equation of Allen-Cahn type owing to proper approx-imations of the singular potential and a suitable definition of solutions. We alsoprove the existence of the finite dimensional global attractor as well as exponentialattractors.

5.1 Introduction

In this article we are interested in the study of the following initial and boundaryvalue problem, considered in a smooth and bounded domain Ω ⊂ R

3 with boundary∂Ω = Γ:

∂tφ = ∆µ− µ = −(−∆+ I)µ = −Aµ, ∂nµ|∂Ω = 0,µ = −∆φ+ f(φ)− λφ, φ|t=0 = φ0,∂tψ = ∆Γψ − g(ψ)− ∂nφ, x ∈ ∂Ω, ψ|t=0 = ψ0,φ|∂Ω = ψ,

(5.1)

where λ ∈ R, ∆Γ is the Laplace-Beltrami operator on the boundary ∂Ω, f andg are given nonlinear interaction functions. In particular, f is the derivative of adouble-well potential whose wells correspond to the phases of the material. A ther-modynamically relevant function f is the following logarithmic (singular) function:

f(s) = −2κ0s+ κ1 ln1 + s

1− s, s ∈ (−1, 1), κ0 > κ1 > 0.

The boundary condition will be interpreted as an additional second-order parabolicequation on the boundary ∂Ω.Equation (5.1) may be viewed as a combination of the well-known Cahn-Hilliardequation

∂tu = −∆(∆u+ f(u)), u(0, x) = u0(x),

5 . Long Time Behavior of an Allen-Cahn Type Equation With a Singular Potential andDynamic Boundary Conditions


∂tu = ∆u+ f(u), u(0, x) = u0(x).

This equation is associated with multiple microscopic mechanisms such as surfacediffusion and adsorption/desorption and was recently derived and studied in Karali& Katsoulakis [30], Katsoulakis & Vlachos [32], Israel [28], Mikhailov, Hildebrand& Ertl [37].This paper is organized as follows. In Section 2, we introduce regularized problemsin which the singular nonlinearity is approximated by regular functions and wederive uniform a priori estimates on the corresponding solutions. In Section 3, weformulate the variational formulation of (5.2), we verify the existence and uniquenessof a solution and we study the further regularity of the solutions. In Section 4,we give sufficient conditions which ensure that solutions are separated from thesingularities of f and that a variational solution coincides with a solution in theusual (distribution) sense. Finally, we study in Section 5 the asymptotic behaviorof the system and we prove the existence of finite-dimensional (both global andexponential) attractors.

5.2 Approximations and uniform a priori estimates

We set f(φ) := f(φ)− λφ and rewrite problem (5.1) in the form:

∂tφ = ∆µ− µ = −(−∆+ I)µ = −Aµ, ∂nµ|∂Ω = 0,

µ = −∆φ+ f(φ), φ|t=0 = φ0,∂tψ = ∆Γψ − g(ψ)− ∂nφ, x ∈ ∂Ω, ψ|t=0 = ψ0,φ|∂Ω = ψ,

(5.2)

where the singular function f satisfies:

f ∈ C2((−1, 1)),f(0) = 0, lim

s→±1f(s) = ±∞,

f ′(s) > 0, lims→±1

f ′(s) = +∞,

f ′′(s) sgn s > 0.

(5.3)

As a consequence, the following properties hold for f :

f ′(s) > −λ and − c 6 F (s) 6 f(s)s+ C, ∀s ∈ (−1, 1), (5.4)

where F (s) =∫ s

0f(r)dr and c, C are strictly positive constants.

The nonlinear function g ∈ C2([−1, 1]) can be extended, without loss of generality,to the whole real line by writing:

g(s) = s+ g0(s), ∀s ∈ R, where ‖g0‖C2(R) := C0 < +∞. (5.5)

We set, for r > 1,

Hr(Ω)⊗Hr(Γ) := v ∈ Hr(Ω), v|Γ ∈ Hr(Γ) ,106


which we endow with the norm:

‖v‖2Hr(Ω)⊗Hr(Γ) = ‖v‖2Hr(Ω) + ‖v‖2Hr(Γ).

Alternatively, the functions in Hr(Ω) ⊗ Hr(Γ) can be viewed as pairs of functions(v, v|Γ).We introduce a family of regular approximating functions: given any N ∈ IN, weset:

fN(s) =

f(−1 + 1/N) + f ′(−1 + 1/N)(s+ 1− 1/N), s < −1 + 1/N,f(s), |s| 6 1− 1/N,f(1− 1/N) + f ′(1− 1/N)(s− 1 + 1/N), 1− 1/N < s.

(5.6)

Then, we denote FN the primitive FN(s) =∫ s

0fN(s)ds and having set fN(s) =

fN(s)−λs, we define FN analogously, with fN instead of fN . We recall the followingproperties (for more details, see Miranville & Zelik [45]), namely, there exist α > 0,c > 0 and C > 0 such that:

fN(s)s > α/2|fN(s)| − c, (5.7)

and1/2FN(s)− C 6 FN(s) 6 2FN(s) + C, (5.8)

∀s ∈ R and for N > N0(λ) large enough, where the constant C only depends on λ.We then consider the approximate problems:

∂tφ = ∆µ− µ = −(−∆+ I)µ = −Aµ, ∂nµ|∂Ω = 0,

µ = −∆φ+ fN(φ), φ|t=0 = φ0,∂tψ = ∆Γψ − g(ψ)− ∂nφ, x ∈ ∂Ω, ψ|t=0 = ψ0,φ|∂Ω = ψ.

(5.9)

It is convenient to rewrite problem (5.9) in an equivalent form by using the inverseof A := (−∆+ I) (endowed with Neumann boundary conditions). Applying A−1 toboth side of (5.9), we obtain:

A−1∂tφ−∆φ+ fN(φ) = 0, x ∈ Ω, φ|t=0 = φ0,∂tψ = ∆Γψ − g(ψ)− ∂nφ, x ∈ ∂Ω, ψ|t=0 = ψ0,φ|∂Ω = ψ.

(5.10)

We start with the usual energy equality.

Lemma 5.2.1. Let the above assumptions hold and let φ be a sufficiently regularsolution of (5.10). Then, the following identities hold:

‖φ(t)‖2H−1(Ω) + ‖ψ(t)‖2L2(Γ)+

∫ t

0

(

‖φ(s)‖2H1(Ω) + ‖ψ(s)‖2H1(Γ) + (FN(φ(s)), 1)Ω

)

ds

6 C(

1 + ‖φ(0)‖2H−1(Ω) + ‖ψ(0)‖2L2(Γ)

)

,

(5.11)

107


‖φ(t)‖2H−1(Ω) + ‖ψ(t)‖2L2(Γ) +

∫ t

0

e−ν(t−s)(

‖φ(s)‖2H1(Ω) + ‖ψ(s)‖2H1(Γ)

)

ds

+

∫ t

0

e−ν(t−s)‖fN(φ(s)‖L1(Ω)ds 6 C(

1 + ‖φ(0)‖2H−1(Ω) + ‖ψ(0)‖2L2(Γ)

)

e−νt,

(5.12)

and∫ t+1

t

(‖φ(s)‖2H1(Ω)+‖ψ(s)‖2H1(Γ) + α‖fN(φ(s)‖L1(Ω))ds

6 C(

1 + (‖φ(0)‖2H−1(Ω) + ‖ψ(0)‖2L2(Γ))e−νt)

,

(5.13)

for some positive constants C and ν which are independent of t.

Proof:Multiplying (5.10) by φ and using the fact that:

‖φ‖H1(Ω) 6 C(‖∇φ‖L2(Ω) + ‖ψ‖H1(Γ)), (5.14)

we find:

1

2

d

dt

(

‖φ(t)‖2H−1(Ω) + ‖ψ(t)‖2L2(Γ)

)

+ ‖φ(t)‖2H1(Ω) + ‖ψ(t)‖2H1(Γ)

+(fN(φ(t)), φ(t))Ω + (g0(ψ(t)), ψ(t))Γ 6 0.(5.15)

Using (5.4), the fact that g0 is globally bounded and that ‖ψ(t)‖L∞(Γ) 6 1, weobtain:

d

dt

(

‖φ(t)‖2H−1(Ω) + ‖ψ(t)‖2L2(Γ)

)

+ ‖φ(t)‖2H1(Ω) + ‖ψ(t)‖2H1(Γ) + (FN(φ(t)), 1)Ω 6 C,

(5.16)for some positive constant C. Integrating (5.16) with respect to t, we deduce:

‖φ(t)‖2H−1(Ω) + ‖ψ(t)‖2L2(Γ)+

∫ t

0

(

‖φ(s)‖2H1(Ω) + ‖ψ(s)‖2H1(Γ) + (FN(φ(s)), 1)Ω

)

ds

6 C(

1 + ‖φ(0)‖2H−1(Ω) + ‖ψ(0)‖2L2(Γ)

)

,

for some constant C.Now, using (5.7) and the fact that g0 is globally bounded, (5.15) gives:

1

2

d

dt

(

‖φ(t)‖2H−1(Ω) + ‖ψ(t)‖2L2(Γ)

)

+ ‖φ(t)‖2H1(Ω) + ‖ψ(t)‖2H1(Γ)

+ α/2‖fN(φ(t)‖L1(Ω) 6 C,(5.17)

for some positive constants α and C. Hence, for ν > 0 small enough, we obtain:

d

dt(‖φ(t)‖2H−1(Ω)+‖ψ(t)‖2L2(Γ)) + ν

(

‖φ(t)‖2H−1(Ω) + ‖ψ(t)‖2L2(Γ)

)

+ cν(‖φ(t)‖2H1(Ω) + ‖ψ(t)‖2H1(Γ) + ‖fN(φ(t)‖L1(Ω))

6C.

(5.18)

108


Applying Gronwall’s lemma, we deduce estimate (5.12). Finally, integrating (5.17)with respect to t over (t, t+ 1) and using (5.12), we obtain:

∫ t+1

t

(‖φ(s)‖2H1(Ω) + ‖ψ(s)‖2H1(Γ)+α‖fN(φ(s)‖L1(Ω))ds

6 C(1 + ‖φ(t)‖2H−1(Ω) + ‖ψ(t)‖2L2(Γ))

6 C(

1 + (‖φ(t)‖2H−1(Ω) + ‖ψ(t)‖2L2(Γ))e−νt)

.

(5.19)

Lemma 5.2.2. Let the assumptions of Lemma 5.2.1 hold. Then, the followingidentity holds:

‖φ(t)‖2H1(Ω) + ‖ψ(t)‖2H1(Γ) + 2(FN(φ(t)), 1)Ω

+

∫ t

0

(‖∂tφ(s)‖2H−1(Ω) + ‖∂tψ(s)‖2L2(Γ))ds

6 C(

1 + ‖φ(0)‖2H1(Ω) + ‖ψ(0)‖2H1(Γ) + 2(FN(φ(0)), 1)Ω

)

,

(5.20)

where the constant C is independent of t and of the initial data.

Proof:Multiplying the first equation of (5.10) by ∂tφ and integrating over Ω,we obtain:

1

2

d

dt

(

‖∇φ(t)‖2L2(Ω) + ‖∇Γψ(t)‖2L2(Γ) + ‖ψ(t)‖2L2(Γ)

)

+ ‖∂tφ(t)‖2H−1(Ω)

+ ‖∂tψ(t)‖2L2(Γ) + (fN(φ(t)), ∂tφ(t))Ω + (g0(ψ(t)), ∂tψ(t))Γ

=1

2

d

dt(‖∇φ(t)‖2L2(Ω) + ‖∇Γψ(t)‖2L2(Γ) + ‖ψ(t)‖2L2(Γ) + 2(FN(φ(t)), 1)Ω

+ 2(G0(ψ(t)), 1)Γ) + ‖∂tφ(t)‖2H−1(Ω) + ‖∂tψ(t)‖2L2(Γ)

=0,

(5.21)

where G0(t) =

∫ t

0

g0(s)ds.

Using (5.14), we find:

1

2

d

dt

(

‖φ(t)‖2H1(Ω) + ‖ψ(t)‖2H1(Γ) + 2(FN(φ(t)), 1)Ω + 2(G0(ψ(t)), 1)Γ

)

+‖∂tφ(s)‖2H−1(Ω) + ‖∂tψ(s)‖2L2(Γ) = 0. (5.22)

Integrating (5.22) with respect of t, taking into account that g0 is globally bounded,we deduce (5.20).

109


Lemma 5.2.3. Let the assumptions of Lemma 5.2.1 hold, φ be a sufficiently regularsolution of (5.10) and N be large enough. Then, for t > 0, the following smoothingproperty holds:

t(

‖φ(t)‖2H1(Ω) + ‖ψ(t)‖2H1(Γ) + 2(FN(φ(t)), 1)Ω

)

+

∫ t

0

s(‖∂tφ(s)‖2H−1(Ω) + ‖∂tψ(s)‖2L2(Γ))ds

6 C(

1 + ‖φ(0)‖2H−1(Ω) + ‖ψ(0)‖2L2(Γ)

)

,

(5.23)

where the constant C is independent of N .

Proof:Multiplying (5.22) by t, we obtain:

1

2

d

dt(t(‖φ(t)‖2H1(Ω) + ‖ψ(t)‖2H1(Γ) + 2(FN(φ(t)), 1)Ω + 2(G0(ψ(t)), 1)Γ))

+ t(‖∂tφ(t)‖2H−1(Ω) + ‖∂tψ(t)‖2L2(Γ))

=1

2(‖φ(t)‖2H1(Ω) + ‖ψ(t)‖2H1(Γ) + 2(FN(φ(t)), 1)Ω + 2(G0(ψ(t)), 1)Γ).

(5.24)

Integrating (5.24) with respect of t from 0 to t, we have:

t(‖φ(t)‖2H1(Ω) + ‖ψ(t)‖2H1(Γ) + 2(FN(φ(t)), 1)Ω + 2(G0(ψ(t)), 1)Γ

+ 2

∫ t

0

s(‖∂tφ(s)‖2H−1(Ω) + ‖∂tψ(s)‖2L2(Γ))ds

6

∫ t

0

(‖φ(s)‖2H1(Ω) + ‖ψ(s)‖2H1(Γ) + 2(FN(φ(s)), 1)Ω + 2(G0(ψ(s)), 1)Γ).

(5.25)

Using (5.8), (5.11) and that g0 is bounded globally, we deduce:

t(

‖φ(t)‖2H1(Ω) + ‖ψ(t)‖2H1(Γ) + 2(FN(φ(t)), 1)Ω

)

+

∫ t

0

s(‖∂tφ(s)‖2H−1(Ω) + ‖∂tψ(s)‖2L2(Γ))ds

6 C(

1 + ‖φ(0)‖2H−1(Ω) + ‖ψ(0)‖2L2(Γ)

)

,

(5.26)

and the proof is complete.

Lemma 5.2.4. Let the assumptions of Lemma 5.2.1 hold. Then, we have, for allt > 1 and N large enough, the following property:

‖φ(t)‖2H1(Ω) + ‖ψ(t)‖2H1(Γ) + 2(FN(φ(t)), 1)Ω

+

∫ t

1

(‖∂tφ(s)‖2H−1(Ω) + ‖∂tψ(s)‖2L2(Γ))ds

6 C(1 + ‖φ(0)‖2H−1(Ω) + ‖ψ(0)‖2L2(Γ)).

(5.27)

Moreover, for any t > 0 and N large enough, the following inequality holds:

‖φ(t)‖2H1(Ω)+‖ψ(t)‖2H1(Γ) + 2(FN(φ(t)), 1)Ω

6 Ct+ 1

t(1 + ‖φ(0)‖2H−1(Ω) + ‖ψ(0)‖2L2(Γ)).

(5.28)

110


Proof:Integrating (5.22) with respect to t from 1 to t, we obtain:

‖φ(t)‖2H1(Ω) + ‖ψ(t)‖2H1(Γ) + 2(FN(φ(t)), 1)Ω

+

∫ t

1

(‖∂tφ(s)‖2H−1(Ω) + ‖∂tψ(s)‖2L2(Γ))ds

6 C(

1 + ‖φ(1)‖2H1(Ω) + ‖ψ(1)‖2H1(Γ) + 2(FN(φ(1)), 1)Ω

)

.

(5.29)

From (5.23) and for t = 1, we find:(

‖φ(1)‖2H1(Ω) + ‖ψ(1)‖2H1(Γ) + 2(FN(φ(1)), 1)Ω

)

+

∫ 1

0

s(‖∂tφ(s)‖2H−1(Ω) + ‖∂tψ(s)‖2L2(Γ))ds

6 C(

1 + ‖φ(0)‖2H−1(Ω) + ‖ψ(0)‖2L2(Γ)

)

.

(5.30)

Estimates (5.29) and (5.30) allow us to find (5.27). Estimate (5.28) follows imme-diately from (5.23) and (5.27).We will now give some additional regularity results on ∂tφ(t). To prove this, wedifferentiate (5.9) and set (u(t), v(t), w(t)) := ∂t(φ(t), µ(t), ψ(t)). Then we have:

∂tu = ∆v − v = −(−∆+ I)v = −Av, ∂nv|Γ = 0,

v = −∆u+ f ′N(φ)u,

∂tw = ∆Γw − g′(ψ)w − ∂nw, x ∈ Γ,u|Γ = w.

(5.31)

Lemma 5.2.5. Let the assumptions of Lemma 5.2.1 hold. Then, the followingestimate is valid for all t > 0:

‖u(t)‖2H−1(Ω) + ‖w(t)‖2L2(Γ)+

∫ t+1

t

(‖u(s)‖2H1(Ω) + ‖w(s)‖2H1(Γ))ds

6 c(‖u(0)‖2H−1(Ω) + ‖w(0)‖2L2(Γ))e−νt + c,

(5.32)

for some positive constants c and ν independent of N . Moreover, for t > 0, we havethe smoothing property:

‖u(t)‖2H−1(Ω) + ‖w(t)‖2L2(Γ) 6 ct2 + 1

t2(1 + ‖φ(0)‖2H−1(Ω) + ‖ψ(0)‖2L2(Γ)). (5.33)

Proof:Multiplying the first equation of (5.31) by A−1u, the second equation by u and thethird one by w and taking the sum of the equations that we obtain, we have thefollowing identity:

1

2

d

dt

(

‖u(t)‖2H−1(Ω) + ‖w(t)‖2L2(Γ)

)

+‖∇u(t)‖2L2(Ω) + ‖∇Γw(t)‖2L2(Γ) + ‖w(t)‖2L2(Γ)

+ (f ′N (φ(t))u(t), u(t))Ω + (g′0(ψ(t))w(t), w(t))Γ = 0.

(5.34)

111


Using (5.4), (5.14) and that g′0 is globally bounded, we find:

d

dt

(

‖u(t)‖2H−1(Ω) + ‖w(t)‖2L2(Γ)

)

+2‖u(t)‖2H1(Ω) + 2‖w(t)‖2H1(Γ)

6 c(‖u(t)‖2L2(Ω) + ‖w(t)‖2L2(Γ)).(5.35)

Using the interpolation inequality ‖u‖2L2(Ω) 6 c‖u‖H1(Ω)‖u‖H−1(Ω), we find:

d

dt

(

‖u(t)‖2H−1(Ω) + ‖w(t)‖2L2(Γ)

)

+‖u(t)‖2H1(Ω) + ‖w(t)‖2H1(Γ)

6 c(‖u(t)‖2H−1(Ω) + ‖w(t)‖2L2(Γ)).(5.36)

Provided that ν > 0 is small enough, the following estimate holds:

d

dt(‖u(t)‖2H−1(Ω) + ‖w(t)‖2L2(Γ)) + ν(‖u(t)‖2H−1(Ω) + ‖w(t)‖2L2(Γ))

+cν(‖u(t)‖2H1(Ω) + ‖w(t)‖2H1(Γ)) 6 c(‖u(t)‖2H−1(Ω) + ‖w(t)‖2L2(Γ)).

(5.37)

We consider the case t > 1. Applying Gronwall’s inequality and using estimate(5.27), we obtain:

‖u(t)‖2H−1(Ω) + ‖w(t)‖2L2(Γ) 6 (‖u(1)‖2H−1(Ω) + ‖w(1)‖2L2(Γ))e−ν(t−1)

+ c

∫ t

1

e−ν(t−s)(‖u(s)‖2H−1(Ω) + ‖w(s)‖2L2(Γ))ds

6 (‖u(1)‖2H−1(Ω) + ‖w(1)‖2L2(Γ))e−ν(t−1) + c

∫ t

1

(‖u(s)‖2H−1(Ω) + ‖w(s)‖2L2(Γ))ds

6 (‖u(1)‖2H−1(Ω) + ‖w(1)‖2L2(Γ))e−ν(t−1) + c(1 + ‖φ(0)‖2H−1(Ω) + ‖ψ(0)‖2L2(Γ)).

(5.38)

Integrating (5.37) over (t, t+ 1), t > 1, we find:

∫ t+1

t

(‖u(s)‖2H1(Ω) + ‖w(s)‖2H1(Γ))ds

6‖u(t)‖2H−1(Ω) + ‖w(t)‖2L2(Γ) + c

∫ t+1

t

(‖u(s)‖2H−1(Ω) + ‖w(s)‖2L2(Γ))ds

6(‖u(1)‖2H−1(Ω) + ‖w(1)‖2L2(Γ))e−ν(t−1) + c(1 + ‖φ(0)‖2H−1(Ω) + ‖ψ(0)‖2L2(Γ)).

(5.39)

We have:

t2(‖u(t)‖2H−1(Ω) + ‖w(t)‖2L2(Γ)) =2

∫ t

0

s(‖u(s)‖2H−1(Ω) + ‖w(s)‖2L2(Γ))ds

+

∫ t

0

s2∂t(‖u(s)‖2H−1(Ω) + ‖w(s)‖2L2(Γ))ds.

112


Taking into account (5.26) and (5.36), we obtain for t ∈ (0, 1]:

t2(‖u(t)‖2H−1(Ω) + ‖w(t)‖2L2(Γ)) 62

∫ t

0

s(‖u(s)‖2H−1(Ω) + ‖w(s)‖2L2(Γ))ds

+ c

∫ t

0

s(‖u(s)‖2H−1(Ω) + ‖w(s)‖2L2(Γ))ds

6(2 + c)

∫ t

0

s(‖u(s)‖2H−1(Ω) + ‖w(s)‖2L2(Γ))ds

6c(1 + ‖φ(0)‖2H−1(Ω) + ‖ψ(0)‖2L2(Γ)),

(5.40)

for some positive constant c. For t = 1, estimate (5.40) gives:

‖u(1)‖2H−1(Ω) + ‖w(1)‖2L2(Γ) 6 c(1 + ‖φ(0)‖2H−1(Ω) + ‖ψ(0)‖2L2(Γ)), (5.41)

then, replacing this estimate in (5.38) and (5.39), we deduce:

‖u(t)‖2H−1(Ω) + ‖w(t)‖2L2(Γ)+

∫ t+1

t

(‖u(s)‖2H1(Ω) + ‖w(s)‖2H1(Γ))ds

6c+ c(‖φ(0)‖2H−1(Ω) + ‖ψ(0)‖2L2(Γ))e−νt,

(5.42)

for all t > 1. From estimates (5.40) and (5.42), we deduce:

‖u(t)‖2H−1(Ω) + ‖w(t)‖2L2(Γ) 6 c(1 +1

t2)(1 + ‖φ(0)‖2H−1(Ω) + ‖ψ(0)‖2L2(Γ)), ∀t > 0.

(5.43)To show estimate (5.32) for t ∈ (0, 1], we consider (5.37), in particular,

d

dt(‖u(t)‖2H−1(Ω) + ‖w(t)‖2L2(Γ)) 6 c(‖u(t)‖2H−1(Ω) + ‖w(t)‖2L2(Γ)). (5.44)

Applying Gronwall’s inequality to (5.44), we find:

‖u(t)‖2H−1(Ω) + ‖w(t)‖2L2(Γ) 6cect(‖u(0)‖2H−1(Ω) + ‖w(0)‖2L2(Γ))

6c′e−νt(‖u(0)‖2H−1(Ω) + ‖w(0)‖2L2(Γ)), t ∈ (0, 1].(5.45)

Integrating (5.37) over (t, t+ 1), t ∈ (0, 1], using (5.42) and (5.45), we obtain:

∫ t+1

t

(‖u(s)‖2H1(Ω) + ‖w(s)‖2H1(Γ))ds

6‖u(t)‖2H−1(Ω) + ‖w(t)‖2L2(Γ) + c

∫ t+1

t

(‖u(s)‖2H−1(Ω) + ‖w(s)‖2L2(Γ))ds

6‖u(t)‖2H−1(Ω) + ‖w(t)‖2L2(Γ) + c

∫ 1

t

(‖u(s)‖2H−1(Ω) + ‖w(s)‖2L2(Γ))ds

+ c

∫ t+1

1

(‖u(s)‖2H−1(Ω) + ‖w(s)‖2L2(Γ))ds

6c+ c(‖u(0)‖2H−1(Ω) + ‖w(0)‖2L2(Γ))e−νt.

(5.46)

Hence, the proof is complete.

113


Theorem 5.2.6. Let the nonlinearities f and g satisfy (5.3) and (5.5) respectivelyand set Ωδ := x ∈ Ω, d(x,Γ) > δ . Denote by n = n(x) some smooth extension ofthe unit normal vector field at the boundary inside the domain Ω. Let also Dτφ :=∇xφ− (∂nφ)n be the tangential part of the gradient ∇xφ. Then, for every δ > 0, thefollowing estimate is valid:

‖φ(t)‖2Cα(Ω) + ‖φ(t)‖2H2(Ωδ)+ ‖ψ(t)‖2H2(Γ) + ‖φ(t)‖2H1(Ω) + ‖∂tφ(t)‖2H−1(Ω)

+‖∂tψ(t)‖2L2(Γ) + ‖∇Dτφ(t)‖2[L2(Ω)]6 + ‖fN(φ(t))‖L1(Ω)

+

∫ t+1

t

(‖∂tφ(s)‖2H1(Ω) + ‖∂tψ(s)‖2H1(Γ))ds

6 C(1 + ‖φ(0)‖2H1(Ω) + ‖ψ(0)‖2H1(Γ) + ‖∂tφ(0)‖2H−1(Ω) + ‖∂tψ(0)‖2L2(Γ))e−γt, (5.47)

where the positive constants α(α > 1/4), γ and C are independent of N.

Proof:We consider the nonlinear elliptic problem:

∆φ(t)− fN(φ(t))− φ(t) = h1(t) := −φ(t) + λφ(t) + A−1∂tφ(t), x ∈ Ω∆Γψ(t)− ψ(t)− ∂nφ(t) = h2(t) := g0(ψ(t)) + ∂tψ(t), x ∈ Γ,

(5.48)

for every fixed t. Note here that the estimates derived above yield the followingcontrol of the right-hand side of (5.48):

‖h1(t)‖2L2(Ω) + ‖h2(t)‖2L2(Γ) 6 C(1 + ‖∂tφ(t)‖2H−1(Ω) + ‖∂tψ(t)‖2L2(Γ)), (5.49)

where C is a positive constant that is independent of N. Due to estimate (5.33), wefind that h1 ∈ L2(Ω) and h2 ∈ L2(Γ). Using estimates (5.20), (5.32) and (5.80), weobtain:

‖φ(t)‖2H1(Ω) + ‖∂tφ(t)‖2H−1(Ω) + ‖∂tψ(t)‖2L2(Γ) + ‖fN(φ(t))‖L1(Ω)

+

∫ t+1

t

(‖∂tφ(s)‖2H1(Ω) + ‖∂tψ(s)‖2H1(Γ))ds

6 C(1 + ‖φ(0)‖2H1(Ω) + ‖ψ(0)‖2H1(Γ) + ‖∂tφ(0)‖2H−1(Ω) + ‖∂tψ(0)‖2L2(Γ))e−γt.

(5.50)

In order to prove the following estimate:

‖φ(t)‖2H2(Ωδ)6 C(1 + ‖h1‖2L2(Ω) + ‖h2‖2L2(Γ)), (5.51)

where C = Cǫ depends on ǫ > 0. We consider a smooth nonnegative cut-off functionθ such that θ(x) = 1 if d(x,Γ) > δ and θ(x) = 0 if d(x,Γ) 6 δ/2 which satisfies, inaddition, the inequality:

|∇xθ(x)| 6 Cθ1/2(x).

Then, we multiply equation (5.48) by∑3

i=3 ∂x1(θ(x)∂xi

u), and we integrate by parts.Using estimate (5.50) and the fact that f ′ > 0, we obtain estimate (5.51). In orderto prove:

‖∇xDτφ‖2L2(Ω)6 + ‖φ‖2H2(Γ) 6 C(1 + ‖h1‖2L2(Ω) + ‖h2‖2L2(Γ)), (5.52)

114


we study the function φ in a small ǫ-neighborhood of the boundary Γ. To do so, letx0 ∈ Γ and y = y(x) be a local coordinates in the neighborhood of x0 such thaty(x0) = 0 and Ω is defined, in these coordinates, by the condition y1 > 0. Then, werewrite problem (5.48) in the variable y and after several transformations we findestimate (5.52). To finish the proof of the theorem, we use the following embedding:

L2(IR, H2(IR2)) ∩H1(IR, H1(IR2)) ⊂ Cα(IR3), α < 1/4,

and we deduce the estimate:

‖φ‖2Cα(Ω) 6 C(1 + ‖h1‖2L2(Ω) + ‖h2‖2L2(Γ)).

Hence, Theorem (5.2.6) is proved (for more details see Miranville & Zelik [45]).In what follows, we will establish the uniform Lipschitz continuity of the solution(φ(t), µ(t), ψ(t)) of problem (5.10) with respect to the initial data.

Proposition 5.2.7. Let the above assumptions hold and let (φ1(t), µ1(t), ψ1(t)) and(φ2(t), µ2(t), ψ2(t)) be two solutions of problem (5.10). Then, the following estimateholds:

‖φ1(t)− φ2(t)‖2H−1(Ω) + ‖ψ1(t)− ψ2(t)‖2L2(Γ)

+

∫ t+1

t

(‖φ1(s)− φ2(s)‖2H1(Ω) + ‖ψ1(s)− ψ2(s)‖2H1(Γ))ds

6 C(

‖φ1(0)− φ2(0)‖2H−1(Ω) + ‖ψ1(0)− ψ2(0)‖2L2(Γ)

)

eKt,

(5.53)

where the constants C and K are independent of t, N and the initial data.

Proof:Let (φ(t), µ(t), ψ(t)) = (φ1(t)−φ2(t), µ1(t)−µ2(t), ψ(t)−ψ2(t)). Then, this functionsatisfies the system:

∂tφ = −Aµ, ∂nµ|∂Ω = 0,

µ = −∆φ+ lN(t)φ, φ|t=0 = φ0,∂tψ = ∆Γψ − ψ −m(t)ψ − ∂nφ, x ∈ ∂Ω, ψ|t=0 = ψ0,φ|∂Ω = ψ,

(5.54)

where

lN(t) :=

∫ t

0

f ′N(sφ1(t) + (1− s)φ2(t))ds and m(t) :=

∫ t

0

g′0(sφ1(t) + (1− s)φ2(t))ds.

Multiplying the first equation of (5.31) by A−1φ, the second equation by φ and thethird one by ψ and taking the sum of the equations that we obtain, we have thefollowing identity:

1

2

d

dt

(

‖φ(t)‖2H−1(Ω) + ‖ψ(t)‖2L2(Γ)

)

+ ‖∇φ(t)‖2L2(Ω) + ‖∇Γψ(t)‖2L2(Γ) + ‖ψ(t)‖2L2(Γ)

+ (lN(t)φ(t), φ(t))Ω + (m(t)ψ, ψ(t))Γ = 0.

(5.55)

115


Using (5.4), (5.14) and the fact that g′0 is globally bounded, we obtain:

d

dt

(

‖φ(t)‖2H−1(Ω) + ‖ψ(t)‖2L2(Γ)

)

+α′(

‖φ(t)‖2H1(Ω) + ‖ψ(t)‖2H1(Ω)

)

6 C(‖φ(t)‖2L2(Ω) + ‖ψ(t)‖2L2(Ω))(5.56)

for some positive constants α′ and C which are independent of N. Using the interpo-lation inequality ‖u‖22 6 C‖u‖H1(Ω)‖u‖H−1(Ω) and applying the Gronwall inequality,we deduce (5.53).

5.3 Variational formulation and well-posedness

This section is devoted to the definition of a suitable notion for a solution to the limitproblem, that is, the problem obtained by letting N → +∞ and which coincideswith (5.2).To this end, we first fix a constant L > 0 such that:

‖∇ϕ‖2L2(Ω) − λ‖ϕ‖2L2(Ω) + L‖ϕ‖2H−1(Ω) > 1/2‖ϕ‖2H1(Ω), (5.57)

for all ϕ ∈ H1(Ω) and introduce the quadratic form:

B(ϕ, ρ) := (∇ϕ,∇ρ)Ω − λ(ϕ, ρ)Ω + L((−∆+ I)−1ϕ, ρ)Ω + (∇Γϕ,∇Γρ)Γ,

∀ϕ, ρ ∈ H1(Ω)⊗H1(Γ). Then, obviously, we have:

B(ϕ, ϕ) > 0, ∀ϕ ∈ H1(Ω)⊗H1(Γ).

The limit problem (5.10), corresponding to N = +∞ formally reads:

A−1∂tφ = ∆φ− f(φ) + λφ, φ|t=0 = φ0,∂tψ = ∆Γψ − g(ψ)− ∂nφ, x ∈ ∂Ω, ψ|t=0 = ψ0,φ|∂Ω = ψ.

(5.58)

Multiplying the first equation of (5.58) by the function φ− ϕ, where ϕ = ϕ(t, x) issmooth, and integrating by parts, we obtain:

(A−1∂tφ, φ− ϕ)Ω + (∂tφ, φ− ϕ)Γ+(∇φ,∇(φ− ϕ))Ω − λ(φ, φ− ϕ)Ω + (∇Γφ,∇Γ(φ− ϕ))Γ

+ (f(φ), φ− ϕ)Ω + (g(φ), φ− ϕ)Γ = 0,

which yields:

(A−1∂tφ, φ− ϕ)Ω + (∂tφ, φ− ϕ)Γ+B(φ, φ− ϕ)Ω + (f(φ), φ− ϕ)Ω

= L(A−1φ, φ− ϕ)Ω − (g(φ), φ− ϕ)Γ,(5.59)

∀ϕ ∈ H1(Ω)⊗H1(Γ). Finally, since B is positive and f is monotone, we have:

B(φ, φ− ϕ) > B(ϕ, φ− ϕ), (f(φ), φ− ϕ)Ω > (f(ϕ), φ− ϕ)Ω. (5.60)

Consequently, (5.59) can be written as follow:

(A−1∂tφ, φ− ϕ)Ω + (∂tφ, φ− ϕ)Γ+B(ϕ, φ− ϕ)Ω + (f(ϕ), φ− ϕ)Ω

6 L(A−1φ, φ− ϕ)Ω − (g(φ), φ− ϕ)Γ,(5.61)

116


∀ϕ ∈ H1(Ω)⊗H1(Γ). If we consider the solutions of problem (5.10) with initial databelonging to:

Φ := (φ, ψ) ∈ L∞(Ω)× L∞(Γ), ‖φ‖L∞(Ω) 6 1, ‖ψ‖L∞(Γ) 6 1

and then pass to the limit N → ∞, we will find functions living in Φ for all time.These functions are not necessarily solutions to (5.2) in the usual sense. For this,we define a variational solution of the limit problem (5.58) as follows.

Definition 5.3.1. Let (φ0, ψ0) ∈ Φ. We say that (φ(t), ψ(t)) is a variational solutionto problem (5.2) originating from (φ0, ψ0) if

1. φ(t)|Γ = ψ(t) for almost all t > 0,

2. φ(0) = φ0, ψ(0) = ψ0,

3. −1 < φ(t, x) < 1 for almost all (t, x) ∈ R+ × Ω,

4. (φ, ψ) ∈ C([0,+∞), H−1(Ω) × L2(Γ)) ∩ L2([0, T ], H1(Ω) × H1(Γ)), for anyT > 0,

5. f(φ) ∈ L1([0, T ]× Ω) for any T > 0,

6. (∂tφ, ∂tψ) ∈ L2([0, T ], H−1(Ω)× L2(Γ)), for any T > 0,

and the variational inequality

(A−1∂tφ(t), φ(t)− ϕ)Ω + (∂tφ(t), φ(t)− ϕ)Γ +B(ϕ, φ(t)− ϕ)Ω + (f(ϕ), φ(t)− ϕ)Ω

6 L(A−1φ(t), φ(t)− ϕ)Ω − (g(φ(t)), φ(t)− ϕ)Γ,

(5.62)

is satisfied for almost all t > 0 and any test function ϕ ∈ H1(Ω)⊗H1(Γ) such thatf(ϕ) ∈ L1(Ω).

We emphasize that we do not assume in the definition that ψ0 is the trace of φ0.In order to show the uniqueness of a variational solution, we consider (5.62) in termsof test functions ϕ = ϕ(t, x) depending on t and x with ϕ satisfying the regularityassumptions in Definition 5.3.1. Then, we write inequality (5.62) with ϕ = ϕ(t, x)for almost all t > 0. Moreover, due to the regularity assumptions (5.3.1) on φ andϕ, we integrate (5.62) with respect to t since all terms are in L1. This gives, for allt > s > 0:

∫ t

s

(

(A−1∂tφ, φ− ϕ)Ω + (∂tφ, φ− ϕ)Γ +B(ϕ, φ− ϕ)Ω + (f(ϕ), φ− ϕ)Ω)

dτ

6

∫ t

s

(

L(φ,A−1(φ− ϕ))Ω − (g(φ), φ− ϕ)Γ)

dτ.

(5.63)

Arguing as in Miranville & Zelik [45], we set ϕα := (1− α)φ+ αϕ, where α ∈ (0, 1].Then, assumption (5.3)4 implies that the function |f(φ)| is convex and

|f(ϕα)| 6 |f(φ)|+ |f(ϕ)|,117


which yields that f(ϕα) ∈ L1(Ω). Consequently, ϕα is an admissible test functionfor (5.63). Inserting ϕ = ϕα in the variational inequality (5.63), simplifying byα and using the fact that (φ, ψ) is absolutely continuous on [s, t] with values inH−1(Ω)× L2(Ω), we get:

∫ t

s

((A−1∂tφ, φ− ϕ)Ω + (∂tφ, φ− ϕ)Γ +B(ϕα, φ− ϕ)Ω + (f(ϕα), φ− ϕ)Ω)dτ

6

∫ t

s

(

L(φ,A−1(φ− ϕ))Ω − (g(φ), φ− ϕ)Γ)

dτ.

(5.64)

Passing to the limit in (5.64) as α → 0 and using the Lebesgue dominated conver-gence theorem for the nonlinear term, we obtain:∫ t

s

((A−1∂tφ, φ− ϕ)Ω + (∂tφ, φ− ϕ)Γ +B(φ, φ− ϕ)Ω + (f(φ), φ− ϕ)Ω)dτ

6

∫ t

s

(

L(φ,A−1(φ− ϕ))Ω − (g(φ), φ− ϕ)Γ)

dτ.

(5.65)

We can now state the following theorem which gives the uniqueness of such varia-tional solutions.

Theorem 5.3.2. Let the nonlinearity f and g satisfy the assumptions of Section1. Then, the variational solution of problem (5.58)(in the sense of Definition 5.3.1)is unique and is independent of the choice of L satisfying (5.57). Furthermore, forevery two variational solutions (φ1, ψ1) and (φ2, ψ2), we have the following estimate:

‖φ1(t)− φ2(t)‖2H−1(Ω) + ‖ψ1(t)− ψ2(t)‖2L2(Γ)

6 ceKt(‖φ1(0)− φ2(0)‖2H−1(Ω) + ‖ψ1(0)− ψ2(0)‖2L2(Γ)),(5.66)

where the positive constants c and K are independent of t.

Proof:We use (5.63), with φ = φ1 and ϕ = φ2, and we obtain:

∫ t

s

((A−1∂tφ1, φ1 − φ2)Ω + (∂tφ1, φ1 − φ2)Γ)dτ

+

∫ t

s

(B(φ2, φ1 − φ2)Ω + (f(φ2), φ1 − φ2)Ω)dτ

6

∫ t

s

(

L(φ1, A−1(φ1 − φ2))Ω − (g(φ1), φ1 − φ2)Γ

)

dτ,

(5.67)

and (5.65) with φ = φ2 and ϕ = φ1, we find:∫ t

s

((A−1∂tφ2, φ2 − φ1)Ω + (∂tφ2, φ2 − φ1)Γ)dτ

+

∫ t

s

(B(φ2, φ2 − φ1)Ω + (f(φ2), φ2 − φ1)Ω)dτ

6

∫ t

s

(

L(φ2, A−1(φ2 − φ1))Ω − (g(φ2), φ2 − φ1)Γ

)

dτ.

(5.68)

118


Summing the two resulting inequalities (5.67) and (5.68) and using the fact that(φi, ψi) are absolutely continuous on [s, t], i = 1, 2, with values in H−1(Ω)× L2(Γ),we obtain:

1

2(‖(φ1(t), ψ1(t))− (φ2(t), ψ2(t))‖2H−1(Ω)×L2(Γ)

− ‖(φ1(s), ψ1(s))− (φ2(s), ψ2(s))‖2H−1(Ω)×L2(Γ))

6

∫ t

s

(L‖φ1(τ)− φ2(τ)‖2H−1(Ω) − (g(φ1(τ))− g(φ2(τ)), φ1(τ)− φ2(τ)))Γdτ.

(5.69)

Using the fact that g is bounded globally and applying the Gronwall inequality to(5.69), we have:

‖(φ1(t), ψ1(t))− (φ2(t), ψ2(t))‖2H−1(Ω)×L2(Γ)

6 ceK(t−s)‖(φ1(s), ψ1(s))− (φ2(s), ψ2(s))‖2H−1(Ω)×L2(Γ),

where the positive constants c and K are independent of t > s > 0 and (φi, ψi), i =1, 2. Passing to the limit as s → 0 and thanks to the continuity of (φi, ψi), i = 1, 2from Definition 5.3.1, condition 4, we get the desired estimate, which in particulargives the uniqueness.Now, we need to prove that the above definition of a solution is independent of thechoice of L. To do so, we assume that (φ1, ψ1) is a variational solution for L = L1

and (φ2, ψ2) is a variational solution for L = L2. Using the following relation:

BL1(φ2, φ1 − φ2)− BL2

(φ2, φ1 − φ2)

= L1(φ1, A−1(φ1 − φ2))Ω − L2(φ1, A

−1(φ1 − φ2))Ω − (L1 − L2)‖φ1 − φ2‖2H−1(Ω),

and arguing as in the proof of (5.66), we find:∫ t

s((A−1(∂tφ1 − ∂tφ2),φ1 − φ2)Ω + (∂tφ1 − ∂tφ2, φ1 − φ2)Γ)dτ

+

∫ t

s(BL1

(φ2, φ1 − φ2)Ω −BL2(φ2, φ1 − φ2)Ω) dτ

6L1

∫ t

s

(

φ1, A−1(φ1 − φ2)

)

Ωdτ + L2

∫ t

s

(

φ2, A−1(φ2 − φ1)

)

Ωdτ

−∫ t

s(g(φ1(τ))− g(φ2(τ)), φ1(τ)− φ2(τ))Γ dτ.

(5.70)

After simplification, (5.70) gives:

1

2(‖(φ1(t), ψ1(t))− (φ2(t), ψ2(t))‖2H−1(Ω)×L2(Γ)

−‖(φ1(s), ψ1(s))− (φ2(s), ψ2(s))‖2H−1(Ω)×L2(Γ))

6

∫ t

s

(L1‖φ1(τ)− φ2(τ)‖2H−1(Ω) − (g(φ1(τ))− g(φ2(τ)), φ1(τ)− φ2(τ)))Γdτ. (5.71)

119


which coincides with (5.69) and also leads to (5.66). Theorem 5.3.2 is thus proven.

Theorem 5.3.3. For every initial data (φ0, ψ0) ∈ Φ, problem (5.58) possesses aunique variational solution (φ, ψ) in the sense of Definition 5.3.1. Such a solutionregularizes as t > 0 and all the uniform estimates obtained above hold. In particular,the following estimate is valid for every δ > 0 and t > 0:

‖φ(t)‖2Cα(Ω) + ‖φ(t)‖2H2(Ωδ)+ ‖ψ(t)‖2H2(Γ) + ‖φ(t)‖2H1(Ω) + ‖∂tφ(t)‖2H−1(Ω)

+‖∂tψ(t)‖2L2(Γ) + ‖∇Dτφ(t)‖2[L2(Ω)]6 + ‖f(φ(t))‖L1(Ω)

+

∫ t+1

t

(‖∂tφ(s)‖2H1(Ω) + ‖∂tψ(s)‖2H1(Γ))ds

6 C(1 + ‖φ(0)‖2L2(Ω) + ‖ψ(0)‖2L2(Γ))e−γt, (5.72)

for some positive constants α and C which are independent of t and φ, where Dτ

denotes the tangential part of the gradient ∇.

Proof:Repeating the derivation of the variational inequality (5.63), we obtain that (φN , ψN)satisfies:

∫ t

s

(A−1∂tφN , φN − ϕ)Ω+(∂tφN , φN − ϕ)Γ +B(ϕ, φN − ϕ)Ω + (fN(ϕ), φN − ϕ)Ωdτ

6

∫ t

s

(

L(φN , A−1(φN − ϕ))Ω − (g(φN), φN − ϕ)Γ

)

dτ,

(5.73)

for every admissible test function ϕ and every t > s > 0. Our aim is to pass to thelimit N → +∞.We start with the case when the initial datum φ0 is smooth and satisfies the addi-tional conditions:

|φ0(x)| 6 1− δ, δ > 0, ψ0 := u0|Γ. (5.74)

Then, by (5.47), we have:

‖φN(t)‖L∞([0,T ],H1(Ω)) + ‖φN(t)‖L∞([0,T ],H2(Ωδ)) + ‖ψN(t)‖L∞([0,T ],H2(Γ)) 6 C,

‖∂tφN(t)‖L∞([0,T ],H−1(Ω)) + ‖∂tψN(t)‖L∞([0,T ],L2(Γ)) + ‖∂tφN(t)‖L2([0,T ],H1(Ω))

+ ‖∂tψN(t)‖L2([0,T ],H1(Γ)) 6 C,(5.75)

‖D2τφN(t)‖L∞([0,T ],L2(Ω)) 6 C,

‖φN(t)‖L∞([0,T ],Cα(Ω)) 6 C, (5.76)

120


where the positive constant C depends on Ω, Γ, φ0, ψ0 and T but is independent ofN and t. From this point on, all convergence relations will be intended to hold up tothe extraction of suitable subsequences, generally not relabeled. Thus, we observethat weak and weak star compactness results applied to the sequence φN entail thatthere exists a function φ such that as N → ∞, the following properties hold:

φN → φ weakly-* in L∞(

[0, T ], (H1(Ω)⊗H2(Γ)) ∩H2(Ωδ))

,

(∂tφN , ∂tψN) → (∂tφ, ∂tψ) weakly-* in L∞(

[0, T ], (H−1(Ω)× L2(Γ)))

,

(∂tφN , ∂tψN) → (∂tφ, ∂tψ) weakly in L2(

[0, T ], (H1(Ω)⊗H1(Γ)))

,

D2τφN → D2

τφ weakly-* in L∞([0, T ], L2(Ω)).

It follows from (5.75) and (5.76), using the compactness theorem of Aubin-Lions,that:

φN → φ strongly in Cγ([0, T ]× Ω) for some γ > 0.

These convergence results allow us to pass to the limit N → +∞ in (5.73) andprove that the limit function satisfies (5.63) for any admissible test function ϕ. Theonly nontrivial term containing the nonlinearity fN can be treated by using theinequality |fN(ϕ)| 6 |f(ϕ)|, the fact that f(ϕ) ∈ L1([0, T ] × Ω) and the Lebesguedominated convergence theorem. The crucial point −1 < φ(t, x) < 1, for almostall (t, x) ∈ R × Ω, can be proven as in Miranville & Zelik [45]. Indeed, taking intoaccount the definition of fN and the fact that the L1−norm of fN(φN) is uniformlybounded, we can conclude:

meas (t, x) ∈ [T, T + 1]× Ω, |φM(t, x)| > 1 + 1/N 6 π(1/N), M > N, (5.77)

where

π(x) :=C

max |f(1− x)|, |f(x− 1)| , (5.78)

for some positive constant C which is independent of T ∈ R+, of N and M, with

M > N . Using the fact that π(x) → 0 as x → 0 and passing to the limit M, N →+∞ in (5.77), we conclude that:

meas (t, x) ∈ [T, T + 1]× Ω, |φ(t, x)| = 1 = 0, (5.79)

so that|φ(t, x)| < 1 for almost all (t, x) ∈ R

+ × Ω. (5.80)

Inequality (5.80) and the convergence φN → φ strongly in Cγ([0, T ]×Ω) imply thealmost everywhere convergence fN(φN) → f(φ). Therefore, Fatou’s lemma gives:

‖f(φ)‖L1([0,T ]×Ω) 6 lim infN→+∞

‖fN(φN)‖L1([0,T ]×Ω) < +∞. (5.81)

Thus, f(φ) ∈ L1([0, T ]×Ω) and (φ, ψ) is a variational solution to problem (5.58). Inparticular, the L1−estimate on f(φ) follows from (5.81). Since the separation fromsingularities is not ensured on the boundary, we are not allowed to pass to the limitin ‖FN(φN(t))‖L1(Γ).Finally, we remove assumption (5.74). In that case, we approximate the initial

121


datum (φ0, ψ0) ∈ Φ by a sequence (φk0, ψ

k0) of smooth functions satisfying (5.74)

such that:

‖φ0 − φk0‖L2(Ω) → 0, ‖ψ0 − ψk

0‖L2(Γ) → 0, as k → +∞. (5.82)

Let (φk(t), ψk(t)) be a sequence of variational solutions of problem (5.58) satisfying(φk(0), ψk(0)) = (φk

0, ψk0), where φk|Γ = ψk. The existence of such a sequence of

solutions was proved above. Then, by estimate (5.66) and assumption (5.82), wecan see that (φk, ψk) is a Cauchy sequence in C([0, T ], H−1(Ω)×L2(Γ)) and therefore,the limit function exists and

(φ, ψ) := limk→+∞

(φk, ψk) ∈ C([0, T ], H−1(Ω)× L2(Γ)).

Then, the proof of the theorem is finished as above.We also have the following result:

Lemma 5.3.4. Let (φ(t), ψ(t)) be a variational solution of problem (5.58). Then,ψ(t) = φ(t)|Γ for t > 0. Moreover, this solution solves (5.58) in the usual sense,that is, for any ϕ ∈ C∞

0 ((0, T )× Ω), the following equation holds:∫

R+

(A−1∂tφ(t), ϕ(t))Ωdt =

∫

R+

((∆φ(t), ϕ(t))Ω − (f(φ(t), ϕ(t))Ω) dt

+

∫

R+

λ(φ(t), ϕ(t))Ωdt.

(5.83)

Furthermore,φ ∈ L∞([τ, T ],W 2,1(Ω)), 0 < τ < T, (5.84)

so that the trace of the normal derivative on the boundary,

[∂nφ]int := ∂nφ|Γ ∈ L∞([τ, T ], L1(Γ)), 0 < τ < T, (5.85)

exists.

Proof:Since φN is uniformly bounded in L∞([τ, T ], H2(Ωδ)), ∀δ > 0, and fN is uniformlycontinuous, the sequence fN(φN) is also uniformly bounded in L∞([τ, T ], H2(Ωδ)).Using this fact and that fN(φN) → f(φ) a.e., we obtain using a weak version of thedominated convergence theorem that fN(φN) → f(φ) weakly in L2([τ, T ], H2(Ωδ)).Thus, we are allowed to pass to the limit in the equation corresponding to (5.83) forφN . We deduce from (5.83), that φ is a solution for:

A−1∂tφ(t)−∆φ(t) + f(φ(t))− λφ(t) = 0, in L2loc((τ, T )× Ωδ).

Moreover, since f(φ) and A−1∂tφ belong to L∞((τ, T ), L1(Ω)), we find that

∆φ ∈ L∞((τ, T ), L1(Ω)).

Having the control of ∇Dτφ, we deduce that φ ∈ L∞((τ, T ),W 2,1(Ω)), which yieldsthe existence of the trace (5.85).

122

5.4 Additional regularity results and separation from the singularities

Concerning the second equation from (5.58), we use Theorem 5.2.6 and we see that:

‖∂tψN‖L∞([τ,T ],L2(Γ)) + ‖ψN‖L∞([τ,T ],H2(Γ)) 6 c, (5.86)

where the constant c is independent of N . We have:

∂nφN = ∂tψN −∆ΓψN + g(ψN). (5.87)

Using (5.86), we deduce that ∂nφN ∈ L∞([τ, T ], L2(Γ)). Passing to the limit asN → +∞, we have the weak-star convergence in L∞([τ, T ], L2(Γ))

[∂nφ]ext := limN→+∞

∂nφN |Γ ∈ L∞([τ, T ], L2(Γ)), T > τ > 0, (5.88)

and∂tψ −∆Γψ + g(ψ) + [∂nφ]ext = 0, on Γ, T > τ > 0.

In order to verify that the variational solution (φ, ψ) satisfies equations (5.58) in theusual sense, there only remains to check that:

[∂nφ]int = [∂nφ]ext, for almost every (t, x) ∈ R+ × Γ. (5.89)

5.4 Additional regularity results and separation fromthe singularities

In this section, we formulate several sufficient conditions which ensure that everyvariational solution satisfies equation (5.58) in the usual sense. We have the followingresult which gives an additional regularity on φ close to the points where |φ(t, x)| < 1.

Proposition 5.4.1. Let the assumptions of Theorem 5.3.2 hold and let (φ, ψ) be avariational solution to (5.58). For any δ, T > 0, we set:

Ωδ(T ) = x ∈ Ω, |φ(T, x)| < 1− δ .

Then, φ ∈ H2(Ωδ(T )) and the following estimate holds:

‖φ‖H2(Ωδ(T )) 6 Qδ,T , (5.90)

where the positive constant Qδ,T depends on T and δ but is independent of the con-crete choice of the solution φ.

Proof:Since the solution φ(T, x) is Hölder continuous with respect to x, there exists asmooth nonnegative cut-off function θ(x) such that:

θ(x) ≡ 1, x ∈ Ωδ(T ),θ(x) ≡ 0, x ∈ Ω\Ωδ/2(T ),‖θ‖C2(R3(Ω)) 6 Kδ,T ,

(5.91)

where Kδ,T is independent of the concrete choice of the solution φ. Let φN(t, x)be a sequence of approximate solutions of problem (5.2) which converges to the

123


variational solution φ(t, x) as N → +∞. Then, since the convergence holds in thespace Cγ([0, T ]× Ω) for some γ > 0, we have:

|φN(T, x)| < 1− δ/4, x ∈ Ωδ/2(T ), (5.92)

for N large enough. Setting vN := θ(x)φN(T, x) and wN := θ(x)ψN(T, x), we have:

∆xφN − fN(φN(T ))− φN = h1(T ) := −φN + λφN + A−1∂tφN ,∆ΓψN − ψN − ∂nφN = h2(T ) := g0(ψN(T )) + ∂tψN .

(5.93)

Multiplying the first equation of (5.93) by θ, we find:

θ∆xφN(T )− θφN(T ) = θh1(T ) + θfN(φN(T ))

⇐⇒ ∆x(θφN(T ))− θφN(T ) = θh1(T ) + θfN(φN(T )) + 2∇xθ · ∇xφN(T )

+ φN(T )∆xθ

⇐⇒ ∆xvN − vN = θh1(T ) + θfN(φN(T )) + 2∇xθ · ∇xφN(T ) + φN(T )∆xθ.

(5.94)

Now, multiplying the second equation of (5.93) by θ, we obtain:

θ∆ΓψN(T )− θψN(T )− θ∂nφN(T ) = θh2(T )

⇐⇒ ∆ΓwN − wN − ∂nwN = θh2(T ) + 2∇Γθ · ∇ΓψN(T ) + ψN(T )∆Γθ

− ψN(T )∂nθ.

(5.95)

Thus, ϕN satisfies the following elliptic boundary value problem:

∆xvN − vN = h1(φN) := θh1(T ) + θfN(φN(T )) + 2∇xθ · ∇xφN(T ) + φN(T )∆xθ,

∆ΓwN − wN − ∂nwN = h2(ψN) := θh2(T ) + 2∇Γθ · ∇ΓψN(T ) + ψN(T )∆Γθ

− ψN(T )∂nθ.(5.96)

Using the estimates (5.33), (5.49), (5.91) and (5.92), we find:

‖h1(φN)‖L2(Ω) + ‖h2(ψN)‖L2(Γ) 6 Qδ,T ,

where the positive constant Qδ,T is independent of N and of the concrete choice ofthe solution φ. Applying an H2−regularity result to problem (5.96) (see [42]), wededuce that:

‖φN(T )‖H2(Ωδ(T )) 6 Qδ,T . (5.97)

By passing to the limit as N → +∞, we deduce that φ(T ) ∈ H2(Ωδ(T )) and that‖φ(T )‖H2(Ωδ(T )) 6 Qδ,T .

Lemma 5.4.2. Let (φ, ψ) a variational solution to (5.58). Assume, in addition,that we have:

|φ(t0, x0)| < 1, (5.98)

for some (t0, x0) ∈ (0,+∞)×Γ. Then, there exists a neighborhood (t0−ε, t0+ε)×Vof (t0, x0) in R× Γ such that:

[∂nφ]int(t, x) = [∂nφ]ext(t, x), ∀(t, x) ∈ (t0 − ε, t0 + ε)× V. (5.99)

124

5.4 Additional regularity results and separation from the singularities

In particular, if φ satisfies:

|φ(t, x)| < 1 for almost all (t, x) ∈ R+ × Γ, (5.100)

then, the equality [∂nφ]int = [∂nφ]ext holds almost everywhere in (0,+∞) × Γ and(φ, ψ) solves (5.58) in the usual sense.

Proof:We know that φ is Hölder continuous with respect to t and x. Thus there exists ε > 0such that |φ(t, x)| 6 1− ε holds for all (t, x) in a neighborhood (t0 − ε, t0 + ε)× Vεof (t0, x0) in (0,+∞) × Ω. Thanks to Proposition 5.4.1, the approximate solutionφN (converging to φ) satisfies:

‖φN‖L∞([t0−ε,t0+ε],H2(Vε)) 6 C,

where the positive constant C is independent of N . Then, we can assume thatφN → φ weakly-* in this space, which yields that ∂nφN |Γ → ∂nφ|Γ weakly in L2([t0−ε, t0 + ε] × V ) for some proper neighborhood V of x0. This convergence result,together with the definition (5.88), leads to equality (5.99) and relation (5.100) is aconsequence of (5.99).Thus, in order to prove that any variational solution φ is a solution in the usualsense, it is sufficient to verify that φ satisfies (5.100).

Corollary 5.4.3. Let the assumptions of Theorem 5.3.2 hold. We assume that:

lims→±1

F (s) = +∞. (5.101)

Then, for every variational solution φ of problem (5.58), relation (5.100) holds andthe potential F verifies:

F (φ(t)) ∈ L1(Γ) and ‖F (φ(t))‖L1(Γ) 6 CT , (5.102)

for almost all t > T > 0.

Proof:Let φN be a sequence of approximate solutions converging to the variational solutionφ. Applying estimate (6.4) in Miranville & Zelik [45], we obtain:

‖FN(φN)‖L1(Γ) 6 CT , t > T, (5.103)

where the constant CT is independent of N . Arguing as in the proof of Theorem5.3.3, we obtain that:

meas (t, x) ∈ [T, T + 1]× Γ, |φM(t, x)| > 1− 1/N 6 π(1/N),

where

π(x) =C

max |F (1− x)|, |F (x− 1)| .

Passing to the limit M,N → +∞, we deduce that:

meas (t, x) ∈ [T, T + 1]× Γ, |φ(t, x)| = 1 = 0.

125


Thus, condition (5.100) holds.Then, using the convergence φN → φ in Cγ([0, T ]×Ω), with γ > 0 and the alreadyproved statement 5.100, we conclude that FN(φN) → F (φ) almost everywhere inR

+ × Γ and therefore, thanks to the Fatou lemma,

‖F (φ)‖L1(Γ) 6 lim infN→+∞

‖FN(φN)‖L1(Γ) 6 CT .

Corollary 5.4.4. Let the assumptions of Theorem 5.3.2 hold. We assume that:

g(−1) + ε 6 0 6 g(1)− ε, (5.104)

for some ε > 0. Then, for every variational solution φ of problem (5.58), estimate(5.100) holds and

‖f(φ)‖L1([t,t+1]×Γ) 6 Cε,T , t > T > 0, (5.105)

where the constant Cε,T is independent of the concrete choice of the variationalsolution φ.

Proof:We consider the nonlinear elliptic-parabolic system:

∆φN(t)− fN(φN(t))− φN(t) = h1(t),φN |Γ = ψN ,∂tψN(t)−∆ΓψN(t) + ∂nφN(t) + g(ψN(t)) = 0.

(5.106)

Arguing as in Miranville & Zelik [45], we have:

g(s) · fN(s) >ε

2|fN(s)|+ Cε, s ∈ R, (5.107)

where the constant Cε depends on g and ε but is independent of N . To derive(5.105), we multiply (5.106) by fN(φN) and we use (5.107). We obtain:

d

dt

∫

Γ

FN(φN(t))ds+ (f ′N(φN(t))∇φN(t),∇φN(t))Ω

+ (f ′N(φN(t))∇ΓφN(t),∇ΓφN(t))Γ + 1/2‖fN(φN(t))‖2L2(Ω) + ε/2‖fN(φN(t))‖L1(Γ)

6 Cε(1 + ‖h1(t)‖2L2(Ω)).

(5.108)

The L2-norm of h1(t) is controlled thanks to (5.32) and we find:

‖h1(t)‖2L2(Ω) 6c(1 + ‖∂tφ(t)‖2H−1(Ω))

6c(1 + ‖∂tφ(0)‖2H−1(Ω) + ‖∂tψ(0)‖2L2(Γ))

<+∞.

126

5.5 Attractors and exponential attractors

Integrating (5.108) with respect to t and using the fact that f ′N > 0, we find:

∫

Γ

FN(φN(t+ 1))dσ+

∫ t+1

t

(‖∇φN(s)‖2L2(Ω) + ‖∇ΓφN(s)‖2L2(Γ)ds

+

∫ t+1

t

(‖fN(φN(s))‖2L2(Ω)) + ε/2‖fN(φN(s))‖L1(Γ))ds

6Cε,T +

∫

Γ

FN(φN(t))dσ.

(5.109)

We deduce from (5.109):

‖fN(φN)‖L1([t,t+1]×Γ) 6 2/ε(‖FN(φN(t))‖L1(Γ) + ‖FN(φN(t+ 1))‖L1(Γ)) + Cε,T

6 C ′ε,T .

Arguing as in Corollary 5.4.3, we finish the proof.


In this section, we study the asymptotic behavior of the system. We denote byΦw := H−1(Ω)× L2(Γ). The space Φw is endowed with the natural norm:

‖ϕ‖2Φw = ‖ϕ‖2H−1(Ω) + ‖ϕ‖2L2(Γ), for all ϕ ∈ Φw.

We endow the phase space Φ with the metric of Φw.We have the following result:

Corollary 5.5.1. Under the assumptions of Theorem 5.3.3, equation (5.58) gener-ates a solution semigroup S(t) : Φ → Φ, where S(t)(φ0, ψ0) := (φ(t), ψ(t)) is theunique variational solution of problem (5.58) departing from (φ0, ψ0). Furthermore,we have the following Lipschitz continuity property in the Φw− topology:

‖S(t)(φ10, ψ

10)− S(t)(φ2

0, ψ20)‖2Φw

+

∫ t+1

t

‖S(s)(φ10, ψ

10)− S(s)(φ2

0, ψ20)‖2H1(Ω)×H1(Γ)ds

6CeKt‖(φ10 − φ2

0, ψ10 − ψ2

0))‖2Φw ,

(5.110)

for all (φ10, ψ

10), (φ

20, ψ

20) ∈ Φ.

This corollary is a direct consequence of Proposition 5.2.7.The following proposition gives the existence of the global attractor A for this semi-group. We recall that, by definition, a set A ⊂ Φ is the global attractor for thesemigroup S(t) if the following properties are satisfied:

1. It is a compact subset of Φ;

2. It is strictly invariant, i.e., S(t)A = A, ∀t > 0;

127


3. It attracts Φ as t → +∞, i.e., for every neighborhood O(A) of A in Φ, thereexists a time T = T (O) such that:

S(t)X ⊂ O(A), t > T.

Proposition 5.5.2. The semigroup S(t) associated with the variational solutionsof problem (5.58) possesses the global attractor A which is bounded in the spaceCα(Ω)× Cα(Γ) for some positive constant α < 1/4.

Proof:The semigroup S(t) is dissipative. Indeed, thanks to estimate (5.27) there existsR0 > 0 such that the ball BH1

(R0) centered on zero with radius R0 in H1(Ω)×H1(Γ)is absorbing in Φ and compact in the topology of Φw. In particular, there exists atime t0 > 1 such that S(t)BH1

(R0) ⊂ BH1(R0), for any t > t0. As a consequence,

the set:B0 := ∪t>t0S(t)BH1

(R0)Φw

(5.111)

is absorbing and positively invariant. Thus the existence of the global attractor Afollows from a proper abstract attractor’s existence theorem (see Temam [54]).In the following theorem, we prove the existence of an exponential attractor whichby definition contains the global attractor and has finite fractal dimension. To dothis, we first recall the definition of the exponential attractor where A is the globalattractor for the semigroup S(t)t>0:

Definition 5.5.3. A set M ⊂ Φ is called an exponential attractor for the semigroupS(t)t>0 if :

1. M is compact in Φ;

2. S(t)M ⊂ M, ∀t > 0;

3. M has finite fractal dimension, dF (M) <∞;

4. M attracts Φ exponentially fast as t→ +∞, i.e.,

distE(S(t)Φ,M) 6 Ce−ct, ∀t > 0,

where the pseudo-distance dist is the standard Hausdorff pseudo-distance be-tween two sets, defined by distE(A,B) = supa∈A infb∈B ‖a− b‖E.

Theorem 5.5.4. The semigroup S(t) possesses an exponential attractor M whichis bounded in Cα(Ω)× Cα(Γ), α < 1/4.

Proof:There exists a positive constant R = R(R0) such that:

‖φ(t)‖Cα([t,t+1]×Ω) + ‖φ(t)‖H2(Γ) + ‖∂tφ(t)‖H−1(Ω) + ‖∂tφ(t)‖L2(Γ)

+ ‖f(φ(t))‖L1(Ω) + ‖∂tφ(t)‖L2([t,t+1],H1(Ω)) + ‖∂tφ(t)‖L2([t,t+1],H1(Γ)) 6 R,

(5.112)

128


for any initial datum in B0 where B0 is the set defined by (5.111). In particular,for every point (φ, ψ) ∈ B0, there holds φ|Γ = ψ. We consider an arbitrary smallε−ball B(ε, φ0; Φ

w) in the space B0 endowed with the metric of Φw and centered onφ0, where 0 < ε 6 ε0 ≪ 1, with the parameter ε0 that will be fixed below. Let alsoφ0(t), t > 0, be the solution starting from φ0. We introduce the sets:

Ωδ(φ0) := x ∈ Ω, |φ0(x)| 6 1− δ ,

Ωδ(φ0) := x ∈ Ω, |φ0(x)| > 1− δ ,where δ is a sufficiently small positive number. Then, thanks to the Hölder continuityof φ0 with respect to x, we have for all x1 ∈ ∂Ωδ1(φ0), x2 ∈ ∂Ωδ2(φ0):

0 < |δ1 − δ2| 6 |φ0(x1)− φ0(x2)| 6 c|x1 − x2|α.

Thus, there is a strict separation between ∂Ωδ1(φ0) and ∂Ωδ2(φ0) for any δ1 6= δ2,i.e.

d(∂Ωδ1(φ0), ∂Ωδ2(φ0)) > Cδ1,δ2 > 0, δ1 6= δ2, (5.113)

where the constant Cδ1,δ2 depends on δ1, δ2.We note that, since φ0(t) is uniformly Hölder continuous with respect to t and x,there exists T = T (δ) such that:

|φ0(t)| 6 1− δ/2, x ∈ Ωδ(φ0), t ∈ [0, T ],

|φ0(t)| > 1− 3δ, x ∈ Ω2δ(φ0), t ∈ [0, T ].

Furthermore, using again the uniform Hölder continuity, we have:

‖φ1(t)− φ2(t)‖C(Ω) 6C‖φ1(t)− φ2(t)‖κΦw‖φ1(t)− φ2(t)‖1−κCα(Ω)

6CT‖φ1(0)− φ2(0)‖κΦw‖φ1(t)− φ2(t)‖1−κCα(Ω)

6CT εκ,

for every φ1(0), φ2(0) in B(ε, φ0,Φw). We can fix ε0 = ε0(δ) such that:

|φ(t)| 6 1− δ/4, x ∈ Ωδ(φ0), t ∈ [0, T ],

|φ(t)| > 1− 4δ, x ∈ Ω2δ(φ0), t ∈ [0, T ],(5.114)

for all trajectories φ(t) starting from the ball B(ε, φ0,Φw), ε 6 ε0.

Due to (5.113), there exists a smooth cut-off function θ ∈ C∞(R3, [0, 1]) such that:

θ(x) =

0, if x ∈ Ωδ(φ0),1, if x ∈ Ω2δ(φ0).

Furthermore, θ satisfies the additional condition:

‖θ‖Ck(R3) 6 Ck,

where k ∈ IN is arbitrary and the constant Ck depends on δ, but is independent ofthe choice of φ0 ∈ B0. The second estimate of (5.114) yields:

f ′(φ(t, x)) > Λ(δ), x ∈ Ω2δ(φ0), t ∈ [0, T ], (5.115)

129


for all trajectories φ(t) starting from the ball B(ε, φ0,Φw), where

Λ(δ) := min f ′(1− 4δ), f ′(−1 + 4δ) . (5.116)

Indeed, for every x ∈ Ω2δ(φ0), we have that |φ(t)| > 1 − 4δ. Then, using the factthat sgn s · f ′′(s) > 0, we obtain:

×if φ(t) > 1− 4δ ⇒ f ′′(φ(t)) > 0 ⇒ f ′(φ(t)) > f ′(1− 4δ),×if φ(t) 6 −1 + 4δ ⇒ f ′′(φ(t)) 6 0 ⇒ f ′(φ(t)) > f ′(−1 + 4δ),

(5.117)

which yields (5.115). Since f ′(s) −−−→s→±1

+∞, then, Λ(δ) −−→δ→0

+∞ and we can fix

δ > 0 close enough to zero such that Λ(δ) is arbitrarily large. The next lemma givessome kind of smoothing property for the difference of two solutions.

Lemma 5.5.5. Let the above assumptions hold. Then, there exists δ > 0 such that:

‖φ1(T )− φ2(T )‖2Φw 6 e−βT‖φ1(0)− φ2(0)‖2Φw + C

∫ T

0

‖θ (φ1(s)− φ2(s)) ‖2L2(Ω)ds,

(5.118)

where the positive constants β and C are independent of φ1(0), φ2(0) ∈ B(ε, φ0,Φw)

and φ0 ∈ B0.

Proof:We set φ(t) = φ1(t)− φ2(t) . Then, φ solves the following problem:

A−1∂tφ = ∆φ− l(t)φ+ λφ, in Ω,∂n(∆φ− l(t)φ+ λφ)|Γ = 0,∂tφ−∆Γφ+ ∂nφ+ φ(t) +m(t)φ = 0, on Γ,

(5.119)

where

l(t) :=

∫ t

0

f ′(sφ1(t) + (1− s)φ2(t))ds and m(t) :=

∫ t

0

g′0(sφ1(t) + (1− s)φ2(t))ds.

Multiplying (5.119) by φ(t) and integrating over Ω, we obtain:

1

2

d

dt(‖φ(t)‖2H−1(Ω) + ‖φ(t)‖2L2(Γ)) + ‖∇φ(t)‖2L2(Ω)

+‖∇Γφ(t)‖2L2(Γ) + ‖φ(t)‖2L2(Γ) + (l(t)φ(t), φ(t))Ω

= λ‖φ(t)‖2L2(Ω) − (m(t)φ(t), φ(t))Γ

6 λ‖φ(t)‖2L2(Ω) + C0‖φ(t)‖2L2(Γ),

(5.120)

where C0 = ‖g′‖C([−1,1]).Due to (5.115), we have:∫

Ω

l(t, x)|φ(t, x)|2dx >

∫

Ω2δ

l(t, x)|φ(t, x)|2dx >Λ‖φ‖2L2(Ω2δ)

=Λ‖φ‖2L2(Ω) − Λ‖φ‖2L2(Ω2δ)

>Λ‖φ‖2L2(Ω) − Λ‖θφ‖2L2(Ω).

(5.121)

130


Thus, we obtain:

d

dt‖φ(t)‖2Φw + 2‖∇φ(t)‖2L2(Ω)+2‖φ(t)‖2H1(Γ) + 2(Λ− λ)‖φ‖2L2(Ω)

6 2C0‖φ(t)‖2L2(Γ) + 2Λ‖θφ‖2L2(Ω).(5.122)

For the first term in the right hand-side of (5.122), we use the following traceinequality:

2‖φ‖2L2(Γ) 6 2C‖φ‖H1(Ω)‖φ‖L2(Ω) 6C√Λ− λ

‖φ‖2H1(Ω) + C√Λ− λ‖φ‖2L2(Ω), (5.123)

and the fact that for some ω ∈ (0, 2), we have:

2‖∇φ(t)‖2L2(Ω) + 2‖φ(t)‖2H1(Γ) > ω‖φ(t)‖2H1(Ω) + ω‖φ(t)‖2H1(Γ). (5.124)

Thus, fixing δ in such a way that C0C 6 ω

√Λ− λ

2and using (5.123) and (5.124)

in (5.122), we find:

d

dt‖φ(t)‖2Φw+ω‖φ(t)‖2H1(Ω) + ω‖φ(t)‖2H1(Γ) + 2(Λ− λ)‖φ(t)‖2L2(Ω)

6C0C√Λ− λ

‖φ(t)‖2H1(Ω) + C0C√Λ− λ‖φ(t)‖2L2(Ω) + 2Λ‖θφ(t)‖2L2(Ω)

6 ω/2‖φ(t)‖2H1(Ω) + ω/2(Λ− λ)‖φ(t)‖2L2(Ω) + 2Λ‖θφ(t)‖2L2(Ω).

(5.125)

Taking β′ > 0 small enough, estimate (5.125) leads to:

d

dt‖φ(t)‖2Φw + ω/2‖φ(t)‖2H1(Ω) + β′

(

‖φ(t)‖2H1(Γ) + ‖φ(t)‖2L2(Ω)

)

6 2Λ‖θφ(t)‖2L2(Ω).

(5.126)

Using the inequalities ‖φ‖H−1(Ω) 6 C‖φ‖L2(Ω) and ‖φ‖L2(Γ) 6 C‖φ‖H1(Γ), we end upwith:

d

dt‖φ(t)‖2Φw + ω/2‖φ(t)‖2H1(Ω) + β

(

‖φ(t)‖2L2(Γ) + ‖φ(t)‖2H−1(Ω)

)

6 2Λ‖θφ(t)‖2L2(Ω),

(5.127)for some positive constant β. Applying the Gronwall lemma, we obtain:

‖φ(T )‖2Φw 6‖φ(0)‖2Φwe−βT + C

∫ T

0

e−β(T−s)‖θφ(s)‖2L2(Ω)ds

6‖φ(0)‖2Φwe−βT + C

∫ T

0

‖θφ(s)‖2L2(Ω)ds,

and estimate (5.118) is proven.

Lemma 5.5.6. Let the nonlinearities f and g satisfy the assumptions of Section 2.Then, there exists positive constants C and K independent of φi(0) in B(ε, φ0,Φ

w),i = 1, 2, and φ0 in B0 such that the following estimate holds:

‖∂t(θ(φ1 − φ2))‖L2([0,T ],H−3(Ω)) + ‖θ(φ1 − φ2)‖L2([0,T ],H1(Ω))

6 CeKT‖φ1(0)− φ2(0)‖Φw .(5.128)

131


Proof:Due to (5.110) and the fact that ∇xθ is uniformly bounded, we have:

‖θ(φ1 − φ2)‖2L2([0,T ],H1(Ω))

=‖θ(φ1 − φ2)‖2L2([0,T ],L2(Ω)) + ‖∇x(θ(φ1 − φ2))‖2L2([0,T ],L2(Ω))

6‖θ(φ1 − φ2)‖2L2([0,T ],L2(Ω)) + ‖∇xθ · (φ1 − φ2))‖2L2([0,T ],L2(Ω))

+ ‖θ · ∇x(φ1 − φ2))‖2L2([0,T ],L2(Ω))

6CeT‖φ1(0)− φ2(0)‖2Φw .

In order to find estimates on the time derivative, we first recall that ∂tφ verifies:

∂tφ = (−∆x + I)(∆xφ− l(t)φ+ λφ),

where φ = φ1 − φ2. Testing this equation by θϕ for any test function ϕ ∈ C∞0 (Ω),

using that supp θ ⊂ Ωδ(φ0), we obtain:

(∂t(θφ(t)), ϕ)Ω =− (∆xφ(t)− l(t)φ(t) + λφ(t),∆x(θϕ(t))− θϕ(t))Ω

=(∇xφ(t),∇x∆x(θϕ(t)))Ω − (l(t)φ(t), θϕ(t))Ω + (λφ(t), θϕ(t))Ω

−(∇xφ(t),∇xθϕ(t))Ω + (l(t)φ(t),∆x(θϕ(t)))Ω − (λφ(t),∆x(θϕ(t)))Ω

6C‖φ‖H1(Ω)‖ϕ‖H3(Ω).

(5.129)

Estimate (5.129) yields that ‖∂t(θ(φ1(t) − φ2(t)))‖H−3(Ω) 6 C‖φ‖H1(Ω) and using(5.110), we obtain (5.128).To conclude the proof of the theorem, we introduce the functional spaces:

H1 := L2([0, T ], H1(Ω)) ∩H1([0, T ], H−3(Ω)),

H := L2([0, T ], L2(Ω)).

We have that H1 is compactly embedded into H. For every φ0 ∈ B0, we define thefollowing operator:

Kφ0: B(ε, φ0,Φ

w) 7−→ H1

φ(0) 7−→ Kφ0φ(0) := θφ(·),

where φ(t) is the variational solution departing from φ(0). Due to Lemma 5.5.6, themap Kφ0

is uniformly Lipschitz continuous:

‖Kφ0(φ1 − φ2)‖H1

6 L‖φ1 − φ2‖Φw , φ1, φ1 ∈ B(ε, φ0,Φw), ε 6 ε0 (5.130)

and thanks to Lemma 5.118, we have:

‖S(T )φ1 − S(T )φ2‖Φw 6 γ‖φ1 − φ2‖Φw + c‖Kφ0(φ1 − φ2)‖H, (5.131)

where γ < 0, c > 0 are independent of φ0 ∈ B0, ε 6 ε0 and φ1, φ1 ∈ B(ε, φ0; Φw).

Arguing as in Miranville & Zelik [45], we have that inequalities (5.130) and (5.131),together with the compactness of the embedding H1 ⊂ H, guarantee the existenceof an exponential attractor Md ⊂ B0 for the discrete semigroup S(nT ) acting on

132


the phase space B0. Since the semigroup S(t) is uniformly Hölder continuous withrespect to time and space in [0, T ]× B0, we deduce the existence of an exponentialattractor M for the continuous semigroup S(t) on B0 which can be obtained by thestandard formula

M := ∪t∈[0,T ]Md.

Acknowledgements. The author wishes to thank Alain Miranville and MadalinaPetcu for many stimulating discussions and useful comments on the subject of thepaper.

133


134

Bibliographie

[1] A. Babin and B. Nicolaenko. Exponential attractors of reaction-diffusion sys-tems in an unbounded domain. J. Dynam. Differential Equations, 7(4) :567–590,1995.

[2] A. V. Babin and M. I. Vishik. Attractors of partial differential evolution equa-tions in an unbounded domain. Proc. Roy. Soc. Edinburgh Sect. A, 116(3-4) :221–243, 1990.

[3] A. Bonfoh and A. Miranville. On Cahn-Hilliard-Gurtin equations. NonlinearAnal., 47(5) :3455–3466, 2001.

[4] M. Carrive, A. Miranville, A. Piétrus, and J. M. Rakotoson. Weakly coupleddynamical systems and applications. Asymptot. Anal., 30(2) :161–185, 2002.

[5] L. Cherfils, S. Gatti, and A. Miranville. Existence of global solutions to theCaginalp phase-field system with dynamic boundary conditions and singularpotentials. J. Math. Anal. Appl., 343(1) :557–566, 2008.

[6] L. Cherfils, S. Gatti, and A. Miranville. Long time behavior of the Caginalpsystem with singular potentials and dynamic boundary conditions. Commun.Pure Appl. Anal., 11(6) :2261–2290, 2012.

[7] L. Cherfils, M. Petcu, and M. Pierre. A numerical analysis of the Cahn-Hilliard equation with dynamic boundary conditions. Discrete Contin. Dyn.Syst., 27(4) :1511–1533, 2010.

[8] R. Chill and M. A. Jendoubi. Convergence to steady states in asymptoticallyautonomous semilinear evolution equations. Nonlinear Anal., 53(7-8) :1017–1039, 2003.

[9] A. Eden, C. Foias, B. Nicolaenko, and R. Temam. Exponential attractors fordissipative evolution equations, volume 37 of RAM : Research in Applied Ma-thematics. Masson, Paris, 1994.

[10] M. Efendiev, A. Miranville, and S. Zelik. Exponential attractors for a non-linear reaction-diffusion system in R

3. C. R. Acad. Sci. Paris Sér. I Math.,330(8) :713–718, 2000.

[11] M. Efendiev, A. Miranville, and S. Zelik. Exponential attractors for a singularlyperturbed Cahn-Hilliard system. Math. Nachr., 272 :11–31, 2004.

135

BIBLIOGRAPHIE

[12] C. M. Elliott, D. A. French, and F. A. Milner. A second order splitting methodfor the Cahn-Hilliard equation. Numer. Math., 54(5) :575–590, 1989.

[13] C.M. Elliott. The Cahn-Hilliard model for the kinetics of phase separation. InMathematical models for phase change problems (Óbidos, 1988), volume 88 ofInternat. Ser. Numer. Math., pages 35–73. Birkhäuser, Basel, 1989.

[14] C.M. Elliott and S. Larsson. Error estimates with smooth and nonsmoothdata for a finite element method for the Cahn-Hilliard equation. Math. Comp.,58(198) :603–630, S33–S36, 1992.

[15] A. Ern and J.L. Guermond. Éléments finis : théorie, applications, mise enœuvre, volume 36 of Mathématiques & Applications (Berlin) [Mathematics &Applications]. Springer-Verlag, Berlin, 2002.

[16] C. Gal. Well-posedness and long time behavior of the non-isothermal viscousCahn-Hilliard equation with dynamic boundary conditions. Dyn. Partial Differ.Equ., 5(1) :39–67, 2008.

[17] C. G. Gal. Global well-posedness for the non-isothermal Cahn-Hilliard equationwith dynamic boundary conditions. Adv. Differential Equations, 12(11) :1241–1274, 2007.

[18] C. G. Gal and A. Miranville. Robust exponential attractors and convergence toequilibria for non-isothermal Cahn-Hilliard equations with dynamic boundaryconditions. Discrete Contin. Dyn. Syst. Ser. S, 2(1) :113–147, 2009.

[19] G. Gilardi, A. Miranville, and G. Schimperna. On the Cahn-Hilliard equationwith irregular potentials and dynamic boundary conditions. Commun. PureAppl. Anal., 8(3) :881–912, 2009.

[20] G. Gilardi, A. Miranville, and G. Schimperna. Long time behavior of the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions.Chin. Ann. Math. Ser. B, 31(5) :679–712, 2010.

[21] C. Giorgi, M. Grasselli, and V. Pata. Uniform attractors for a phase-field modelwith memory and quadratic nonlinearity. Indiana Univ. Math. J., 48(4) :1395–1445, 1999.

[22] G. R. Goldstein and A. Miranville. A Cahn-Hilliard-Gurtin model with dynamicboundary conditions. Discrete Contin. Dyn. Syst. Ser. S, 6(2) :387–400, 2013.

[23] G. R. Goldstein, A. Miranville, and G. Schimperna. A Cahn-Hilliard model ina domain with non-permeable walls. Phys. D, 240(8) :754–766, 2011.

[24] Alain Haraux and Mohamed Ali Jendoubi. On a second order dissipative ODEin Hilbert space with an integrable source term. Acta Math. Sci. Ser. B Engl.Ed., 32(1) :155–163, 2012.

[25] S. Injrou and M. Pierre. Stable discretizations of the Cahn-Hilliard-Gurtinequations. Discrete Contin. Dyn. Syst., 22(4) :1065–1080, 2008.

136

BIBLIOGRAPHIE

[26] S. Injrou and M. Pierre. Error estimates for a finite element discretization of theCahn-Hilliard-Gurtin equations. Adv. Differential Equations, 15(11-12) :1161–1192, 2010.

[27] H Israel. Long time behavior of an Allen-Cahn type equation with a singularpotential and dynamic boundary conditions. J. Appl. Anal. Comput., 2(1) :29–56, 2012.

[28] H. Israel. Well-posedness and long time behavior of an Allen-Cahn type equa-tion. Commun. Pure Appl. Anal., 12(6) :2811–2827, 2013.

[29] H. Israel, A. Miranville, and M. Petcu. Well-posedness and long time behaviorof a perturbed Cahn-Hilliard system with regular potentials. J. AsymptoticAnal., to appear, 84 :147–179, 2013.

[30] G. Karali and M.A. Katsoulakis. The role of multiple microscopic mechanismsin cluster interface evolution. J. Differential Equations, 235(2) :418–438, 2007.

[31] G. Karali and T. Ricciardi. On the convergence of a fourth order evolutionequation to the Allen-Cahn equation. Nonlinear Anal., 72(11) :4271–4281, 2010.

[32] M. A. Katsoulakis and D. G. Vlachos. From microscopic interactions to ma-croscopic laws of cluster evolution. Phys. Rev. Lett., 84 :1511–1514, 2000.

[33] R. Kenzler, F. Eurich, P. Maass, B. Rinn, J. Schropp, E. Bohl, and W. Dieterich.Phase separation in confined geometries : solving the Cahn-Hilliard equationwith generic boundary conditions. Comput. Phys. Comm., 133(2-3) :139–157,2001.

[34] Mihály Kovács, Stig Larsson, and Ali Mesforush. Finite element approximationof the Cahn-Hilliard-Cook equation. SIAM J. Numer. Anal., 49(6) :2407–2429,2011.

[35] Stig Larsson and Ali Mesforush. Finite-element approximation of the linearizedCahn-Hilliard-Cook equation. IMA J. Numer. Anal., 31(4) :1315–1333, 2011.

[36] B. Merlet and M. Pierre. Convergence to equilibrium for the backward Eulerscheme and applications. Commun. Pure Appl. Anal., 9(3) :685–702, 2010.

[37] A. S. Mikhailov, M. Hildebrand, and G. Ertl. Nonequilibrium nanostructures incondensed reactive systems. In Coherent structures in complex systems (Sitges,2000), volume 567 of Lecture Notes in Phys., pages 252–269. Springer, Berlin,2001.

[38] A. Miranville. Some generalizations of the Cahn-Hilliard equation. Asymptot.Anal., 22(3-4) :235–259, 2000.

[39] A. Miranville. Long-time behavior of some models of Cahn-Hilliard equationsin deformable continua. Nonlinear Anal. Real World Appl., 2(3) :273–304, 2001.

137

BIBLIOGRAPHIE

[40] A. Miranville, A. Piétrus, and J. M. Rakotoson. Dynamical aspect of a genera-lized Cahn-Hilliard equation based on a microforce balance. Asymptot. Anal.,16(3-4) :315–345, 1998.

[41] A. Miranville and S. Zelik. Robust exponential attractors for Cahn-Hilliard typeequations with singular potentials. Math. Methods Appl. Sci., 27(5) :545–582,2004.

[42] A. Miranville and S. Zelik. Exponential attractors for the Cahn-Hilliard equa-tion with dynamic boundary conditions. Math. Methods Appl. Sci., 28(6) :709–735, 2005.

[43] A. Miranville and S. Zelik. Exponential attractors for the Cahn-Hilliard equa-tion with dynamic boundary conditions. Math. Methods Appl. Sci., 28(6) :709–735, 2005.

[44] A. Miranville and S. Zelik. Attractors for dissipative partial differential equa-tions in bounded and unbounded domains. In Handbook of differential equa-tions : evolutionary equations. Vol. IV, Handb. Differ. Equ., pages 103–200.Elsevier/North-Holland, Amsterdam, 2008.

[45] A. Miranville and S. Zelik. The Cahn-Hilliard equation with singular potentialsand dynamic boundary conditions. Discrete Contin. Dyn. Syst., 28(1) :275–310,2010.

[46] Alain Miranville and Arnaud Rougirel. Local and asymptotic analysis of theflow generated by the Cahn-Hilliard-Gurtin equations. Z. Angew. Math. Phys.,57(2) :244–268, 2006.

[47] A. Novick-Cohen. The Cahn-Hilliard equation. In Handbook of differentialequations : evolutionary equations. Vol. IV, Handb. Differ. Equ., pages 201–228. Elsevier/North-Holland, Amsterdam, 2008.

[48] J. Prüss, R. Racke, and S. Zheng. Maximal regularity and asymptotic behaviorof solutions for the Cahn-Hilliard equation with dynamic boundary conditions.Ann. Mat. Pura Appl. (4), 185(4) :627–648, 2006.

[49] R. Racke and S. Zheng. The Cahn-Hilliard equation with dynamic boundaryconditions. Adv. Differential Equations, 8(1) :83–110, 2003.

[50] J. C. Robinson. Infinite-dimensional dynamical systems. Cambridge Textsin Applied Mathematics. Cambridge University Press, Cambridge, 2001. Anintroduction to dissipative parabolic PDEs and the theory of global attractors.

[51] W. Rudin. Functional analysis. International Series in Pure and Applied Ma-thematics. McGraw-Hill Inc., New York, second edition, 1991.

[52] Batoul Saoud. Pullback exponential attractors for the viscous Cahn-Hilliardequation in bounded domains. J. Appl. Anal. Comput., 2(1) :73–90, 2012.

138

BIBLIOGRAPHIE

[53] J. Shen and X. Yang. Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Discrete Contin. Dyn. Syst., 28(4) :1669–1691, 2010.

[54] R. Temam. Infinite-dimensional dynamical systems in mechanics and physics,volume 68 of Applied Mathematical Sciences. Springer-Verlag, New York, secondedition, 1997.

[55] V. Thomée. Galerkin finite element methods for parabolic problems, volume 25of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, se-cond edition, 2006.

[56] H. Wu and S. Zheng. Convergence to equilibrium for the Cahn-Hilliard equationwith dynamic boundary conditions. J. Differential Equations, 204(2) :511–531,2004.

[57] J. Zhang and Q. Du. Numerical studies of discrete approximations to the Allen-Cahn equation in the sharp interface limit. SIAM J. Sci. Comput., 31(4) :3042–3063, 2009.

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Comportement asymptotique de modèles en séparation de phase

Résumé : Dans cette thèse, on étudie l’existence, l’unicité et la régularité des so-lutions d’équation de type Cahn-Hilliard ainsi que son comportement asymptotiqueen termes d’existence de l’attracteur global et d’un attracteur exponentiel. Cetteéquation est considérée dans un domaine borné et régulier pour différents types denonlinéarités et de conditions au bord.D’abord, on étudie l’équation avec des conditions de type Dirichlet sur le bord etune nonlinéarité régulière. Après, on considère une perturbation du problème et ondémontre l’existence d’une famille robuste d’attracteurs exponentiels lorsque ε tendvers 0.Ensuite, on étudie l’équation avec des conditions dynamiques sur le bord. On consi-dère tout d’abord une nonlinéarité régulière et on donne une étude théorique etnumérique. Après, on illustre ces résultats par des simulations numériques en di-mension deux d’espace qui permettent d’étudier l’influence des différents paramètres.On termine par une étude du modèle considéré avec une nonlinéarité singulière quel’on approche par des fonctions régulières et on introduit une notion de solutionappropriée.

Mot-clefs : Equation de Cahn-Hilliard, comportement asymtotique des solutions,attracteur global, attracteur exponentiel, analyse numérique.

Asymptotic Behaviour of Some Phase Separation Models

Abstract : This thesis is devoted to the study of the existence, uniqueness andregularity of solutions for a Cahn-Hilliard type equation, as well as the asymptoticbehavior in terms of existence of the global attractor and of an exponential attrac-tor. This equation is considered in a bounded and smooth domain under variousassumptions on the nonlinear terms and with different boundary conditions.We start by studying the equation with Dirichlet boundary conditions and a regularnonlinearity. Then, we consider a perturbation of the problem and we prove theexistence of a robust family of exponential attractors as ε tends to 0.For the equation endowed with dynamic boundary conditions, we first consider aregular nonlinearity and we treat the theoretical and numerical analysis. Then, weillustrate the results by numerical simulations in two space dimension which allow usto study the influence of different parameters. Finally, we treat the problem consi-dered with a singular nonlinearity which is approximated by regular functions andwe give a suitable notion of solutions.

Keywords : Cahn-Hilliard Equation, well posedness, dissipativity, asymptotic be-havior of solutions, global attractor, exponential attractor, numerical analysis.

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Comportement asymptotique de modèles