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Textos de
Metodos Matematicos
26An introduction to nonlinear
Schrodinger equations
Thierry Cazenave
Universidade Federal do Rio de Janeiro
Centro de Ciencias Matematicas e da Natureza
INSTITUTO DE MATEMATICA
An introduction to nonlinear Schrodinger equations
by Thierry Cazenave
Analyse Numerique
Universite Pierre et Marie Curie
4, place Jussieu
75252 Paris Cedex 05 France
Third Edition
Instituto de Matematica–UFRJ
Rio de Janeiro, RJ
1996
This monograph contains all the material taht was treated in a course of lectures
given by Dr. Cazenave at the Instituto de Matematica da Universidade Federal do Rio de
Janeiro (IM-UFRJ) and Instituto de Matematica Pura e Aplicada (IMPA) in August and
September of 1989. We are grateful to him for that profitable course, and it is a pleasure
to see it now enriching our collection of “Textos de Metodos Matematicos”.
Rio de Janeiro, December 1989.
Carlos Isnard IMPA
Jose da Silva Ferreira IM
Rafael Iorio IMPA
Rolci Cipolatti IM
Table of contents
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2. Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1. Functional analysis . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2. Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3. Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4. Elliptic equations . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.5. Semigroups of linear operators . . . . . . . . . . . . . . . . . . . . 28
3. The linear Schrodinger equation . . . . . . . . . . . . . . . . . . . . . . 32
3.1. Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2. Dispersive properties in Rn . . . . . . . . . . . . . . . . . . . . . 33
3.3. Smoothing effect in Rn . . . . . . . . . . . . . . . . . . . . . . . 41
3.4. Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4. The local Cauchy problem . . . . . . . . . . . . . . . . . . . . . . . . 48
4.1. An abstract nonlinear Schrodinger equation . . . . . . . . . . . . . . 48
4.2. The main result . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.3. The nonlinear Schrodinger equation in Rn . . . . . . . . . . . . . . . 65
4.4. The nonlinear Schrodinger equation in one dimension . . . . . . . . . . 68
4.5. The nonlinear Schrodinger equation in two dimensions . . . . . . . . . 70
4.6. Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5. Regularity and smoothing effect . . . . . . . . . . . . . . . . . . . . . . 77
5.1. A general result . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.2. H2 regularity in Rn . . . . . . . . . . . . . . . . . . . . . . . . 79
5.3. W 1,p smoothing effect in Rn . . . . . . . . . . . . . . . . . . . . . 84
5.4. The C∞ smoothing effect in Rn . . . . . . . . . . . . . . . . . . . 87
5.5. Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6. Global existence and blow up . . . . . . . . . . . . . . . . . . . . . . . 94
6.1. Energy estimates and global existence . . . . . . . . . . . . . . . . . 94
1
6.2. Global existence for small data . . . . . . . . . . . . . . . . . . . . 99
6.3. L2 solutions in Rn . . . . . . . . . . . . . . . . . . . . . . . . 102
6.4. Blow up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.5. Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7. Asymptotic behavior in the repulsive case . . . . . . . . . . . . . . . . . 118
7.1. The pseudo-conformal conservation law . . . . . . . . . . . . . . . 118
7.2. Decay of solutions in the weighted L2 space . . . . . . . . . . . . . 121
7.3. Scattering theory in the weighted L2 space . . . . . . . . . . . . . . 127
7.4. Morawetz’ estimate . . . . . . . . . . . . . . . . . . . . . . . . 132
7.5. Decay of solutions in the energy space . . . . . . . . . . . . . . . 139
7.6. Scattering theory in the energy space . . . . . . . . . . . . . . . . 146
7.7. Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
8. Stability of bound states in the attractive case . . . . . . . . . . . . . . . 157
8.1. Nonlinear bound states . . . . . . . . . . . . . . . . . . . . . . 157
8.2. An instability result . . . . . . . . . . . . . . . . . . . . . . . 177
8.3. A stability result . . . . . . . . . . . . . . . . . . . . . . . . . 183
8.4. Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
9. Further results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
9.1. The nonlinear Schrodinger equation with a magnetic field . . . . . . . 196
9.2. The nonlinear Schrodinger equation with a quadratic potential . . . . . 202
9.3. The logarithmic Schrodinger equation . . . . . . . . . . . . . . . . 207
9.4. Existence of weak solutions for large nonlinearities . . . . . . . . . . 217
9.5. Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
Index of subjects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
2
Notation.
Re(z) the real part of the complex number z;
Im(z) the imaginary part of the complex number z;
1E the function defined by 1E(x) = 1 if x ∈ E and 1E(x) = 0 if x 6∈ E;
E the closure of the subset E of the topological space X;
C(E,F ) the space of continuous functions from the topological space E to the topological
space F ;
Cb(E,F ) the Banach space of continuous, bounded functions from the topological space
E to the Banach space F , equipped with the topology of uniform convergence;
Cc(E,F ) the space of continuous functions E → F compactly supported in E.
L(E,F ) the Banach space of linear, continuous operators from the Banach space E to
the Banach space F , equipped with the norm topology;
L(E) the space L(E,E);
X ′ the topological dual of the space X;
X → Y if X ⊂ Y with continuous injection;
Ω an open subset of Rn;
Ω the closure of Ω in Rn;
∂Ω the boundary of Ω, that is ∂Ω = Ω \ Ω;
ω ⊂⊂ Ω if ω ⊂ Ω and ω is compact;
∂tu = ut =∂u
∂t=du
dt;
∂iu = uxi=
∂u
∂xi;
∂ru = ur =∂u
∂r=
1
rx · ∇u, where r = |x|;
Dα =∂α1
∂xα11
· · · ∂αn
∂xαnn
;
∇u = (∂1u, · · · , ∂nu);
3
4 =
n∑
i=1
∂2
∂x2i
;
F the Fourier transform in Rn, defined by* Fu(ξ) =
∫
Rn
e−2πix·ξu(x) dx;
F = F−1 given by Fv(x) =
∫
Rn
e2πiξ·xv(ξ) dξ;
u = Fu;
Cc(Ω) = Cc(Ω,R) (or Cc(Ω,C));
Cb(Ω) = Cb(Ω,R) (or Cb(Ω,C));
Cmb (Ω) = u ∈ Cb(Ω); Dαu ∈ Cb(Ω) for all α ∈ Nn such that |α| ≤ m, equipped with
the norm ‖u‖Cmb
(Ω) =∑
|α|≤m
‖Dαu‖L∞ ;
C(Ω) the space of continuous functions Ω → R (or Ω → C). When Ω is bounded, C(Ω)
is a Banach space when equipped with the L∞ norm;
Cb,u(Ω) the Banach space of uniformly continuous and bounded functions Ω → R (or
Ω → C) equipped with the topology of uniform convergence;
Cmb,u(Ω) the Banach space of functions u ∈ Cb,u(Ω) such that Dαu ∈ Cb,u(Ω), for every
multi-index α such that |α| ≤ m. Cmb,u(Ω) is equipped with the norm of Wm,∞(Ω);
C0(Ω) the closure of D(Ω) in L∞(Ω);
Cm,α(Ω) for 0 ≤ α ≤ 1, the Banach space of functions u ∈ Cmb,u(Ω) such that
‖u‖Cm,α = ‖u‖W m,∞ + sup
|Dβu(x)−Dβu(y)||x− y|α ; x, y ∈ Ω, |β| = m
<∞;
D(Ω) = C∞c (Ω) the Frechet space of C∞ functions Ω → R (or Ω → C) compactly
supported in Ω, equipped with the topology of uniform convergence of all derivatives on
compact subsets of Ω;
D′(Ω) the space of distributions on Ω, that is the topological dual of D(Ω);
* with this definition of the Fourier transform, ‖F‖L(L2) = 1, F(u ? v) = FuFv and
F(Dαu) = (2πi)|α|∏n
j=1 xαj
j Fu.
4
S(Rn) the Schwartz space, that is the space of u ∈ C∞(Rn,R) (or C∞(Rn,C)) such
that for every nonnegative integer m and every multi-index α,
pm,α(u) = supx∈Rn
(1 + |x|2)m/2|Dαu(x)| <∞.
S(Rn) is a Frechet space when equipped with the seminorms pm,α;
S ′(Rn) the space of tempered distributions on Rn, that is the topological dual of S(Rn).
S ′(Rn) is a subspace of D′(Rn);
p′ the conjugate of p given by1
p+
1
p′= 1;
Lp(Ω) the Banach space of (classes of) measurable functions u : Ω → R (or C) such that∫
Ω
|u(x)|p dx <∞ if 1 ≤ p <∞, or Ess supΩ
|u| <∞ if p = ∞. Lp(Ω) is equipped with the
norm
‖u‖Lp =
(∫
Ω
|u(x)|p dx)1/p
, if p <∞;
Ess supΩ
|u|, if p = ∞;
Wm,p(Ω) the Banach space of (classes of) measurable functions u : Ω → R (or Ω → C)
such that Dαu ∈ Lp(Ω) in the sense of distributions, for every multi-index α with |α| ≤ m.
Wm,p(Ω) is equipped with the norm
‖u‖W m,p =∑
|α|≤m
‖Dαu‖Lp ;
Wm,p0 (Ω) the closure of D(Ω) in Wm,p(Ω);
W−m,p′(Ω) the dual of Wm,p0 (Ω);
Hm(Ω) = Wm,2(Ω) Hm(Ω) is equipped with the equivalent norm
‖u‖Hm =
∑
|α|≤m
∫
Ω
|Dαu(x)|2 dx
1/2
.
Hm(Ω) is a Hilbert space for the scalar product
(u, v)Hm =
∫
Ω
Re(u(x)v(x)) dx;
Hm0 (Ω) = Wm,2
0 (Ω);
5
H−m(Ω) = W−m,2(Ω);
D(I,X) = C∞c (I,X) the Frechet space of C∞ functions I → X compactly supported
in I, equipped with the topology of uniform convergence of all derivatives on compact
subintervals of I;
D′(I,X) the space of X-valued distributions on I, that is the space of linear, continuous
mappings D(I) → X, where X is equipped with the weak topology;
Lp(I,X) the Banach space of (classes of) measurable functions u : I → X such that∫
I
‖u(t)‖pX dt <∞ if 1 ≤ p <∞, or Ess sup
I‖u(t)‖X <∞ if p = ∞. Lp(I,X) is equipped
with the norm
‖u‖Lp =
(∫
I
‖u(t)‖pX dt
)1/p
, if p <∞;
Ess supI
‖u(t)‖X , if p = ∞;
Wm,p(I,X) the Banach space of (classes of) measurable functions u : I → X such thatdju
dtj∈ Lp(I,X) for every 0 ≤ j ≤ m. Wm,p(I,X) is equipped with the norm
‖u‖W m,p =
m∑
j=1
‖dju
dtj‖Lp ;
Cb,u(I,X) the Banach space of uniformly continuous and bounded functions I → X,
equipped with the topology of uniform convergence;
Cmb,u(I,X) the Banach space of functions u ∈ Cb,u(I,X) such that
dju
dtj∈ Cb,u(I,X), for
every 0 ≤ j ≤ m. Cmb,u(I,X) is equipped with the norm of Wm,∞(I,X);
Cm,α(I,X) for 0 ≤ α ≤ 1, the Banach space of functions u ∈ Cmb,u(I,X) such that
‖u‖Cm,α = ‖u‖W m,∞ + sup
|d
mu(t)dtm − dmu(s)
dtm ||t− s|α ; s, t ∈ I
<∞;
C(I,X) the space of continuous functions I → X. When I is bounded, C(I,X) is a
Banach space when equipped with the norm of L∞(I,X);
T (t) except otherwise specified, the group of isometries on L2(Ω) generated by the skew-
adjoint operator iA, where A is the Laplacian with Dirichlet boundary condition on ∂Ω.
6
1. Introduction. The purpose of these notes is to give a self-contained presentation
of some basic results concerning nonlinear Schrodinger equations. The reader is supposed
to be familiar with elementary notions of functional analysis, real and vector integration,
Sobolev spaces, elliptic equations and linear semigroups.
Nonlinear Schrodinger equations have received a great deal of interest from the math-
ematicians in the past ten years or so, due in particular to their applications in nonlinear
optics. Indeed, simplified versions or limits of Zakharov’s system lead to certain nonlin-
ear Schrodinger equations. See for example Kelley [1], Lam, Lippmann and Tappert [1],
Schochet and Weinstein [1], Suydam [1]. However, the litterature on this subject is too
abundant to be mentioned extensively here and the reader interested in a more complete
bibliography should consult the physics journals. Nonlinear Schrodinger equations or sys-
tems also arise in quantum field theory, and in particular in the Hartree-Fock theory.
See for example Avron, Herbst and Simon [1,2,3], Bialinycki-Birula and Mycielski [1,2],
Combes, Schrader and Seiler [1], Eboli and Marques [1], Gogny and Lions [1], Kato [2],
Lebowitz, Rose and Speer [1], Lieb and Simon [1], Reed and Simon [1], B. Simon [1].
From the mathematical point of view, Schrodinger’s equation appears as a delicate
problem, since it posesses a mixture of the properties of parabolic and hyperbolic equa-
tions. Indeed, it is “almost” reversible, it has conservation laws and also some dispersive
properties like the Klein-Gordon equation, but it has an infinite speed of propagation. On
the other hand, Schrodinger’s equation has a kind of smoothing effect shared by parabolic
problems but the time-reversibility prevents it from generating an analytic semigroup.
For these reasons, even the local Cauchy problem (initial value problem) is delicate
for the nonlinear equation and satisfactory partial results are rather recent, compared to
the heat or to the Klein-Gordon equation. It is worthwile to notice that the problem is still
widely open. The conservation laws and the reversibility of the equation force to study
the Canchy problem in the energy space, which is constructed on L2 and therefore is not
appropriate for most nonlinearities. When the equation is considered in Rn and when
the nonlinearity is subcritical (essentially in the sense that the energy space is H1), the
dispersive properties of the linear equation allow a satisfactory solution of the local initial
value problem. When the nonlinearity is supercritical or when the linear equation is not
dispersive (for example if the equation is considered in a bounded subset of Rn), then the
7
problem is essentially open.
As for the study of the global behavior of the solutions, the situation is much worse.
The only satisfactory results concern the dispersive case with a subcritical, repulsive non-
linearity, when all solutions are asymptotically free (i.e. they behave like the solutions
of the linear equation). In the dispersive case with a subcritical, attractive nonlinearity,
only some very partial results are known, such as the stability or instability of some pe-
riodic solutions (standing waves). In the nondispersive case, nothing seems to be known
for any kind of nonlinearity. The nature of the singularities developped by the blowing-up
solutions has been studied numerically in great details, but up to now only very partial
analytical results are known. A very particular situation is the one-dimensional cubic
Schrodinger equation, which is a completely integrable Hamiltonian system. It can be
solved by the inverse scattering method, and a soliton behavior is observed (see Zakharov
and Shabat [1]).
The notes are organised as follows. In Chapter 2, we recall the basic properties of
functional analysis, complex and vector integration, Sobolev spaces, elliptic equations and
linear semigroups that we use throughout the text.
In Chapter 3, we establish the most important properties of the (linear) Schrodinger
equation iut +4u = 0, in R× Ω,
u = 0, on R× ∂Ω,
where Ω is an open subset of Rn. The case Ω = Rn is studied in details, and in particular
the dispersive properties and the smoothing effects.
Chapter 4 is devoted to the local Cauchy problem for the nonlinear equationiut +4u+ g(u) = 0, in R× Ω,
u = 0, on R× ∂Ω,
where g is a nonlinear self-interaction. A first section is devoted to a simple abstract
result. Then, a preliminary existence theorem is established for stronger nonlinearities.
In the following sections, that result is applied to solve the local Cauchy problem in the
energy space in the case Ω = Rn, and in some special cases in strict subdomains of Rn for
n = 1, 2.
In Chapter 5, we study the H2 regularity of the solutions and the W 1,p and C∞
smoothing effects when Ω = Rn, similar to those observed in the linear problem.
8
Chapter 6 is devoted to the study of global existence and finite-time blowup of so-
lutions, and a section is devoted to the study of L2-solutions when Ω = Rn. Chapter 7
concerns the asymptotic behavior of solutions in the repulsive case, based in particular on
the “pseudo-conformal” conservation law and Morawetz’ estimate. Both chapters concern
the case Ω = Rn.
In Chapter 8 (still in the case Ω = Rn), we study the stability of standing waves,
in the attractive case. Finally, Chapter 9 is devoted to some further comments concern-
ing generalized nonlinear Schrodinger equations. Every chapter ends with a section of
comments, concerning further developments and references. The notes are followed by an
extensive bibliography.
9
2. Preliminaries. In this Chapter, we recall the basic properties of functional analysis,
complex and vector integration, Sobolev spaces, elliptic equations and linear semigroups
that we use throughout the text.
2.1. Functional analysis. See for example Yosida [1], Brezis [2], Cazenave [5], Cazenave
and Haraux [2], Brezis and Cazenave [1], Strauss [1]. For convenience, we recall below some
useful properties.
Proposition 2.1.1. Let X and Y be two Banach spaces such that X → Y , with dense
embedding. The following properties hold.
(i) Y ? → X?, where the embedding is defined by 〈f, x〉X?,X = 〈f, x〉Y ?,Y , for all x ∈ X
and f ∈ Y ?;
(ii) if X is reflexive, then the embedding Y ′ → X ′ is dense.
Lemma 2.1.2. Let X and Y be two Banach spaces such that X → Y , and consider
x ∈ X and a sequence (xn)n∈N ⊂ X. If xn x in X, as n → ∞, then, xn x in Y , as
n→∞.
Proof. The embedding is continuous X → Y ; and so, it is also continuous X → Y for the
weak topologies. Hence the result.
Lemma 2.1.3. Let X and Y be two Banach spaces such that X → Y . Assume X is
reflexive and consider y ∈ Y and a bounded sequence (xn)n∈N ⊂ X. If xn y in Y as
n→∞, then y ∈ X and xn y in X, as n→∞.
Proof. Let us first prove that y ∈ X. There exists x ∈ X and a subsequence nk such that
xnk x in X, as k →∞. Therefore, by Lemma 2.1.2 xnk
x in Y , as k →∞. It follows
that y = x ∈ X.
Let us prove that xn y in X by contradiction. If not, there exists x′ ∈ X ′, ε > 0
and a subsequence nk such that |〈x′, xnk− y〉| ≥ ε, for every k ∈ N. On the other hand,
there exists x ∈ X and a subsequence nkjsuch that xnkj
x in X as j → ∞. By the
preceding argument, it follows that x = y; and so xnkj y in X as j → ∞, which is a
contradiction.
10
Corollary 2.1.4. Let X → Y be two Banach spaces, X being reflexive, and let I be a
bounded, open interval of R. Consider a weakly continuous function u : I → Y . If there
exists a dense subset E of I such that
(i) u(t) ∈ X, for all t ∈ E,
(ii) sup‖u(t)‖X , t ∈ E = K <∞,
then u(t) ∈ X, for all t ∈ I, and u : I → X is weakly continuous.
Proof. Let t ∈ I and let (tn)n∈N ⊂ E converge to t, as n→∞. Since u(tn) u(t) in Y ,
it follows from Lemma 2.1.3 that u(t) ∈ X and that
‖u(t)‖X ≤ lim infn→∞
‖u(tn)‖X ≤ K.
Let now t ∈ I and let (tn)n∈N ⊂ I converge to t, as n→∞. Since u(tn) u(t) in Y and
u(tn) is bounded in X, it follows from Lemma 2.1.3 that u(tn) u(t) in X. Hence the
result.
The following two lemmas, whose proofs are elementary are also quite useful in the
theory of evolution equation.
Lemma 2.1.5. Let X be a uniformly convex Banach space, let I be a bounded, open
interval of R and let u : I → X be weakly continuous. If the function t 7→ ‖u(t)‖X is
continuous I → R, then u ∈ C(I,X).
Lemma 2.1.6. Let X be a Banach space, let I be a bounded, open interval of R and
let u : I → X be weakly continuous. If there exists a Banach space B such that X → B
with compact embedding, then u ∈ C(I, B).
The following compactness result is very helpful for passing to the limit in nonlinear
evolution equations. Its proof is quite simple.
Proposition 2.1.7. Let X → Y be two Banach spaces, X being reflexive and let I be a
bounded, open interval of R. Let (fn)n∈N be a bounded sequence in C(I, Y ). Assume that
fn(t) ∈ X, for all (n, t) ∈ N×I and sup‖fn(t)‖Y , (n, t) ∈ N×I = K <∞, and that fn is
11
uniformly equicontinuous in Y (i.e. ∀ε > 0, ∃δ > 0, ∀n, s, t ∈ N×I×I, ‖fn(t)−fn(s)‖Y ≤ ε
if |t− s| ≤ δ). The following properties hold.
(i) There exists a function f ∈ C(I, Y ) which is weakly continuous I → X and a subse-
quence nk such that fnk(t) f(t) in X as k →∞, for all t ∈ I;
(ii) if there exists a uniformly convex Banach space B such that X → B → Y and if
(fn)n∈N ⊂ C(I, B) and ‖fnk(t)‖B → ‖f(t)‖B as k → ∞, uniformly on I, then also
f ∈ C(I, B) and fnk→ f in C(I, B) as k →∞.
Proof. (i) Let (tn)n∈N be a representation of Q∩I. It follows easily from the reflexivity
of X and the diagonal procedure that there exists a subsequence nk and a function f :
Q ∩ I → X such that fnk(tj) f(tj) in X (hence in Y ) as k → ∞, for all j ∈ N. It
follows from the uniform equicontinuity of (fn)n∈N and the weak lower semicontinuity of
the norm that f can be extended to a function of C(I, Y ). Furthermore, by Lemma 2.1.3
and Corollary 2.1.4, f : I → X is weakly continuous and sup‖f(t)‖, t ∈ I ≤ K. Consider
now t ∈ I, let (tj)j∈N ⊂ Q ∩ I converge to t and let y′ ∈ Y ′. We have
|〈y′, fnk(t)− f(t)〉Y ′,Y | ≤ |〈y′, fnk
(t)− fnk(tj)〉Y ′,Y |
+ |〈y′, f(t)− f(tj)〉Y ′,Y |+ |〈y′, fnk(tj)− f(tj)〉Y ′,Y |.
Given ε > 0, it follows from the uniform equicontinuity that the first and second terms of
the right hand side are less than ε/4 for j large enough. Given such a j, the third term is
less than ε/2 for k large enough; and so
|〈x′, fnk(t)− f(t)〉Y ′,Y | → 0, as k →∞.
It follows that fnk(t) f(t) in Y ; and so fnk
(t) f(t) in X, by Lemma 2.1.3. Hence (i).
(ii) Note first that f : I → B is weakly continuous. Also, ‖f‖B : I → R is continuous;
and so (Lemma 2.1.5) f ∈ C(I, B). It remains to prove that fnk→ f in C(I, B). We argue
by contradiction. If the conclusion did not hold, there would exist a sequence (tk)k∈N ⊂ I
and ε > 0 such that ‖fnk(tk) − f(tk)‖B ≥ ε, for every k ∈ N. We may assume that
tk → t ∈ I, as k → ∞. It follows from (i) and uniform continuity that fnk(tk) f(t) in
Y , as k →∞. Since (fn)n∈N is bounded in C(I, B), we have as well fnk(tk) f(t) in B,
as k →∞. Furthermore,
| ‖fnk(tk)‖B − ‖f(t)‖B | ≤ | ‖fnk
(tk)‖B − ‖f(tk)‖B |+ | ‖f(tk)‖B − ‖f(t)‖B |.
12
It follows that ‖fnk(tk)‖B → ‖f(t)‖B; and so, fnk
(tk) → f(t) in B, as k →∞, which is a
contradiction.
Finally, we will use some properties of the intersection and sum of Banach spaces.
Consider two Banach spaces X1 and X2 that are subsets of a Hausdorff topological vector
space X . Let
X1 ∩X2 = x ∈ X ; x ∈ X1, x ∈ X2,
and
X1 +X2 = x ∈ X ; ∃x1 ∈ X1, ∃x2 ∈ X2, x = x1 + x2.
Define
‖x‖X1∩X2= ‖x‖X1
+ ‖x‖X2, for x ∈ X1 ∩X2,
and
‖x‖X1.X2= Inf‖x1‖X1
+ ‖x2‖X2; x = x1 + x2, for x ∈ X1 +X2.
Then, we have the following result (see Bergh and Lofstrom [1], Lemma 2.3.1 and Theo-
rem 2.7.1).
Proposition 2.1.8. (X1 ∩X2, ‖ ‖X1∩X2) and (X1 +X2, ‖ ‖X1+X2
) are Banach spaces. If
furthermore X1 ∩ X2 is a dense subset of both X1 and X2, then (X1 ∩ X2)′ = X ′
1 + X ′2
and (X1 +X2)′ = X ′
1 ∩X ′2.
2.2. Integration. For real and complex integration, consult for example Brezis [2],
Dunford and Schwartz [1], Yosida [1]. For vector integration, see Brezis and Cazenave [1],
Cazenave [5], Cazenave and Haraux [2], Diestel and Uhl [1], Dinculeanu [1], Dunford and
Schwartz [1], J. Simon [1], Yosida [1], and the appendix of Brezis [1].
Throughout these notes, we consider Lp spaces of complex valued functions. Ω being
an open subset of Rn, Lp(Ω) (or Lp, when there is no risk of confusion) denotes the space
of (classes of) measurable functions u : Ω → C such that
∫
Ω
|u(x)|p dx <∞, if p ∈ [1,∞);
Ess supΩ
|u| <∞, if p = ∞.
13
Lp(Ω) is equipped with the norm ‖ ‖Lp(Ω) (or ‖ ‖Lp , when there is no risk of confusion)
defined by
‖u‖Lp =
(∫
Ω
|u(x)|p dx) 1
p
, if p ∈ [1,∞);
Ess supΩ
|u|, if p = ∞.
L2(Ω) is a real Hilbert space when equipped with the scalar product
(u, v) = Re
(∫
Ω
u(x)v(x)dx
).
Below is a useful result of Strauss [2] concerning Lp spaces.
Proposition 2.2.1. Let Ω be an open subset of Rn and let 1 < p ≤ ∞. Consider
u : Ω → R and a bounded sequence (un)n∈N of Lp(Ω). If un → u almost everywhere as
n → ∞, then u ∈ Lp(Ω) and un → u as n → ∞ in Lq(Ω′), for every Ω′ ⊂ Ω of finite
measure and every q ∈ [1, p). In particular, un → u as n → ∞, in Lp(Ω) weak if p < ∞,
and in L∞(Ω) weak-? if p = ∞.
Proof. By extending the functions by 0 outside Ω, we may assume that Ω = Rn. Observe
that by Fatou’s lemma, we have u ∈ Lp(Rn). Let Ω′ ⊂ Rn have a finite measure and let
q ∈ [1, p). Consider ε > 0. By Egorov’s theorem, there exists a measurable subset E of Ω′
such that un → u uniformly on Ω′ \E and
|E|p−q
p supn≥0
(∫
Rn
|un − u|p) q
p
≤ ε/2.
Let n0 be large enough so that |un − u|q ≤ ε/2 on Ω′ \E, for n ≥ n0. It follows that∫
Ω′
|un − u|q =
∫
E
|un − u|q +
∫
Ω′\E|un − u|q
≤ |E|p−q
p
(∫
E
|un − u|p) q
p
+ |Ω′ \E| supΩ′\E
|un − u|q
≤ ε.
Hence the result, since ε is arbitrary.
Consider now an open interval I ⊂ R and a Banach space X, equipped with the norm
‖ ‖. A function f : I → X is measurable if there exists a set N ⊂ I of measure 0 and a
sequence (fn)n∈N ⊂ Cc(I,X) such that
limn→∞
fn(t) = f(t), for all t ∈ I \N.
14
It follows easily from the definition that if f : I → X is measurable, then ‖ f ‖: I → R is
also measurable. Also, if f : I → X is measurable and if Y is a Banach space such that
X → Y , then f : I → Y is measurable.
Remark 2.2.2. Pettis’ theorem asserts that a function f is measurable if and only if
f is weakly measurable (i.e. for every x′ ∈ X ′, the function t 7→ 〈x′, f(t)〉 is measurable
I → X) and there exists a set N ⊂ I of measure 0 such that f(I \N) is separable. The
following properties follow easily from that characterization.
(i) Let f : I → X be weakly continuous (i.e. continuous from I to X equipped with its
weak topology). Then f is measurable.
(ii) Let (fn)n∈N be a sequence of measurable functions I → X and let f : I → X. Assume
that for almost all t ∈ I, fn(t) f(t) in X as n→∞. Then f is measurable.
A measurable function f : I → X is integrable if there exists a sequence (fn)n∈N ⊂Cc(I,X) such that
limn→∞
∫
I
‖ fn(t)− f(t) ‖ dt = 0
If f : I → X is integrable, there exists x(f) ∈ X such that for any sequence (fn)n∈N ⊂Cc(I,X) verifying
limn→∞
∫
I
‖ fn(t)− f(t) ‖ dt = 0,
one has
limn→∞
∫
I
fn(t) dt = x(f),
the above limit being for the strong topology of X. The element x(f) is called the integral
of f on I. We note
x(f) =
∫f =
∫
I
f =
∫
I
f(t) dt.
If I = (a, b), we also note
x(f) =
∫ b
a
f =
∫ b
a
f(t) dt.
As for real-valued functions, it is convenient to note
∫ β
α
f(t) dt = −∫ α
β
f(t) dt,
15
if β < α. Bochner’s Theorem asserts that if f : I → X is measurable, then f is integrable
if and only if ‖ f ‖: I → R is integrable. In addition, we have
‖∫
I
f(t) dt ‖≤∫
I
‖ f(t) ‖ dt.
Bochner’s Theorem allows one to deal with vector valued integrable functions like one deals
with real valued integrable functions. It suffices in general to apply the usual convergence
theorems to ‖ f ‖. For example, one can easily establish the following result (the dominated
convergence theorem). Let (fn)n∈N be a sequence of integrable functions I → X, let
g ∈ L1(I) and let f : I → X. Assume that
‖ fn(t) ‖≤ g(t), for almost all t ∈ I and all n ∈ N,
limn→∞
fn(t) = f(t) for almost all t ∈ I.
Then f is integrable and ∫
I
f(t) dt = limn→∞
∫
I
fn(t) dt.
For p ∈ [1,∞], one denotes by Lp(I,X) the set of (classes of) measurable functions
f : I → X such that the function t 7→‖ f(t) ‖ belongs to Lp(I). For f ∈ Lp(I,X), one
defines
‖ f ‖Lp(I,X) =
∫
I
‖ f(t) ‖p dt
1p
, if p <∞,
‖ f ‖Lp(I,X) = Ess supt∈I
‖ f(t) ‖ if p = ∞.
When there is no risk of misunderstanding, we denote ‖ ‖Lp(I,X) by ‖ ‖Lp(I) or ‖ ‖Lp or
‖ ‖p.
Remark 2.2.3. The space Lp(I,X) enjoys most of the properties of the space Lp(I) =
Lp(I,R), with essentially the same proofs. In particular, one obtains easily the following
results.
(i) ‖ ‖Lp(I,X) is a norm on the space Lp(I,X). Lp(I,X) equipped with that norm is
a Banach space. If p < ∞, then D(I,X) is dense in Lp(I,X) (apply the classical
procedure by truncation and regularization).
(ii) A measurable function f : I → X belongs to Lp(I,X) if and only if there exists a
function g ∈ Lp(I) such that ‖ f ‖≤ g almost everywhere on I.
16
(iii) If f ∈ Lp(I,X) and ϕ ∈ Lq(I) with1
p+
1
q=
1
r≤ 1, then ϕf ∈ Lr(I,X), and we have
the inequality
‖ ϕf ‖Lr(I,X)≤‖ f ‖Lp(I,X) ‖ ϕ ‖Lq(I) .
In particular, if f ∈ Lp(I,X) and if J is an open sub-interval of I, then f|J ∈ Lp(J,X).
(iv) If f ∈ Lp(I,X) and g ∈ Lq(I,X ′) with1
p+
1
q=
1
r≤ 1, then the function h defined
by h(t) = 〈g(t), f(t)〉X′,X is in Lr(I) and ‖ h ‖Lr≤‖ f ‖Lp(I,X) ‖ g ‖Lq(I,X′).
(v) If f ∈ Lp(I,X) ∩ Lq(I,X) with p < q, then for every r ∈ [p, q] we have f ∈ Lr(I,X),
and ‖ f ‖Lr(I,X)≤‖ f ‖θLp(I,X) ‖ f ‖1−θ
Lq(I,X), where1
r=θ
p+
1− θ
q.
(vi) If I is bounded and p ≤ q, then Lq(I,X) → Lp(I,X) and ‖ f ‖Lp(I,X)≤ |I|q−ppq ‖
f ‖Lq(I,X).
(vii) If Y is a Banach space and if A ∈ L(X,Y ), then for every f ∈ Lp(I,X) we have
Af ∈ Lp(I, Y ) and ‖ Af ‖Lp(I,Y )≤‖ A ‖L(X,Y )‖ f ‖Lp(I,X). In particular, if X → Y
and if f ∈ Lp(I,X), then f ∈ Lp(I, Y ) (take A to be the embedding).
(viii) If Y is a Banach space and if A ∈ L(X,Y ) then for every f ∈ L1(I,X), we have
∫
I
Af(t) dt = A
(∫
I
f(t) dt
).
In particular, if X → Y and if f ∈ L1(I,X), then the integral of f in the sense of X
is also the integral of f in the sense of Y (take A to be the embedding).
The following result is essential for its applications to the theory of evolution equations.
Theorem 2.2.4. Let 1 ≤ p ≤ ∞. Let (fn)n∈N be a bounded sequence in Lp(I,X). If
there exists f : I → X such that for almost all t ∈ I, fn(t) f(t) in X as n → ∞, then
f ∈ Lp(I,X) and ‖ f ‖Lp(I,X)≤ lim infn→∞
‖ fn ‖Lp(I,X).
2.3. Sobolev spaces. For Sobolev spaces of real (or complex) valued functions, see
for example Adams [1], Bergh and Lofstrom [1], Brezis [2], Gilbarg and Trudinger [1],
J.-L. Iions [1], Lions and Magenes [1]. For vector valued Sobolev spaces, see the appendix of
Brezis [1], Brezis and Cazenave [1], Cazenave [5], Cazenave and Haraux [2], J.-L. Lions [1],
Lions and Magenes [1].
17
Consider an open subset Ω of Rn. We recall that D(Ω) (= D(Ω,C)) is equipped with
the topology induced by the family of seminorms dK,m where K is a compact subset of Ω
and m ∈ N, defined by
dK,m(ϕ) = supx∈K
∑
|α|=m
|Dαϕ(x)|, for all ϕ ∈ D(Ω).
The set of distributions on Ω, D′(Ω), is the dual space of D(Ω). If T ∈ D′(Ω) and if α ∈ Nn
is a multi-index, one defines the distribution
DαT =∂α1
∂xα11
· · · ∂αn
∂xαnnT ∈ D′(Ω)
by
〈DαT, ϕ〉 = (−1)|α|〈T,Dαϕ〉, for all ϕ ∈ D(Ω).
A function f ∈ L1loc(Ω) defines a distribution Tf ∈ D′(Ω) by
〈Tf , ϕ〉 = Re
(∫
Ω
f(x)ϕ(x)dx
), for all ϕ ∈ D(Ω).
It is well known that if Tf = Tg, then f = g almost everywhere. A distribution T ∈ D′(Ω)
is said to belong to Lp(Ω) if there exists f ∈ Lp(Ω) such that T = Tf . In this case, f is
unique.
For m ∈ N and 1 ≤ p ≤ ∞, the Sobolev space Wm,p(Ω) is defined by
Wm,p(Ω) = u ∈ Lp(Ω), Dαu ∈ Lp(Ω) for |α| ≤ m.
Wm,p(Ω) is a Banach space when equipped with the norm ‖ ‖W m,p=‖ ‖W m,p(Ω) defined by
‖ u ‖W m,p=∑
0≤|α|≤m
‖ Dαu ‖Lp(Ω) .
One defines the closed subset Wm,p0 (Ω) of Wm,p(Ω) as the closure in Wm,p(Ω) of D(Ω).
When p = 2, one sets Wm,p(Ω) = Hm(Ω) and Wm,p0 (Ω) = Hm
0 (Ω) and one rather
equips Hm(Ω) with the equivalent norm
‖ u ‖Hm(Ω)=‖ u ‖Hm=
∑
0≤|α|≤m
∫
Ω
|Dαu(x)|2 dx
12
.
18
Hm(Ω) (hence Hm0 (Ω)) is then a Hilbert space with the scalar product
(u, v)Hm =∑
0≤|α|≤m
Re
(∫
Ω
Dαu(x)Dαv(x) dx
).
Remark 2.3.1 The following properties are well known.
(i) If 1 < p <∞, then the spaces Wm,p(Ω) and Wm,p0 (Ω) are reflexive.
(ii) If (un)n∈N is a bounded sequence of W 1,p(Ω), 1 ≤ p ≤ ∞, then ((un)|ω)n∈N is a
relatively compact subset of L1(ω), for every ω ⊂⊂ Ω. In particular, there exists a
subsequence (unk)k∈N converging almost everywhere in ω. Therefore, one constructs
easily a subsequence of (un)n∈N converging almost everywhere in Ω.
(iii) Assume m ≥ 1 and 1 < p ≤ ∞. Consider a bounded sequence (un)n∈N of Wm,p(Ω).
Then there exists u ∈ Wm,p(Ω) and a subsequence (unk)k∈N such that unk
→ u
almost everywhere as k →∞, and
‖ u ‖W m,p≤ lim infn→∞
‖ un ‖W m,p .
If p < ∞, then also unk u in Wm,p. If p < ∞ and (un)n∈N ⊂ Wm,p
0 (Ω), then
u ∈Wm,p0 (Ω).
(iv) Let m ≥ 0 and 1 < p ≤ ∞. Consider a bounded sequence (un)n∈N of Wm,p(Ω) and
assume that there exits u : Ω → R such that un → u almost everywhere as n → ∞.
Then u ∈Wm,p(Ω) and
‖ u ‖W m,p≤ lim infn→∞
‖ un ‖W m,p .
If p < ∞, then also un u in Wm,p. If p < ∞ and (un)n∈N ⊂ Wm,p0 (Ω), then
u ∈Wm,p0 (Ω).
(v) Let F : C → C be a Lipschitz continuous function such that F (0) = 0. We may
consider F as a function R2 → R2, so that F ′(u) = DF (u) (which is defined for
almost all u ∈ C) is a 2×2 real matrix, hence a linear operator C → C. Let p ∈ [1,∞].
Then, for every u ∈W 1,p(Ω), we have F (u) ∈W 1,p(Ω) and ∂iF (u) = F ′(u)∂iu almost
everywhere, for every 1 ≤ i ≤ n. In particular, if L is the Lipschitz constant of F ,
19
then ‖ ∇F (u) ‖Lp≤ L ‖ ∇u ‖Lp . (Note that the formula ∂iF (u) = F ′(u)∂iu almost
everywhere makes sense, since if A ⊂ R is a set of measure 0, then ∇u = 0 almost
everywhere on the set x ∈ Ω; u(x) ∈ A.) If p < ∞ and if u ∈ W 1,p0 (Ω), then
F (u) ∈ W 1,p0 (Ω); and the mapping u 7→ F (u) is continuous W 1,p
0 (Ω) →W 1,p0 (Ω). On
these questions, see Marcus and Mizel [1,2,3].
(vi) In particular, let p ∈ [1,∞] and let u ∈ W 1,p(Ω) (respectively u ∈ W 1,p0 (Ω)). Then
|u| ∈ W 1,p(Ω) (respectively |u| ∈ W 1,p0 (Ω), if 1 < p < ∞ and | ∇|u| | ≤ |∇u| almost
everywhere. Moreover, the mapping u 7→ |u| is continuous W 1,p(Ω) → W 1,p(Ω)
(W 1,p0 (Ω) →W 1,p
0 (Ω) if p <∞).
(vii) Let F : C → C satisfy F (0) = 0, and assume that there exists α ≥ 0 such that
|F (v) − F (u)| ≤ L(|v|α + |u|α)|v − u|, for all u, v ∈ C. Let 1 ≤ p, q, r ≤ ∞ be such
that1
r=α
p+
1
q. Let u ∈ Lp(Ω) be such that ∇u ∈ Lq(Ω). Then, ∇F (u) ∈ Lr(Ω) and
‖∇F (u)‖Lr ≤ L‖u‖αLp‖∇u‖Lq . In particular, if p = α+2, then for every u ∈W 1,p(Ω)
(respectively u ∈ W 1,p0 (Ω)) we have F (u) ∈ W 1,p′(Ω) (respectively F (u) ∈ W 1,p′
0 (Ω))
and ‖ ∇F (u) ‖Lp′≤ L‖u‖αLp ‖ ∇u ‖Lp .
(viii) Let 1 ≤ p, q <∞ and let m, j be nonnegative integers. Then D(Rn) is a dense subset
of Wm,p(Rn) ∩W j,q(Rn). In particular, Wm,p0 (Rn) = Wm,p(Rn).
(ix) Hm(Rn) can be alternatively defined as the set of u ∈ S ′(Rn) such that (1 +
|ξ|2)m/2u ∈ L2(Rn), and the corresponding norm is equivalent to the usual one.
More generaly, for any s ∈ R, one can define Hs(Rn) as the set of u ∈ S ′(Rn) such
that (1 + |ξ|2)s/2u ∈ L2(Rn).
We recall below some well known inequalities and embedding results.
Theorem 2.3.2. (Poincare’s inequality) Assume that |Ω| is finite (or that Ω is bounded
in one direction) and let 1 ≤ p ≤ ∞. There exists a constant C such that
‖u‖Lp ≤ C‖∇u‖Lp , for every u ∈W 1,p0 (Ω).
In particular, ‖∇u‖Lp(Ω) is an equivalent norm to ‖u‖W 1,p(Ω) on W 1,p0 (Ω).
20
Theorem 2.3.3. (Sobolev’s embedding theorem) If Ω has a Lipschitz continuous
boundary, then the following properties hold.
(i) If 1 ≤ p < n, then W 1,p(Ω) → Lq(Ω), for every q ∈ [p, npn−p ];
(ii) if p = n > 1, then W 1,p(Ω) → Lq(Ω), for every q ∈ [p,∞);
(iii) if p = n = 1, then W 1,p(Ω) → Lq(Ω), for every q ∈ [p,∞];
(iv) if p > n, then W 1,p(Ω) → L∞(Ω).
If Ω has a uniformly Lipschitz continuous boundary, then also
(v) if p > n, then W 1,p(Ω) → C0,α(Ω), where α = p−np
.
Theorem 2.3.4. (Rellich’s compactness theorem) If Ω is bounded and has a Lipschitz
continuous boundary, then the following properties hold.
(i) If 1 ≤ p ≤ n, then the embedding W 1,p(Ω) → Lq(Ω) is compact, for every q ∈ [p, npn−p
);
(ii) if p > n, then the embedding W 1,p(Ω) → L∞(Ω) is compact.
If we assume further that Ω has a uniformly Lipschitz continuous boundary, then
(iii) if p > n, then the embedding W 1,p(Ω) → C0,λ(Ω) is compact, for every λ ∈ (0, p−np
).
Theorem 2.3.5. The conclusions of Theorems 2.3.3 and 2.3.4 remain valid without any
smoothness assumption on Ω if one replaces W 1,p(Ω) by W 1,p0 (Ω) (note that Ω still needs
to be bounded for the compact embedding).
Remark 2.3.6. If p = n > 1, then W 1,p(Ω) → Lq(Ω) for every p < q < ∞, but
W 1,p(Ω) 6→ L∞(Ω). However, Sobolev’s embedding theorem can be improved by Trudin-
ger’s inequality. In particular, if n = 2, then for every M < ∞ there exists µ > 0 and
K <∞ such that ∫
Ω
(eµ|u|2 − 1
)≤ K,
for every u ∈ H10 (Ω) with ‖u‖H1 ≤M (see Adams [1]).
Theorem 2.3.7. (Gagliardo-Nirenberg’s inequality) Let 1 ≤ p, q, r ≤ ∞ and let j,m
be two integers, 0 ≤ j < m. If
1
p=j
n+ a
(1
r− m
n
)+
(1− a)
q,
21
for some a ∈ [j
m, 1] (a < 1 if r > 1 and m− j− n
r= 0), then there exists C(n,m, j, a, q, r)
such that
∑
|α|=j
‖Dαu‖Lp ≤ C
∑
|α|=m
‖Dαu‖Lr
a
‖u‖1−aLq ,
for every u ∈ D(Rn).
For 1 ≤ p < ∞ and m ∈ N, one defines W−m,p′(Ω) as the (topological) dual of
Wm,p0 (Ω). One defines H−m(Ω) = W−m,2(Ω), so that H−m(Ω) = (Hm
0 (Ω))′.
Remark 2.3.8. Here are some useful properties of the spaces W−m,p′(Ω).
(i) From the dense embedding D(Ω) → Wm,p0 (Ω), it follows that W−m,p′(Ω) is a space
of distributions on Ω. Furthermore, it follows from the dense embedding Wm,p0 (Ω) →
Lp(Ω) that Lp′(Ω) →W−m,p′(Ω). If p > 1, then the embedding is dense. In particu-
lar, D(Ω) is dense in W−m,p′(Ω).
(ii) Assume that that 1 ≤ q ≤ ∞ is such that Wm,p0 (Ω) → Lq(Ω). Then Lq′(Ω) →
W−m,p′(Ω). Furthermore, if p, q > 1, then the embedding is dense.
(iii) Even though Hm0 (Ω) is a Hilbert space, one generally does not identify H−m(Ω)
with Hm0 (Ω). One rather identifies L2(Ω) with its dual, so that H−m(Ω) becomes a
subspace of D′(Ω) containing L2(Ω). In particular, if u ∈ Hm0 (Ω) and v ∈ L2(Ω), then
〈u, v〉Hm0 ,H−m = Re
(∫
Ω
u(x)v(x)dx
)
It follows that ‖u‖2L2 ≤ ‖u‖Hm
0‖u‖H−m
, for all u ∈ Hm0 (Ω).
(iv) Like any distribution, an element of H−m(Ω) can be localized. Indeed, if T ∈ H−m(Ω)
and Ω′ is an open subset of Ω, then one defines T|Ω′ as follows. Let ϕ ∈ D(Ω′) and let
ϕ ∈ D(Ω) be equal to ϕ on Ω′ and to 0 on Ω \ Ω′. Then
Ψ(ϕ) = 〈ϕ, T 〉Hm0 (Ω),H−m(Ω)
defines a distribution Ψ ∈ D′(Ω′). Since ‖ϕ‖Hm0 (Ω′) ≤ ‖ϕ‖Hm
0 (Ω), it follows that
Ψ ∈ H−m(Ω′), and one sets T|Ω′ = Ψ. It is clear that the operator
PΩ′ :
H−m(Ω) → H−m(Ω′)
T 7→ T|Ω′
22
is linear and continuous, and is consistent with the usual restriction of functions.
(v) For every multi-index α of length j, then Dα is a bounded operator from H−m(Ω)
to H−m−j(Ω), for every m ∈ N. Since also Dα is bounded from Hk(Ω) to Hk−j(Ω),
for every k ≥ j, it follows easily that for every k ∈ Z, Dα is bounded from Hk0 (Ω) to
Hk−j(Ω).
(vi) In particular, 4 defines a linear, continuous operator from H1(Ω) to H−1(Ω). Note
that for u ∈ H1(Ω), the linear form 4u ∈ H−1(Ω) on H10 (Ω) is defined by
〈4u, v〉 = −Re
(∫
Ω
∇u(x)∇v(x) dx), for v ∈ H1
0 (Ω).
This is clear for v ∈ D(Ω) and follows by density for v ∈ H10 (Ω).
Consider now an open interval I ⊂ R and a Banach space X, equipped with the norm
‖ ‖. We denote by D′(I,X) the space of linear, continuous mappings D(I) → X, where X
is equipped with the weak topology. It is called the space of X-valued distributions on I.
An element f ∈ L1loc(I,X) defines a distribution Tf ∈ D′(I,X) by the formula
〈Tf , ϕ〉 =
∫
I
f(t)ϕ(t) dt, for every ϕ ∈ D(I).
One defines the nth derivative T (n) (ordnT
dtn) of a distribution T by the formula
〈Tn, ϕ〉 = (−1)n
∫
I
f(t)dnϕ(t)
dtndt, for every ϕ ∈ D(I).
For 1 ≤ p ≤ ∞, we denote by W 1,p(I,X) the set of (classes of) functions f ∈ Lp(I,X)
such that f ′ ∈ Lp(I,X), in the sense of D′(I,X). For f ∈ W 1,p(I,X), we set
‖ f ‖W 1,p(I,X)=‖ f ‖Lp(I,X) + ‖ f ′ ‖Lp(I,X) .
When there is no risk of misunderstanding, we denote ‖ ‖W 1,p(I,X) by ‖ ‖W 1,p(I) or ‖ ‖W 1,p .
The space W 1,p(I,X) enjoys most properties of the space W 1,p(I) = W 1,p(I,R), with
essentially the same proofs. In particular, ‖ ‖W 1,p(I,X) is a norm on the space W 1,p(I,X).
The space W 1,p(I,X) equipped with the norm ‖ ‖W 1,p(I,X) is a Banach space. If Y is a
Banach space and if A ∈ L(X,Y ) then for every f ∈W 1,p(I,X), Af ∈W 1,p(I, Y ) and
‖ Af ‖W 1,p(I,Y )≤‖ A ‖L(X,Y ) ‖ f ‖W 1,p(I,X) .
23
In particular, if X → Y and if f ∈ W 1,p(I,X), then f ∈ W 1,p(I, Y ) (take A to be the
embedding). The following characterization is quite useful.
Theorem 2.3.9. If 1 ≤ p ≤ ∞ and f ∈ Lp(I,X), then the following properties are
equivalent.
(i) f ∈ W 1,p(I,X);
(ii) for almost all s, t ∈ I we have f(t) = f(s) +∫ t
sf ′(σ) dσ;
(iii) there exists g ∈ Lp(I,X) such that for almost all s, t ∈ I we have f(t) = f(s) +∫ t
sg(σ) dσ;
(iv) there exist g ∈ Lp(I,X), t0 ∈ I and x0 ∈ X such that for almost all t ∈ I we have
f(t) = x0 +∫ t
t0g(s) ds;
(v) f is absolutely continuous, differentiable almost everywhere and f ′ (in the sense of
the almost everywhere derivative) is in Lp(I,X);
(vi) f is weakly absolutely continuous (hence weakly differentiable almost everywhere) and
f ′ (in the sense of the almost everywhere weak derivative) is in Lp(I,X).
In addition, if f satisfies these properties, then the derivatives of f in the senses of D′(I,X)
and almost everywhere coincide almost everywhere.
Remark 2.3.10. It follows easily from the above result that W 1,1(I,X) → Cb,u(I,X)
and that if p > 1, then W 1,p(I,X) → C0,α(I,X), with α = p−1p .
The following result is also quite useful.
Proposition 2.3.11. Assume X is reflexive and let f ∈ Lp(I,X). Then f ∈W 1,p(I,X)
if and only if there exists ϕ ∈ Lp(I) and a set N of measure 0 such that
‖ f(t)− f(s) ‖≤ |∫ t
s
ϕ(σ) dσ|, for all t, s ∈ I \N.
In this case, we have ‖ f ′ ‖Lp(I,X)≤‖ ϕ ‖Lp(I).
Remark 2.3.12. Applying Proposition 2.3.11, one can show the following results.
24
(i) Assume that X is reflexive and let f : I → X be Lipschitz continuous and bounded.
Then f ∈W 1,∞(I,X) and ‖ f ′ ‖L∞(I,X)≤ L, where L is the Lipschitz constant of f .
(ii) Assume that X is reflexive and that 1 < p ≤ ∞. Let (fn)n∈N be a bounded sequence
of W 1,p(I,X) and let f : I → X be such that fn(t) f(t) in X as n→∞, for almost
all t ∈ I. Then f ∈W 1,p(I,X) and ‖ f ‖W 1,p(I,X)≤ lim infn→∞
‖ fn ‖W 1,p(I,X).
Proposition 2.3.13. Let I be a bounded interval of R, let m be a nonnegative integer,
let Ω be an open subset of Rn and let (fn)n∈N be a bounded sequence of L∞(I,H10(Ω))∩
W 1,∞(I,H−m(Ω)). Then,
(i) there exists f ∈ L∞(I,H10(Ω))∩W 1,∞(I,H−m(Ω)) and a subsequence (fnk
)k∈N such
that for every t ∈ I, fnk(t) f(t) in H1
0 (Ω) as k →∞;
(ii) if ‖fnk(t)‖L2 → ‖f(t)‖L2 as k →∞, uniformly on I, then also fnk
→ f in C(I, L2(Ω))
as k →∞;
(iii) if (fn)n∈N ⊂ C(I,H10 (Ω)) and ‖fnk
(t)‖H1 → ‖f(t)‖H1 as k → ∞, uniformly on I,
then also f ∈ C(I,H10 (Ω)) and fnk
→ f in C(I,H10 (Ω)) as k →∞.
Proof. (i) follows from Remark 2.3.12, (ii), Theorem 2.3.5 and Proposition 2.1.7 (i) ap-
plied with X = H10 (Ω) and Y = H−m(Ω). (ii) follows from Proposition 2.1.7 (ii) applied
with X = H10 (Ω), Y = H−m(Ω) and B = L2(Ω). (iii) follows from Proposition 2.1.7 (ii)
applied with X = B = H10 (Ω) and Y = H−m(Ω).
One can define higher order vector valued Sobolev spaces as follows. For m ∈ N, one
sets
Wm,p(I,X) = f ∈ Lp(I,X),djf
dtj∈ Lp(I,X) for all j ∈ 1, · · · ,m.
It is clear that
Wm,p(I,X) = f ∈W 1,p(I,X),djf
dtj∈W 1,p(I,X) for all j ∈ 1, · · · ,m− 1,
and it follows that Wm,1(I,X) → Cm−1b,u (I,X) and that Wm,p(I,X) → Cm−1,α(I,X)
with α =p− 1
p, if p > 1.
25
2.4. Elliptic equations. There is an extensive litterature on the subject. Consult for
example Agmon, Douglis and Nirenberg [1], Brezis [2], Brezis and Cazenave [1], Gilbarg
and Trudinger [1], J.-L. Lions [1], Lions and Magenes [1], Nirenberg [1].
We recall below some of the results that we will use in the following sections. In all
this section, we consider an open subset Ω of Rn and we denote by 〈·, ·〉 the duality pairing
between H10 (Ω) and H−1(Ω) or the scalar product in L2(Ω) and by (·, ·) the scalar product
in H10 (Ω). We equip H−1(Ω) with the dual norm, that is
‖u‖H−1 = sup〈u, v〉, v ∈ H10 (Ω), ‖v‖H1
0= 1,
and we denote by ((·, ·)) the scalar product of H−1(Ω).
We recall that, essentially by Lax-Milgram’s lemma, for every f ∈ H−1(Ω), there
exists a unique element u ∈ H10 (Ω) such that
−4u+ u = f, in H−1(Ω).
In addition, we have
‖f‖H−1 = ‖u‖H10.
It follows in particular that 4− I defines an isometry from H10 (Ω) onto H−1(Ω).
By the same method, one shows also that for every λ > 0 and every f ∈ H−1(Ω),
there exists a unique element u ∈ H10 (Ω) such that
−4u+ λu = f, in H−1(Ω).
‖|f‖| = ‖u‖H10 (Ω) defines an equivalent norm on H−1(Ω) and λ‖u‖H−1 ≤ ‖f‖H−1 . If f ∈
L2(Ω), then 4u ∈ L2(Ω), the equation makes sense in L2(Ω) and λ‖u‖L2(Ω) ≤ ‖f‖L2(Ω).
One shows also that if Ω has a C2 boundary and if f ∈ L2(Ω), then f ∈ H2(Ω) and
‖u‖H2 ≤ C‖f‖L2 . In particular, −4 + I is an isomorphism from H2(Ω) ∩ H10 (Ω) onto
L2(Ω). Concerning Lp estimates, we have the following result.
Proposition 2.4.1. Let λ > 0, u ∈ H10 (Ω) and f ∈ H−1(Ω) verify −4u + λu = f . If
f ∈ Lp(Ω) for some p ∈ [1,∞), then u ∈ Lp(Ω) and λ‖u‖Lp ≤ ‖f‖Lp.
Proof. Let ϕ ∈ C1(R2,R2), considered as a function C → C have a bounded derivative,
and let v = ϕ(u). If for every x ∈ C the 2× 2 real matrix ϕ′(x) verifies ϕ′(x)y · y ≥ 0 for
26
every y ∈ R2, it follows easily that 〈4u, ϕ(u)〉 ≥ 0. If ϕ has the form ϕ(u) = uθ(|u|2), then
ϕ′(x)y·y = (y21+y2
2)θ(x21+x2
2)+2(x1y1+x2y2)2θ′(x2
1+x22), for u = u1+iu2 and y = (y1, y2).
In particular, if θ(s) ≥ 0 and θ′(s) ≥ 0 for s ≥ 0, then ϕ′(x)y ·y ≥ 0. On the other hand, if
θ(s) ≥ 2s|θ′(s)| for s ≥ 0, then ϕ′(x)y ·y ≥ (y21+y2
2)[θ(x21+x2
2)−2(x21+x2
2)|θ′(x21+x2
2)|] ≥ 0.
If ϕ is as above, it follows that
λ
∫
Ω
|u|2θ(|u|2) ≤∫
Ω
|f | |u|θ(|u|2).
If for some p ∈ [1,∞) we have θ(s) ≤ sp/2−1, then |u|θ(|u|2) ≤ (|u|2θ(|u|2))1−1/p; and so
λp
∫
Ω
|u|2θ(|u|2) ≤∫
Ω
|f |p.
For p ≤ 2 and ε > 0, take θ(s) = (ε+s)p/2−1. It follows from the preceding argument that
λp
∫
Ω
|u|2(ε+ |u|2)p/2−1 ≤∫
Ω
|f |p.
Letting ε ↓ 0 and applying Fatou’s Lemma, we obtain u ∈ Lp and λ‖u‖Lp ≤ ‖f‖Lp. For
2 ≤ p <∞ and ε > 0, take θ(s) = ( s1+εs)p/2−1. It follows that
λp
∫
Ω
|u|p(1 + ε|u|2)p/2−1
≤∫
Ω
|f |p.
Letting ε ↓ 0 and applying Fatou’s Lemma, we obtain u ∈ Lp and λ‖u‖Lp ≤ ‖f‖Lp .
Finally, we recall some convergence results. Given ε > 0, we define the operator Jε
on H−1(Ω) by
Jεu = (I − ε4)−1.
In other words, for every f ∈ H−1(Ω), uε = Jεf ∈ H10 (Ω) is the unique solution of
uε − ε4uε = f . It follows from what precedes that ‖Jε‖L(X,X) ≤ 1, whenever X =
H10 (Ω), L2(Ω), H−1(Ω), or X = Lp(Ω) for 1 ≤ p <∞. Furthermore, we have the following
result.
Proposition 2.4.2. If X is either of the spaces H10 (Ω), L2(Ω), H−1(Ω), then
(i) Jεf → f in X as ε ↓ 0, for every f ∈ X;
(ii) if fε is bounded in X as ε ↓ 0, then Jεfε − fε 0 in X as ε ↓ 0.
27
Proof. (i) By density, we may assume f ∈ H10 (Ω). Let uε = Jεf . One verifies easily
that
uε − f = Jε(f − (I − ε4)f) = εJε4f ;
and so,
‖uε − f‖H−1 ≤ ε‖4f‖H−1 −→ε↓0
0.
Hence (i) for X = H−1(Ω). Furthermore, uε is bounded in H10 (Ω), and it follows from
Remark 2.3.8 (iii) that
‖uε − f‖2L2 ≤ ‖uε − f‖H1‖uε − f‖H−1 ≤ Cε1/2−→
ε↓00.
Hence (i) for X = L2(Ω). Finally, one verifies easily that (I−4)uε = (I−4)f . Therefore,
from the result for X = H−1(Ω), it follows that (I −4)uε → (I −4)f in H−1(Ω), which
implies that uε → u in H10 (Ω). This completes the proof of (i).
(ii) Let uε = Jεfε. We know that uε is bounded in X; and so it suffices to show that
uε − fε → 0 in D′(Ω). Given ϕ ∈ D(Ω), we have
〈uε − fε, ϕ〉 = ε〈uε,4ϕ〉−→ε↓0
0.
Hence the result.
2.5. Semigroups of linear operators. Consult for example Brezis [2], Brezis and
Cazenave [1], Cazenave [5], Cazenave and Haraux [2], Haraux [2], Pazy [1].
Consider a complex Hilbert space X, with the norm ‖ · ‖X and the sesquilinear form
〈·, ·〉X . We consider X as a real Hilbert space with the scalar product (x, y)X = Re〈u, v〉X .
Let A : D(A) ⊂ X → X be a C-linear operator. Assume that A is self-adjoint
(so that D(A) is a dense subset of X) and that A ≤ 0 (i.e. (Ax, x) ≤ 0 for all x ∈D(A)). A generates a self-adjoint semigroup of contractions (S(t))t≥0 on X. D(A) is a
Hilbert space when equipped with the scalar product (x, y)D(A) = (Ax,Ay)X + (x, y)X ,
corresponding to the norm ‖u‖2D(A) = ‖Au‖2
X + ‖u‖2X . We have D(A) → X → (D(A))′,
all the embeddings being dense. We denote by XA the completion of D(A) for the norm
‖x‖2A = ‖x‖2
X − (Ax, x)X . XA is also a Hilbert space with the scalar product defined by
(x, y)A = (x, y)X − (Ax, y)X , for x, y ∈ D(A). We have
D(A) → XA → X → X ′A → (D(A))′,
28
all the embeddings being dense. Furthermore, it is easily shown that A can be extended
to a self-adjoint operator A on (D(A))′ with domain X. We have A|D(A) = A, and
A|D(A) ∈ L(D(A), X), A|XA∈ L(XA, X
′A), A|X ∈ L(X, (D(A))′).
Since A is self-adjoint, then, iA : D(A) ⊂ X → X defined by (iA)x = iAx, for
x ∈ D(A) is also C-linear, and is skew-adjoint. In particular, iA generates a group of
isometries (T (t))t∈R on X. It follows easily from the skew-adjointness of iA that
T (t)∗ = T (−t), for every t ∈ R.
It follows easily from the preceding observations that (T (t))t∈R can be extended to a
group of isometries (T (t))t∈R on (D(A))′, which is the group generated by the skew-
adjoint operator iA. T (t) coincides with T (t) on X, and (T (t))t∈R restricted to any of
the spaces X ′A, X, XA, D(A) is a group of isometries. For convenience, we use the same
notation for T (t) and T (t). We know that for every x ∈ X, u(t) = T (t)x is the unique
solution of the problem
u ∈ C(R, X) ∩ C1(R, (D(A))′);
idu
dt+ Au = 0, ∀t ∈ R;
u(0) = x.
In addition, we have the following regularity properties. If x ∈ XA, then u ∈ C(R, XA) ∩C1(R, X ′
A); if x ∈ D(A), then u ∈ C(R, D(A)) ∩ C1(R, X).
Concerning the non-homogeneous problem, we recall that for every x ∈ X and every
f ∈ C([0, T ], X) (where T ∈ R), there exists a unique solution of the problem
u ∈ C([0, T ], X)∩ C1([0, T ], (D(A))′);
idu
dt+ Au+ f = 0, ∀t ∈ [0, T ];
u(0) = x.
(2.5.1)
Indeed, u ∈ C([0, T ], X) is a solution of the above problem if and only if u satisfies the
following equation.
u(t) = T (t)x+ i
∫ t
0
T (t− s)f(s) ds, for all t ∈ [0, T ]. (2.5.2)
It is well known that if in addition f ∈ W 1,1((0, T ), X) (or if f ∈ L1((0, T ), D(A))), then
u ∈ C([0, T ], D(A))∩ C1([0, T ], X).
29
Remark 2.5.1. For every x ∈ (D(A))′ and f ∈ L1((0, T ), (D(A))′), (2.5.2) defines a
function u ∈ C([0, T ], (D(A))′). A natural question is to ask under what additional condi-
tions u satisfies an equation of type (1.5.1). Here are some answers.
(i) If in addition x ∈ X and if u ∈ W 1,1((0, T ), (D(A))′)) or u ∈ L1((0, T ), X), then u
satisfies (1.5.2) if and only if u satisfies
u ∈ L1((0, T ), X)∩W 1,1((0, T ), (D(A))′);
idu
dt+Au+ f = 0, almost everywhere on [0, T ];
u(0) = x.
(ii) If x ∈ X, f ∈ C([0, T ], (D(A))′) and if u ∈ C1([0, T ], (D(A))′)) or u ∈ C([0, T ], X),
then u satisfies (1.5.2) if and only if u satisfies (1.5.1).
(iii) Similarly, if x ∈ XA, f ∈ L1([0, T ], X ′A) and if u ∈ W 1,1((0, T ), X ′
A)) or if u ∈L1((0, T ), XA), then u satisfies (1.5.2) if and only if u satisfies
u ∈ L1((0, T ), XA) ∩W 1,1((0, T ), X ′A);
idu
dt+Au+ f = 0, almost everywhere on [0, T ];
u(0) = x.
(iv) If x ∈ XA, f ∈ C([0, T ], X ′A) and if u ∈ C1([0, T ], X ′
A)) or u ∈ C([0, T ], XA), then u
satisfies (1.5.2) if and only if u satisfies
u ∈ C([0, T ], XA) ∩ C1([0, T ], X ′A);
idu
dt+ Au+ f = 0, ∀t ∈ [0, T ];
u(0) = x.
(v) As well, if x ∈ D(A), f ∈ L1([0, T ], X) and if u ∈ W 1,1((0, T ), X)) or if u ∈L1((0, T ), D(A)), then u satisfies (1.5.2) if and only if u satisfies
u ∈ L1((0, T ), D(A))∩W 1,1((0, T ), X);
idu
dt+Au+ f = 0, almost everywhere on [0, T ];
u(0) = x.
30
(vi) If x ∈ D(A), f ∈ C([0, T ], X) and if u ∈ C1([0, T ], X)) or u ∈ C([0, T ], D(A)), then u
satisfies (1.5.2) if and only if u satisfies
u ∈ C([0, T ], D(A))∩ C1([0, T ], X);
idu
dt+Au+ f = 0, ∀t ∈ [0, T ];
u(0) = x.
31
3. The linear Schrodinger equation. This chapter is devoted to ths study of the
most important properties of the (linear) Schrodinger equation. We study in particular
the dispersive properties and the smoothing effect of the equation in Rn.
3.1. Basic properties. Let Ω be an open subset of Rn (Ω does not need to be smooth
or bounded). We define the operator A on L2(Ω) by
D(A) = u ∈ H1
0 (Ω),4u ∈ L2(Ω),
Au = 4u, for u ∈ D(A).
Evidently, D(A) = H2(Ω) ∩ H10 (Ω) if Ω is smooth enough. It is well known that A is
self-adjoint and ≤ 0; and so we can apply the results of Section 2.5. Observe that the
space XA is nothing else than H10 (Ω). Indeed ‖ · ‖A = ‖ · ‖H1
0, and D(Ω) ⊂ D(A). It
follows that H10 (Ω) ⊂ XA. Since also D(A) is a subset of H1
0 (Ω), it follows that XA ⊂H1
0 (Ω); and so XA = H10 (Ω), with equality of the norms. It follows that X ′
A = H−1(Ω).
On the other hand, note that D(A) 6= H20 (Ω); and so D(A)′ 6= H−2(Ω). The operator
A ∈ L(L2(Ω), (D(A))′) is simply defined by
〈Au, v〉(D(A))′,D(A) = (u,4v)L2, for u ∈ L2(Ω) and v ∈ D(A).
Let us denote by (T (t))t∈R the group of isometries generated by iA in any of the spaces
D(A), H10 (Ω), L2(Ω), H−1(Ω), (D(A))′. We have the following result.
Proposition 3.1.1. Let ϕ ∈ L2(Ω). Then u(t) = T (t)ϕ is the unique solution of the
problem
u ∈ C(R, L2(Ω)) ∩ C1(R, (D(A))′),
iut +4u = 0, in (D(A))′, for every t ∈ R,
u(0) = ϕ.
We have ‖u(t)‖L2 = ‖ϕ‖L2 , for every t ∈ R. If ϕ ∈ H10 (Ω), then u ∈ C(R, H1
0 (Ω)) ∩C1(R, H−1(Ω)) and ‖∇u(t)‖L2 = ‖∇ϕ‖L2 , for every t ∈ R.
Remark 3.1.2. It follows from Section 2.5 that T (t)∗ = T (−t), for every t ∈ R. On the
other hand, with the notation of Proposition 3.1.1, let v(t) = u(−t). It follows that
ivt +4v = 0,
v(0) = ϕ;
32
and so, T (−t)ϕ = T (t)ϕ, for every ϕ ∈ L2(Ω).
3.2. Dispersive properties in Rn. In this section, we consider the case Ω = Rn. We
begin with the following well-known result.
Proposition 3.2.1. If p ∈ [2,∞] and t 6= 0, then T (t) maps continuously Lp′(Rn) to
Lp(Rn) and
‖T (t)ϕ‖Lp(Rn) ≤ (4π|t|)−n(12− 1
p)‖ϕ‖Lp′(Rn),
for all ϕ ∈ Lp′(Rn).
The proof of Proposition 3.2.1 relies on the following lemma.
Lemma 3.2.2. Given t 6= 0, define the function Kt by
Kt(x) =
(1
4πit
)n2
ei|x|2
4t ,
for x ∈ Rn. Then
T (t)ϕ = Kt ∗ ϕ, (3.2.1)
for all t 6= 0 and all ϕ ∈ S(Rn).
Proof. Let ϕ ∈ S(Rn) and let u ∈ C(R,S(Rn)) be defined by
u(t)(ξ) = e−4π2i|ξ|2tϕ(ξ), for ξ ∈ Rn. (3.2.2)
It follows that
iut − 4π2|ξ|2u = 0, in R×Rn;
and so
iut +4u = 0, in R×Rn.
Since u(0) = ϕ, it follows that u(t) = T (t)ϕ. On the other hand, it is well known that
Kt(ξ) = e−i4π2|ξ|2t. Hence the result.
Proof of Proposition 3.2.1. Let ϕ ∈ S(Rn). It follows from Lemma 3.2.2 that
‖T (t)ϕ‖L∞ ≤ (4π|t|)−n2 ‖ϕ‖L1 .
33
By density, it follows that T (t) ∈ L(L1(Rn), L∞(Rn)) and that
‖T (t)‖L(L1(Rn),L∞(Rn)) ≤ (4π|t|)−n2 .
The general case is obtained by interpolation between the cases p = 2 and p = ∞ (use
Riesz’ interpolation theorem).
Remark 3.2.3.
(i) It follows from formula (3.2.1) that
T (t)ϕ(x) =
(1
4πit
)n/2
ei|x|2
4t
∫e−
ix·y2t e
i|y|2
4t ϕ(y) dy.
In other words, up to a rescaling and to multiplication by a function of modulus
1, T (t) is nothing else but the Fourier transform. In particular, the estimates of
Proposition 3.2.1 are optimal in the sense that if T (t) ∈ L(Lq, Lp), then necessarily
2 ≤ p ≤ ∞ and q = p′.
(ii) It follows in particular from formula (3.2.2) that for every ϕ ∈ S(Rn), we have T (·)ϕ ∈C(R,S(Rn)). By duality, T (t) can be extended to S ′(Rn), and T (·)ϕ ∈ C(R,S ′(Rn))
for every ϕ ∈ S ′(Rn).
(iii) If ϕ ∈ Hs(Rn) for some s ∈ R, then u(t) = T (t)ϕ verifies
u ∈⋂
0≤j<∞Cj(R, Hs−2j(Rn)),
as follows immediately from formula (3.2.2) and the definition of the Sobolev spaces
Hs(Rn) by the Fourier transform.
The estimates of Proposition 3.2.1 are remarkable but they are not quite handy for
solving the nonlinear problems, since the Lp spaces are not stable by T (t). However,
we will derive from those estimates, space-time estimates that are essential for solving
the nonlinear Schrodinger equations. The first estimates of that kind were obtained by
Strichartz [1] by Fourier transform methods. Strichartz’ estimates were generalized by
Ginibre and Velo [5], who gave a remarkable, elementary proof. Strichartz’ estimate for
the nonhomogeneous problem was generalized by Yajima [1] and then by Cazenave and
34
Weissler [3], following essentially the proof of Ginibre and Velo. The main estimate is the
following.
Definition 3.2.4. In all what follows, we say that a pair (q, r) is admissible if the fol-
lowing holds.
(i) 2 ≤ r <2n
n− 2(2 ≤ r ≤ ∞ if n = 1, 2 ≤ r ≤ ∞ if n = 2);
(ii)2
q= n
(1
2− 1
r
).
Remark. It follows that if (q, r) is an admissible pair, then 2 < q ≤ ∞. The pair (∞, 2)
is always admissible.
Theorem 3.2.5. The following properties hold.
(i) For every ϕ ∈ L2(Rn) and for every admissible pair (q, r), the function t 7→ T (t)ϕ
belongs to Lq(R, Lr(Rn)) ∩ C(R, L2(Rn)). Furthermore, there exists a constant C,
depending only on q such that
‖T (·)ϕ‖Lq(R,Lr) ≤ C‖ϕ‖L2 ,
for every ϕ ∈ L2(Rn);
(ii) let I be an interval of R (bounded or not), let J = I and let t0 ∈ J . Let (γ, ρ) be an
admissible pair, and let f ∈ Lγ′(I, Lρ′(Rn)). Then, for every admissible pair (q, r),
the function
t 7→ Φf (t) =
∫ t
t0
T (t− s)f(s) ds, for t ∈ I,
belongs to Lq(I, Lr(Rn)) ∩ C(J, L2(Rn)). Furthermore, there exists a constant C,
depending only on γ and q such that
‖Φf‖Lq(I,Lr) ≤ C‖f‖Lγ′(I,Lρ′ ),
for every f ∈ Lγ′(I, Lρ′(Rn)).
Remark 3.2.6. Theorem 3.2.5 deserves a few comments. One is easily convinced that
property (i) describes a quite remarkable smoothing effect. Indeed, for all t ∈ R, T (t)L2 =
L2. In particular, given t 6= 0 and p > 2, there exists a dense subset Ep of L2 such that
35
T (t)ϕ 6∈ Lp, for every ϕ ∈ Ep. However, it follows from property (i) that for every ϕ ∈ L2,
T (t)ϕ ∈ Lp, for almost all t ∈ R. Note that by the preceding observation, the restriction
“for almost all t ∈ R” cannot be reduced to “for all t 6= 0” in general. Note also that by
considering a sequence (pn)n∈N ⊂ [2,2n
n− 2) such that pn →
2n
n− 2as n → ∞, it follows
that given ϕ ∈ L2, there exists a set Nϕ ⊂ R of measure 0 such that for every t ∈ R \Nϕ,
one has T (t)ϕ ∈ Lp for every p ∈ [2,2n
n− 2) (p ∈ [2,∞], if n = 1).
Concerning property (ii), note that the definition of Φf makes sense. Indeed, Lρ′ →H−1, and so f ∈ L2(I ′, H−1), for every bounded interval I ′ ⊂ I. In particular, we have
Φf ∈ C(I ′, H−1). Evidently, properties (i) and (ii) give an estimate of the solution of the
nonhomogeneous Schrodinger equation
iut +4u+ f = 0,
u(0) = ϕ,
in terms of f and ϕ.
Proof of Theorem 3.2.5. We divide the proof into seven steps, and we first establish
property (ii). For convenience, we assume that I = [0, T ), for some T ∈ (0,∞) and that
t0 = 0, the proof being the same in the general case. It is convenient to define, in the same
way as Φ, the operators Ψ and Θt (where t ∈ (0, T ) is a parameter) by
Ψf (s) =
∫ T
s
T (s− t)f(t) dt, ∀s ∈ [0, T ),
and
Θt,f (s) =
∫ t
0
T (s− σ)f(σ) dσ, ∀s ∈ [0, T ).
It is clear that both Ψ and Θt map continuously L1loc([0, T );H−1) to C([0, T ), H−1).
Step 1. For every admissible pair (q, r), Φ ∈ L(Lq′(0, T ;Lr′(Rn)), Lq(0, T ;Lr(Rn))),
with a norm depending only on q. By density, it is sufficient to consider the case f ∈Cc([0, T ), Lr′). In this case, Proposition 3.2.1 shows that Φf ∈ C([0, T ), Lr), and that
‖Φf (t)‖Lr ≤∫ t
0
|t− s|−n( 12− 1
r )‖f(s)‖Lr′ ds ≤∫ T
0
|t− s|−2q ‖f(s)‖Lr′ ds.
It follows from the classical Riesz’ potential inequalities (cf Stein [1], Theorem 1, p.119)
that
‖Φf‖Lq(0,T ;Lr) ≤ C‖f‖Lq′(0,T ;Lr′),
36
where C depends only on q.
Step 2. By the same argument, one shows that both Ψ and Θt are continuous from
Lq′(0, T ;Lr′(Rn)) to Lq(0, T ;Lr(Rn)), with norms depending only on q.
Step 3. For every admissible pair (q, r), Φ ∈ L(Lq′(0, T ;Lr′(Rn)), C([0, T ], L2(Rn))),
and its norm depends only on q. By density, it is sufficient to consider the case f ∈Cc([0, T ), Lr′). By using the embedding Lr′ → H−1 and applying the operator (I −ε4)−1, one may thus assume that f ∈ Cc([0, T ), Lr′) ∩ Cc([0, T ), L2). It follows that
Φf ∈ C([0, T ), L2); and so,
‖Φf (t)‖2L2 = (
∫ t
0
T (t− s)f(s) ds,
∫ t
0
T (t− σ)f(σ) dσ)L2
=
∫ t
0
∫ t
0
(T (t− s)f(s), T (t− σ)f(σ))L2 dσ ds
=
∫ t
0
∫ t
0
(f(s), T (s− σ)f(σ))L2 dσ ds =
∫ t
0
(f(s),Θt,f(s))L2 ds,
where we used the property T (t)∗ = T (−t). Applying Holder’s inequality in space, then
in time, and applying Step 2, it follows that
‖Φf (t)‖2L2 ≤ ‖f‖Lq′(0,T ;Lr′)‖Θt,f‖Lq(0,T ;Lr) ≤ C(q)‖f‖2
Lq′(0,T ;Lr′).
Hence the result, since t is arbitrary.
Step 4. By the same argument, one shows that both Ψ and Θt are continuous from
Lq′(0, T ;Lr′(Rn)) to C([0, T ], L2(Rn)), with norms depending only on q.
Step 5. For every admissible pair (q, r), Φ ∈ L(L1(0, T ;L2(Rn)), Lq(0, T ;Lr(Rn))), and
its norm depends only on q. Let f ∈ L1(0, T ;L2) and consider ϕ ∈ Cc([0, T ),D(Rn)).
We have ∫ T
0
(Φf (t), ϕ(t))L2 dt =
∫ T
0
∫ t
0
(T (t− s)f(s), ϕ(t))L2 ds dt
=
∫ T
0
∫ T
s
(f(s), T (s− t)ϕ(t))L2 dt ds
=
∫ T
0
(f(s),Ψϕ(s))L2 ds;
and so, by Cauchy-Schwartz and Step 4,
|∫ T
0
(Φf (t), ϕ(t))L2 dt| ≤ ‖f‖L1(0,T ;L2)‖Ψϕ‖L∞(0,T ;L2)
≤ C(q)‖f‖L1(0,T ;L2)‖ϕ‖Lq′(0,T ;Lr′).
(3.2.3)
37
On the other hand, one verifies easily by choosing appropriate test functions that for every
g ∈ L1(0, T ;L2(Rn)), one has
‖g‖Lq(0,T ;Lr(Rn)) = sup∫ T
0
(g(t), ϕ(t))L2 dt;
ϕ ∈ Cc([0, T ),D(Rn)), ‖ϕ‖Lq′(0,T ;Lr′(Rn)) = 1.
The result follows from (3.2.3), and the above relation applied with g = Φf .
Step 6. Proof of (ii). Let (γ, ρ) be an admissible pair. It follows from steps 1 and 3 that
Φ is continuous from Lγ′(0, T ;Lρ′(Rn)) to L∞(0, T ;L2(Rn)) and from Lγ′(0, T ;Lρ′(Rn))
to Lγ(0, T ;Lρ(Rn)). Consider an admissible pair (q, r) for which 2 ≤ q ≤ ρ, and let
θ ∈ [0, 1] be such that1
r=θ
ρ+
1− θ
2, and
1
q=θ
γ+
1− θ
∞ .
By applying Holder’s inequality in space, then in time, we obtain
‖Φf‖Lq(0,T ;Lr) ≤ ‖Φf‖θLγ(0,T ;Lρ)‖Φf‖1−θ
L∞(0,T ;L2) ≤ C‖f‖Lγ′ (0,T ;Lρ′),
where C depends only on γ and q. Therefore, Φ maps continuously Lγ′(0, T ;Lρ′(Rn)) to
Lq(0, T ;Lr(Rn)).
Let now (q, r) be an admissible pair for which ρ < r and let µ ∈ [0, 1] be such that
1
γ′=µ
1+
1− µ
q′and
1
ρ′=µ
2+
1− µ
r′.
By steps 1 and 5, Φ is continuous from Lq′(0, T ;Lr′(Rn)) to Lq(0, T ;Lr(Rn)) and from
L1(0, T ;L2(Rn)) to Lq(0, T ;Lr(Rn)). By Interpolation (see Bergh and Lofstrom [1], The-
orem 5.1.2 p.107), Φ is continuous Lσ(0, T ;Lδ(Rn)) → Lq(0, T ;Lr(Rn)) for every pair
(σ, δ) such that, for some θ ∈ [0, 1],
1
σ=θ
1+
1− θ
q′and
1
δ=θ
2+
1− θ
r′.
The result follows by choosing θ = µ.
Step 7. Proof of (i). The proof is parallel to the proof of (ii), and we describe only the
main steps. Let
Λf (t) =
∫ +∞
−∞T (t− s)f(s) ds, and Γf =
∫ +∞
−∞T (−t)f(t) dt.
38
One shows (see Step 1) that
‖Λf‖Lq(0,T ;Lr) ≤ C(q)‖f‖Lq′(0,T ;Lr′),
for every admissible pair (q, r). It follows (see Step 3) that
‖Γf‖L2 ≤ C(q)‖f‖Lq′(0,T,Lr′),
from which one obtains easily that
|∫ +∞
−∞(T (t)ϕ, ψ(t))L2 dt| = (ϕ,
∫ +∞
−∞T (−t)ψ(t) dt)L2 ≤ C(q)‖ϕ‖L2‖ψ‖Lq′(0,T ;Lr′),
for every ϕ ∈ L2(Rn) and ψ ∈ Cc([0, T ),D(Rn)). (ii) follows easily (see Step 5). This
completes the proof.
Corollary 3.2.7. Let I = (T,∞), for some T ≥ −∞ (respectively I = (−∞, T ), for
some T ≤ ∞) and let J = I. Let (γ, ρ) be an admissible pair, and let f ∈ Lγ′(I, Lρ′(Rn)).
Then, the function
t 7→ Φf (t) =
∫ ∞
t
T (t− s)f(s) ds, for every t ∈ J,
(respectively Φf (t) =
∫ −∞
t
T (t− s)f(s) ds, for every t ∈ J,)
makes sense as the uniform limit in L2(Rn), as m → +∞ (respectively, as m → −∞) of
the functions
Φmf (t) =
∫ m
t
T (t− s)f(s) ds, for every t ∈ J.
In addition, for every admissible pair (q, r), Φf ∈ Lq(I, Lr(Rn))∩C(J, L2(Rn)). Further-
more, there exists a constant C, depending only on γ and q such that
‖Φf‖Lq(I,Lr) ≤ C‖f‖Lγ′ (I,Lρ′ ),
for every f ∈ Lγ′(I, Lρ′(Rn)).
Proof. We consider for example the case I = (T,∞). Let j,m be two integers, T < j < m.
For every t ∈ J , we have
‖Φmf (t)− Φj
f (t)‖L2 = ‖T (m− t)(Φmf (t)− Φj
f (t))‖L2 = ‖∫ m
j
T (m− s)f(s) ds‖L2.
39
By Theorem 3.2.5, there exists a constant C(γ), such that
‖Φmf (t)− Φm
f (t)‖L2 ≤ C‖f‖Lγ′ (j,∞;Lρ′).
It follows that Φm is a Cauchy sequence in L∞(I, L2(Rn)). Therefore, Φf ∈ C(J, L2(Rn)),
and
‖Φf‖L∞(I,L2) ≤ C‖f‖Lγ′ (I,Lρ′ ). (3.2.4)
Finally, given any admissible pair (q, r), it follows from Theorem 3.2.5 that there exists a
constant C(γ, q) such that
‖Φmf ‖Lq(I,Lr) ≤ C‖f‖Lγ′ (I,Lρ′ ). (3.2.5)
For j ∈ N, j ≥ T , define fj ∈ Lγ′(I, Lρ′(Rn)) by
fj(t) =
f(t), if t ≤ j;
0, if t > j.
We have fj → f in Lγ′(I, Lρ′(Rn)), as j →∞. In particular, by (3.2.4),
Φfj→ Φf in L2(Rn), uniformly in t ∈ J. (3.2.6)
Note that for m ≥ j, Φmfj
is independent of m. It follows from (3.2.5) that Φfj∈
Lγ′(I, Lρ′(Rn)). Furthermore, taking T ≤ j ≤ k, it follows from (3.2.5) that
‖Φfj− Φfk
‖Lq(I,Lr) ≤ C‖fj − fk‖Lγ′ (I,Lρ′ ) ≤ C‖f‖Lγ′ (j,k;Lρ′).
In particular, Φfjis a Cauchy sequence in Lγ′(I, Lρ′(Rn)), which posesses a limit ψ such
that, by (3.2.5),
‖ψ‖Lq(I,Lr) ≤ C‖f‖Lγ′ (I,Lρ′ ). (3.2.7)
Therefore, there exists a subsequence, which we still denote by fj , such that Φfj(t) → ψ(t)
in Lr(Rn), for almost all t ∈ I. Applying (3.2.6), it follows that ψ(t) = Φf (t) almost
everywhere on I, and the result follows from (3.2.7).
Corollary 3.2.8. Let ϕ ∈ H1(Rn) and let r ∈ (2,2n
n− 2) (r ∈ (2,∞), if n = 2, r ∈ (2,∞],
if n = 1). Then, ‖T (t)ϕ‖Lr → 0, as t→ ±∞.
40
Proof. Let q be such that (q, r) is an admissible pair. It follows from Theorem 2.3.7 that
there exists C such that for every t, s ∈ R,
‖u(t)− u(s)‖Lr ≤ C‖u(t)− u(s)‖2q
H1‖u(t)− u(s)‖q−2
q
L2 .
Since ϕ ∈ H1(Rn), u(t) is bounded in H1(Rn); and so,
‖u(t)− u(s)‖Lr ≤ C‖u(t)− u(s)‖q−2
q
L2 .
Furthermore, by Proposition 3.1.1, ut is bounded in H−1(Rn); and so, u is globally Lip-
schitz continuous R → H−1(Rn) (see Theorem 2.3.9). It follows (see Remark 2.3.8, (iii))
that there exists C such that
‖u(t)− u(s)‖L2 ≤ C|t− s|1/2.
Therefore,
‖u(t)− u(s)‖Lr ≤ C|t− s|q−22q .
In particular, u : R → Lr(Rn) is uniformly continuous. The result now follows from the
property u ∈ Lq(R, Lr(Rn)) (Theorem 3.2.5), since q <∞.
3.3. Smoothing effect in Rn. We still assume in this section that Ω = Rn. We have
seen in Proposition 3.2.1 and Theorem 3.2.5 that T (t) has a smoothing effect in some Lp
spaces. On the other hand, one verifies easily with the formula of Lemma 3.2.2 that for
every ϕ ∈ L1(Rn) supported in a compact subset Ω of Rn, the function (t, x) 7→ T (t)ϕ(x) is
analytic in (0,+∞)×Rn. In other words, T (t) being essentially the Fourier transform (see
Remark 3.2.3 (i)), maps functions having a nice decay as |x| → ∞ to smooth functions. In
this section, we establish precise estimates describing this smoothing effect, which enable
us to prove similar results in the nonlinear case. Let us first introduce some notation. For
j ∈ 1, · · · , n, let Pj be the partial differential operator on Rn+1 defined by
Pju(t, x) = (xj + 2it∂j)u(t, x) = xju(t, x) +∂ju
∂xj(t, x). (3.3.1)
For a multi-index α, we define the partial differential operator Pα on Rn+1 by
Pα =
n∏
i=1
Pαi
i . (3.3.2)
41
Furthermore, for x ∈ Rn, we set
xα =n∏
i=1
xαi
i . (3.3.3)
Consider a smooth function u : Rn+1 → C. An easy calculation shows that
Pju(t, x) = 2itei|x|2
4t∂
∂xj
(e−i
|x|2
4t u
),
from which it follows by an obvious recurrence argument that
Pαu(t, x) = (2it)|α|ei|x|2
4t Dα
(e−i
|x|2
4t u
). (3.3.4)
On the other hand, a formal calculation shows that
[Pα, i∂t +4] = 0, (3.3.5)
where [·, ·] is the commutator bracket. In other words, if u is a smooth solution of the
linear Schrodinger equation, then so is Pαu. In particular, if we consider ϕ ∈ S(Rn) and
if we set u(t) = T (t)ϕ, then uα = Pαu is a solution of Schrodinger’s equation. It follows
that
uα(t) = T (t)uα(0) = T (t)xαϕ; (3.3.6)
and so ‖uα‖L2 = ‖xαϕ‖L2 . By (3.3.4), this means that
(2|t|)|α|‖Dα
(e−i
|x|2
4t u(t)
)‖L2 = ‖xαϕ‖L2 . (3.3.7)
By density, we obtain immediately the following result.
Proposition 3.3.1. Let α be a multi-index. Let ϕ ∈ S ′(Rn) be such that xαϕ ∈ L2(Rn),
and let u(t) = T (t)ϕ ∈ C(R,S ′(Rn)). Then, Dαe−i|x|2
4t u(t) ∈ C(R \ 0, L2(Rn)) and
formula (3.3.7) holds for every t 6= 0.
Corollary 3.3.2. Let ϕ ∈ L2(Rn), and assume that for some nonnegative integer m,
we have (1 + |x|m)ϕ ∈ L2(Rn). Then, e−i|x|2
4t u(t) ∈ C(R \ 0, Hm(Rn)), and if k is the
integer part of m/2, we have
u ∈⋂
0≤j≤k
Cj(R \ 0, Hm−2jloc (Rn)).
42
In particular, if (1 + |x|m)ϕ ∈ L2(Rn) for every nonnegative integer m, then u ∈ C∞(R \0 ×Rn).
Proof. The Hm-regularity of e−i|x|2
4t u(t) follows from Proposition 3.3.1. Since ei|x|2
4t ∈C∞(R \ 0 ×Rn), it follows that u ∈ C(R \ 0, Hm
loc(Rn)). The regularity of the time
derivatives follows from the equation.
Remark 3.3.3. Formula (3.3.6) means that PαT (t)ϕ = T (t)(xαϕ), or alternatively, by
setting ϕ = T (−t)ψ, T (−t)Pαψ = xαT (−t)ψ. In particular, (x + 2it∇)T (t) = T (t)x, or
equivalently T (−t)(x+ 2it∇) = xT (−t).
Corollary 3.3.4. Let ϕ ∈ L2(Rn) be such that | · |ϕ(·) ∈ L2(Rn), and let u(t) = T (t)ϕ.
Then,
(i) the function t 7→ (x+2it∇)u(t, x) belongs to Lq(R, Lr(Rn)) for every admissible pair
(q, r);
(ii) for every r ∈ [2,2n
n− 2] (r ∈ [2,∞), if n = 2, r ∈ [2,∞), if n = 1), we have u ∈
C(R/0, Lr(Rn)) and there exists C, depending only on r and n such that
‖u(t)‖Lr ≤ C(‖ϕ‖L2 + ‖xϕ‖L2)|t|−n( 12− 1
r ),
for every t 6= 0.
Proof. By (3.3.1) and (3.3.6), we have (x+2it∇)u(t, x) = T (t)ψ, where ψ(x) = xϕ(x); and
so, (i) follows from Theorem 3.2.5. Consider now the function v(t, x) = e−i
(|x|2
4t
)u(t, x).
It follows from (3.3.6) and (3.3.7) that ∇v ∈ C(R/0, L2(Rn)) and that
‖∇v(t)‖L2 ≤ C|t|−1‖xϕ‖L2 .
The result follows from Gagliardo-Nirenberg’s inequality, since |u(t, x)| ≡ |v(t, x)|.
3.4. Comments. As we will see in Chapter 4, Theorem 3.2.5 is the crux for proving
the local well-posedness of the Cauchy problem in the energy space or in L2(Ω), when
Ω = Rn. Therefore, it is natural to ask whether Theorem 3.2.5 can be generalized to a
43
wider class of equations. In fact, a careful analysis of the proof shows that it uses only
two properties. The first one is the identity T (t)∗ = T (−t), which is valid for every skew-
adjoint generator. The second one is the estimate of Proposition 3.2.1, which itself follows
from Lemma 3.2.2. Therefore, such an inequality holds whenever T (t) has a kernel K(t)
whose L∞-norm behaves like |t|−n/2 (at least near 0). More precisely, we have the following
result.
Theorem 3.4.1. Let A be a self-adjoint, ≤ 0 operator on X = L2(Ω). Assume that
there exists t0 > 0 such that for every t ∈ (−t0, 0) ∪ (0, t0), T (t) = eitA maps L1(Ω) to
L∞(Ω), with a norm less than K|t|−n/2. The following properties hold.
(i) For every ϕ ∈ L2(Ω) and for every admissible pair (q, r), the function t 7→ T (t)ϕ
belongs to Lqloc(R, L
r(Ω)). Furthermore, there exists a constant C, depending only
on K and q such that
‖T (·)ϕ‖Lq(−T,T ;Lr) ≤ C
(1 + T
t0
)1/q
‖ϕ‖L2 ,
for every ϕ ∈ L2(Rn) and every T > 0;
(ii) let 0 < |T | <∞, let (γ, ρ) be an admissible pair, and let f ∈ Lγ′(0, T ;Lρ′(Ω)). Then,
for every admissible pair (q, r), the function
t 7→ Φf (t) =
∫ t
0
T (t− s)f(s) ds
belongs to Lq(0, T ;Lr(Ω)). Furthermore, there exists a constant C, depending only
on K, γ and q such that
‖Φf‖Lq(0,T ;Lr) ≤ C
(1 + |T |t0
)1/q
‖f‖Lγ′(0,T ;Lρ′),
for every f ∈ Lγ′(0, T ;Lρ′(Ω)). In addition, Φf ∈ C([0, T ], L2(Ω)).
Proof. Following the proof of Theorem 3.2.5, one shows that estimates (i) and (ii) hold
for T = t0. In particular, assuming q <∞,
∫ t0
0
‖T (t)ϕ‖qLr dt ≤ C‖ϕ‖q
L2 .
44
It follows that for every positive integer k,
∫ (k+1)t0
kt0
‖T (t)ϕ‖qLr dt ≤ C‖T (kt0)ϕ‖q
L2 ≤ C‖ϕ‖qL2 .
In particular, ∫ kt0
−kt0
‖T (t)ϕ‖qLr dt ≤ Ck‖ϕ‖q
L2 .
Hence (i). One proves (ii) by the same argument.
In view of Theorem 3.4.1, it is interesting to know when eitA satisfies the estimate of
Lemma 3.2.2 (for possibly small times). In the following remarks, we collect some results
in that direction.
Remark 3.4.2. The estimates of Proposition 3.2.1 do not hold if Ω is a bounded set, for
any p > 2. The reason is that in this case, L2(Ω) → Lp′(Ω); and so, if such an estimate
would hold, then I = T (t)T (−t) would map L2(Ω) → L2(Ω) → Lp′(Ω) → Lp(Ω). This is
absurd, since this would mean that L2(Ω) → Lp(Ω). However, note that the estimates of
Proposition 3.2.1 might hold if for example Ω is the complement of a star-shaped domain.
Unfortunately, such a result is apparently unknown (see Hayashi [2]).
Remark 3.4.3. The estimates of Proposition 3.2.1 hold when one replaces the Laplacian
by a more general pseudo differential operator on Rn (see Balabane [1,2], Balabane and
Emami Rad [1,2]).
Remark 3.4.4. Assume that Ω = Rn and consider A = 4 − V , where V : Rn →R is a given potential. If the negative part of V is not too large, then A defines a
self-adjoint opearator on L2(Rn) (see for example Kato [2]). If V is small enough in
L1∩L∞, then it follows from a perturbation method that T (t) = eitA verifies the estimate
of Proposition 3.2.1 (see Schonbek [1]). More general cases are considered in Journe, Soffer
and Sogge [1].
As well, if V ∈ C∞(Rn) is nonnegative and if DαV ∈ L∞(Rn) for all |α| ≥ 2 (the
model case is V (x) = |x|2), then also T (t) = eitA verifies the estimate of Proposition 3.2.1
(see Fujiwara [1,2], A. Weinstein [1], Zelditch [1], Oh [1,2]).
On the other hand, such estimates do not hold in general, for several reasons. First of
all, A may have eigenvalues. Therefore, if λ is an eigenvalue of A and if ϕ is a corresponding
45
eigenvector, then T (t)ϕ = eiλtϕ, and so T (t)ϕ does not decay as |t| → ∞. But there is
a more subtle reason that prevents the estimates of Proposition 3.2.1 to hold. Even if
one removes the eigenvectors, that is if one works in the supplement (in L2) of the space
spanned by the eigenvectors, then a resonnance effect can occur, even for short range (i.e.
localized) potentials. The reader should consult on that subject the very interesting papers
of Rauch [1], Jensen and Kato [1], Murata [1].
Remark 3.4.5. Such estimates as those of Proposition 3.2.1 hold for the Schrodinger
equation with an external magnetic potential. See Chapter 9.
Remark 3.4.6. The estimates of Proposition 3.2.1 (hence these of Theorem 3.2.5) hold
for certain subdomains Ω of Rn. For example, they hold if Ω = Rn+. To see this, consider
ϕ ∈ D(Rn+), and let ϕ be defined by
ϕ(x1, · · · , xn) =
ϕ(x1, · · · , xn), if xn > 0;
− ϕ(x1, · · · ,−xn), if xn < 0.
It follows that ϕ ∈ D(Rn). Let u = T (t)ϕ, where T (t) is the group of isometries generated
by i4 in Rn. One verifies easily, by uniqueness, that u|Rn+
= T (t)ϕ, where T (t) is the
group of isometries generated by i4 in Rn+. The result follows, by Proposition 3.2.1 and
density. This applies in particular to the case where Ω ⊂ R is a half-line.
More generally, one can repeat this argument and obtain the estimates of Propo-
sition 3.2.1 when Ω is a cone of Rn of a certain type. For example, the estimates of
Proposition 3.2.1 hold when Ω ⊂ R2 is defined by Ω = ρeiθ; ρ > 0, 0 < θ < π/2m, for
some nonnegative integer m.
Remark 3.4.7. In addition to the smoothing effects of Sections 3.2 and 3.3, a third kind
of smoothing effect was discovered. It says that for every ϕ ∈ L2(Rn), then u(t) = T (t)ϕ
belongs to H1/2loc (Rn) for almost all t ∈ R. It was discovered independently by Constantin
and Saut [1,2,3], Sjolin [1] and Vega [1]. See also Ben Artzi and Devinatz [1], Ben Artzi and
Klainerman [1], Kato and Yajima [1] for further developments, as well as Kenig, Ponce and
Vega [1] for a related smoothing effect. A typical result in this direction is the following
(see Ben Artzi and Klainerman [1] for a rather simple proof): There exists a constant C
46
such that for every ϕ ∈ L2(Rn), u(t) = T (t)ϕ verifies
∫ +∞
−∞
∫
Rn
1
(1 + |x|2) |Pu(t, x)|2 dx dt ≤ C‖ϕ‖2
L2 ,
where P = (I − 4)1/4 is the pseudo-differential operator defined by P u(ξ) = (1 +
4π2|ξ|2)1/4u(ξ). One can obtain similar estimates for the nonhomogeneous problem. More
precisely, if f ∈ L2([0, T ], L2(Rn)), then u ∈ L2([0, T ], H1/2(B)) for every bounded open
set B ⊂ Rn (see Constantin and Saut [1,2,3]). Therefore, there is locally a gain of half a
derivative. As a matter of fact, if one is willing to reverse the time and space integrations,
then the gain is one derivative. More precisely, if Qαα∈Zn is a family of disjoint open
cubes of size R such that Rn = ∪α∈Zn
Qα, then (see Kenig, Ponce and Vega [2])
supα∈Zn
(∫
Qα
∫ +∞
−∞|∇u(t, x)|2dt dx
)1/2
≤ CR∑
α∈Zn
(∫
Qα
∫ +∞
−∞|f(t, x)|2dt dx
)1/2
.
See also Ruiz and Vega [1] for related estimates. Such results also hold for A = 4−V , for
some potentials V (see Constantin and Saut [3]), and also for some nonlinear Schrodinger
equations (see Constantin and Saut [1,2], Kenig, Ponce and Vega [2]).
Remark 3.4.8. We do not know if the estimates of Theorem 3.2.5 hold in the limiting
case r =2n
n− 2, q = 2. However, a similar estimate holds with the space and time
integration reversed. More precisely, we have
(∫
Rn
(∫ +∞
−∞|u(t, x)|2 dt
) nn−2
dx
)n−22n
≤ C‖ϕ‖L2 ,
for every ϕ ∈ L2(Rn), that is ‖u‖L
2nn−2 (Rn,L2(R))
≤ C‖ϕ‖L2 (see Ruiz and Vega [1]).
47
4. The local Cauchy problem. Our goal in this chapter is to give a (relatively) simple
proof of local existence of solutions in the energy space, which can be adapted to most
situation where local existence in that space is known. There is a wide litterature on this
subject. The study of the Cauchy problem in the energy space was initiated by Ginibre
and Velo [1] and [2] for local nonlinearities, and by Ginibre and Velo [3] for nonlocal
nonlinearities of Hartree type. Further results can be found in Ginibre and Velo [5,8],
Kato [1,3], Cazenave [5,2], Cazenave and Haraux [2,1], Cazenave and Weissler [1,2,3].
In Section 4.1, we establish a preliminary abstract result. In Section 4.2, we prove
our main local existence result, which we apply in the following sections to the nonlin-
ear Schrodinger equation in Rn and to some special cases in R and R2. The results of
Section 4.2 will also be applied in Chapter 9 to some generalizations.
4.1. An abstract nonlinear Schrodinger equation. In this section, we continue the
notation as introduced in Section 2.5. X is a complex Hilbert space with the real scalar
product (·, ·)X . A is a C-linear, self-adjoint ≤ 0 operator on X with domain D(A). XA
is the completion of D(A) for the norm ‖x‖2A = ‖x‖2
X − (Ax, x)X , X ′A = (XA)′ and A
is the extention of A to (D(A))′. Finally, T (t) is the group of isometries generated on
either (D(A))′, X ′A, X, XA or D(A) by the skew-adjoint operator iA. We establish a local
existence result for the problem
iut + Au+ g(u) = 0,
u(0) = x;
where x is a given initial datum and g is a (weak) nonlinearity. More precisely, we assume
that
g ∈ C(X,X) is Lipschitz continuous on bounded sets of X; (4.1.1)
there exists a functional G ∈ C1(XA,R) such that
G′(x) = g(x), for every x ∈ XA;
(4.1.2)
(g(x), ix)X = 0, for every x ∈ X. (4.1.3)
For x ∈ XA, we define
E(x) =1
2(‖x‖2
A − ‖x‖2X)−G(x) = −(Ax, x)X −G(x). (4.1.4)
48
It follows easily that E ∈ C1(XA,R), and that
E′(x) = −Ax− g(x) ∈ X ′A, for every x ∈ XA.
The main result of this section is the following.
Theorem 4.1.1. Let g satisfy assumptions (4.1.1) through (4.1.3). For every x ∈ X,
there exists a unique solution u of the problem
u ∈ C(R, X) ∩ C1(R, (D(A))′);
idu
dt+ Au+ g(u) = 0, ∀t ∈ R;
u(0) = x.
(4.1.5)
In addition, the following properties hold.
(i) ‖u(t)‖X = ‖x‖X , for every t ∈ R (conservation of charge);
(ii) if furthermore x ∈ XA, then u ∈ C(R, XA) ∩ C1(R, X ′A) and E(u(t)) = E(x), for
every t ∈ R (conservation of energy);
(iii) if in addition x ∈ XA, then also u ∈ C(R, D(A)) ∩ C1(R, X).
Proof. Step 1. It is well known that for every x ∈ X, there exists a unique, maxi-
mal solution u ∈ C((−T1, T2), X) of (4.1.5). u is maximal in the sense that if Ti = ∞(for i = 1, 2) then ‖u(t)‖X → ∞, as t → Ti. In addition, if x ∈ D(A), then u ∈C((−T1, T2), D(A)) ∩ C1((−T1, T2), X). Furthermore, u depends continuously on x in
X, uniformly on compact subsets of the maximal existence interval. This follows essen-
tially from Segal [1] (see Cazenave [5], Cazenave and Haraux [2], Brezis and Cazenave [1],
Pazy [1]).
Step 2. Assume x ∈ D(A), and take the scalar product of the equation with iu. It follows
that
(ut, u)X = (iut, iu)X = −(Au, iu)x − (g(u), iu).
The first term of the right hand side is 0 by self-adjointness, and the second is also 0 by
(4.1.3). Therefore,d
dt‖u(t)‖2
X = 2(ut, u)X = 0.
49
Hence the conservation of charge. Multiplying the equation by ut, we obtain
0 = (iut, ut)X = (−Au, ut)X − (g(u), ut)X .
It follows thatd
dtE(u(t)) = 0.
Hence the conservation of energy.
Step 3. By Step 2 and continuous dependence, we obtain conservation of charge when
x ∈ X. Therefore ‖u(t)‖X is uniformly bounded on the maximal existence interval; and
so the solution exists on (−∞,∞). Hence (i) and (ii)
Step 4. Let x ∈ XA, and let xn ∈ D(A) converge to x in XA, as n → ∞. We denote by
un the solution of (4.1.5) with initial datum xn. By (i), un is bounded in L∞(R, X); and
so, G(un) is uniformly bounded. It follows from conservation of energy that un is bounded
in L∞(R, XA), and by the equation, (un)t is bounded in L∞(R, X ′A). On the other hand,
it follows from continuous dependence that for every t ∈ R, un(t) → u(t) as n → ∞,
strongly in X, hence weakly in XA. It follows that u ∈ L∞(R, XA) ∩W 1,∞(R, X ′A) and
that
E(u(t)) ≤ E(x), for every t ∈ R.
Step 5. Let t ∈ R, let y = u(t) and let v be the solution of (4.1.5) with initial datum y.
By Step 4, we have in particular
E(v(−t)) ≤ E(y).
On the other hand, we have by uniqueness v(−t) = x. It follows that
E(u(t)) = E(x), for every t ∈ R.
Hence conservation of energy. It follows in particular that the function t 7→ ‖u(t)‖2X is
cuntinuous. Since u : R → XA is weakly continuous, we obtain u ∈ C(R, XA); and so
u ∈ C1(R, X ′A), by the equation. Hence (ii)
Remark 4.1.2. Note that assumption (4.1.3) is only needed to ensure that all solu-
tions of (4.1.5) are global. Without that assumption, we would have a local version of
Theorem 4.1.1.
50
4.2. The main result. Theorem 4.1.1 is not applicable in general for solving the local
Cauchy problem in the energy space for the nonlinear Schrodinger equation, since in that
case, we must take X = L2(Ω); and so, we need g to be locally Lipschitz continuous on
L2(Ω). If g is of the form g(u)(x) = f(u(x)) for some function f : C → C, then f needs
to be globally Lipschitz continuous, and in particular sublinear. Thus, we have to improve
Theorem 4.1.1, under weaker assumptions on g.
We use the notation of Chapter 3. In particular, Ω is an open subset of Rn, A is
the Laplacian with Dirichlet boundary conditions; and so, X = L2(Ω), XA = H10 (Ω) and
X ′A = H−1(Ω). We want to go as far as possible under fairly general assumptions on g. The
main result of this section is Theorem 4.2.8, where we establish the local well posedness of
the Cauchy problem in H10 (Ω), provided we have the “a priori” information that solutions
are unique. The reason why we proceed that way is that in order to apply Theorem 4.2.8
we will only have to show uniqueness, and that the techniques that are known for proving
uniqueness depend heavily on the the type of nonlinearity and on geometric properties of
Ω (see the following sections).
We make the following assumptions on the nonlinearity g.
g ∈ C(H10 (Ω), H−1(Ω)), and furthermore there exists
G ∈ C1(H10 (Ω),R) such that g = G′;
(4.2.1)
there exists r, ρ ∈ [2,2n
n− 2) (r, ρ ∈ [2,∞] if n = 1, r, ρ ∈ [2,∞) if n = 2)
such that g : H10 (Ω) → Lρ′(Ω) → H−1(Ω);
(4.2.2)
for every M > 0, there exists C(M) <∞ such that
‖g(v)− g(u)‖Lρ′ ≤ C(M)‖v − u‖Lr , for every u, v ∈ H10 (Ω)
such that ‖u‖H1 + ‖v‖H1 ≤M ;
(4.2.3)
Im(g(u)u) = 0 almost everywhere on Ω, for every u ∈ H10 (Ω). (4.2.4)
We define the energy E by
E(u) =1
2
∫
Ω
|∇u|2 dx−G(u), for every u ∈ H10 (Ω). (4.2.5)
It follows that E ∈ C1(H10 (Ω),R) and that
E′(u) = −4u− g(u), for every u ∈ H10 (Ω).
51
Remark 4.2.1. Assumptions (4.2.1) through (4.2.4) deserve some comments. The en-
ergy space being here H10 (Ω), it is natural to require that g : H1
0 (Ω) → H−1(Ω), as the
Laplacian does. The assumption that g is the gradient of some functional G is a little bit
stronger. It allows us to define the energy, and the conservation of energy is essential in our
proof of local existence. Note that most of the classical examples from theoretical physics
posess this property. However, in the case of local nonlineartites, local existence can be
proved without conservation of energy (see Kato [1,3]). Assumptions (4.2.2) require that
g is slightly better than a mapping H10 (Ω) → H−1(Ω), and assumption (4.2.3) is a form of
local Lipschitz condition. Finally, assumption (4.2.4) implies the conservation of charge.
It is essential for our proof, but can be replaced by other hypotheses on g with different
proofs (see Kato [1,3], Cazenave and Weissler [3]).
Remark 4.2.2. Before proceeding further, we describe some classical examples of non-
linearities in order to motivate our assumptions.
Example 1, the external potential. Consider a real-valued potential V : Ω → R. Assume
that V ∈ Lp(Ω), for some p ≥ 1, p > n/2. Then, g defined by
g(u) = V u, for every u ∈ H10 (Ω),
satisfies assumptions (4.2.1) through (4.2.4). Indeed, (4.2.4) follows from the fact that V
is real-valued, (4.2.2) and (4.2.3) follow from Sobolev’s inequality with r = ρ =2p
p− 1and
(4.2.1) is verified with G defined by
G(u) =1
2
∫
Ω
V (x)|u(x)|2 dx,
for all u ∈ H10 (Ω).
Example 2, the local nonlinearity in one dimension. Assume that n = 1 and consider
a function f : Ω × [0,∞) → R. Assume that f(x, u) is measurable in x and continuous
in u, and that f(x, 0) = 0, almost everywhere on Ω. Assume that f is uniformly locally
Lipschitz continuous in the sense that for every M , there exists L(M) such that
|f(x, v)− f(x, u)| ≤ L(M)|v − u|, for almost all x ∈ Ω
and all u, v ∈ R such that |u|+ |v| ≤M.
52
Extend f to Ω×C by setting
f(x, z) =z
|z|f(x, |z|), for all z ∈ C, z 6= 0, and almost all x ∈ Ω.
Then, g defined by
g(u)(x) = f(x, u(x)) almost everywhere, for every u ∈ H10 (Ω),
satisfies assumptions (4.2.1) through (4.2.4). Indeed, (4.2.4) is obvious, (4.2.2) and (4.2.3)
follow from the local Lipschitz assumption and Sobolev’s embedding H10 (Ω) → L∞(Ω),
with r = ρ = 2, and (4.2.1) is verified with
G(u) =
∫
Ω
F (x, u(x)) dx,
where F (x, z) =
∫ |z|
0
f(x, s) ds, for all z ∈ C and almost all x ∈ Ω.
Example 3, the local nonlinearity in several dimensions. Assume that n ≥ 2 and consider
a function f : Ω × [0,∞) → R. Assume that f(x, u) is measurable in x and continuous
in u, and that f(x, 0) = 0, almost everywhere on Ω. Assume that there exist constants C
and α ∈ [0,4
n− 2) (α ∈ [0,∞) if n = 2) such that
|f(x, v)− f(x, u)| ≤ C(|u|α + |v|α)|v − u|, for almost all x ∈ Ω and all u, v ∈ R.
Extend f to Ω×C by setting
f(x, z) =z
|z|f(x, |z|), for all z ∈ C, z 6= 0, and almost all x ∈ Ω.
Then, g defined by
g(u)(x) = f(x, u(x))almost everywhere, for every u ∈ H10 (Ω),
satisfies assumptions (4.2.1) through (4.2.4). Indeed, (4.2.4) is obvious, (4.2.2) and (4.2.3)
follow easily from the local Lipschitz assumption and Sobolev’s embedding H10 (Ω) →
Lα+2(Ω), with r = ρ = α+ 2, and (4.2.1) is verified with
G(u) =
∫
Ω
F (x, u(x)) dx,
53
where F (x, z) =
∫ |z|
0
f(x, s) ds, for all z ∈ C and almost all x ∈ Ω.
Example 4, the Hartree type nonlinearity in Rn. Assume that Ω = Rn and consider an
even, real-valued potential W : Rn → R. Assume that W ∈ Lp(Rn), for some p ≥ 1,
p > n/4. Then, g defined by
g(u) = (W ∗ |u|2)u, for every u ∈ H10 (Ω),
where ∗ denotes the convolution, satisfies assumptions (4.2.1) through (4.2.4). Indeed,
(4.2.4) follows from the fact that W is real-valued, (4.2.2) and (4.2.3) follow from Sobolev’s
and Young’s inequalities, with r = ρ =4p
2p− 1and (4.2.1) is verified with
G(u) =1
4
∫
Rn
(W ∗ |u|2)(x) |u(x)|2 dx.
We begin with the following proposition.
Proposition 4.2.3. Let g satisfy assumptions (4.2.1) through (4.2.4). For every M > 0,
there exists T (M) > 0 with the following property. For every ϕ ∈ H10 (Ω) such that
‖ϕ‖H1 ≤M , there exists a solution u ∈ L∞(I,H10 (Ω))∩W 1,∞(I,H−1(Ω)) of the problem
iut +4u+ g(u) = 0, for almost all t ∈ I;
u(0) = ϕ,(4.2.6)
with I = (−T (M), T (M)). In addition,
‖u‖L∞(−T (M),T (M);H1) ≤ 2M. (4.2.7)
Furthermore,
‖u(t)‖L2 = ‖ϕ‖L2 ; (4.2.8)
E(u(t)) ≤ E(ϕ); (4.2.9)
for all t ∈ (−T (M), T (M)).
Remark. Note that the conclusions of the above proposition make sense. Indeed, since
u ∈ W 1,∞(−T (M), T (M);H−1(Ω)), u is continuous [−T (M), T (M)] → H−1(Ω); and so,
since u ∈ L∞(−T (M), T (M);H10(Ω)), it follows that u : [−T (M), T (M)] → H1
0 (Ω) is
54
weakly continuous. Furthermore, it follows from inequality ‖v‖2L2 ≤ ‖v‖H−1‖v‖H1
0that
u ∈ C([−T (M), T (M)], L2(Ω)). Therefore, the first equation in (4.2.6) makes sense in
H−1(Ω), the second equation makes sense in H10 (Ω), and (4.2.8) and (4.2.9) make sense.
Before proceeding to the proof of the proposition, we establish two elementary lemmas.
Lemma 4.2.4. Let I ⊂ R be an interval. Then, for every u ∈ L∞(I,H10(Ω)) ∩
W 1,∞(I,H−1(Ω)), we have
‖u(t)− u(s)‖L2(Ω) ≤ C|t− s|1/2, for all s, t ∈ I,
where C = max‖u‖L∞(I,H1), ‖u′‖L∞(I,H−1).
Proof. The result follows from Remark 2.3.10 applied with X = H10 (Ω) and p = ∞ and
inequality ‖v‖2L2 ≤ ‖v‖H−1‖v‖H1
0(see Remark 2.3.8 (iii)).
Lemma 4.2.5. Let g satisfy assumptions (4.2.1) through (4.2.4). Then, after possibly
modifying the function C(M),
(i) ‖g(v)−g(u)‖Lρ′ ≤ C(M)‖v−u‖αL2 , for every u, v ∈ H1
0 (Ω) such that ‖u‖H1 +‖v‖H1 ≤M ;
(ii) |G(v)−G(u)| ≤ C(M)‖v−u‖βL2 , for every u, v ∈ H1
0 (Ω) such that ‖u‖H1+‖v‖H1 ≤M ;
with α = 1− n
(1
2− 1
r
)and β = 1− n
(1
2− 1
ρ
).
Proof. (i) follows from (4.2.3) and Sobolev’s inequality ‖w‖Lr ≤ C‖w‖1−αH1
0‖w‖α
L2 (see
Theorem 2.3.7). (ii) follows from the identity
G(v)−G(u) =
∫ 1
0
d
dsG(sv + (1− s)u) ds =
∫ 1
0
〈g(sv + (1− s)u), v − u〉H−1,H10ds,
and inequality ‖w‖Lρ ≤ C‖w‖1−βH1
0‖w‖β
L2 .
Proof of Proposition 4.2.3. The proof proceeds in three steps. We first approximate g by
a family of nicer nonlinearities, for which we can apply Theorem 4.1.1 in order to construct
approximate solutions. Next, we obtain uniform estimates on the approximate solutions,
55
by using the conservation laws. Finally, we use these estimates to pass to the limit in the
approximate equation.
Note that the proof of Proposition 4.2.3 requires at some stage a regularization pro-
cedure. Indeed, construction of solutions could be made, under appropriate assumptions
on g and in the case Ω = Rn, by a fixed point argument (see Kato [1,3], Cazenave and
Weissler [4]). However, the energy inequality (4.2.9) is obtained, at least formally, by tak-
ing the scalar product of the equation with iut. Note that for a solution with values in
H10 (Ω), ut is only in H−1(Ω); and so one cannot multiply the equation by ut. Hence the
necessity of the regularization.
Now in principle, we have the choice on the type of regularization. For a given
type of nonlinearity, a natural regularization appears, but which is of a different nature
according to the nonlinearity. For example, for a local nonlinearity (Examples 2 and 3),
the most appropriate thing to do would be to truncate f for large values of u. For a linear
potential (Example 1), it would be natural to truncate the potential; and for a Hartree
type nonlinearity (Example 4) it would be natural to use the convolution with a sequence
of mollifiers. Since we want a proof that applies to these different nonlinearities, and that
works as well when Ω = Rn or when Ω is bounded, we find it convenient to regularize the
nonlinearity by applying (I − ε4)−1.
We obtain estimates on the approximate solutions by using essentially the conservation
of energy for the approximate problem. For that purpose, we need g to be the gradient of
some functional G (assumption (4.2.1)).
As usual, the difficulty is to pass to the limit in the nonlinearity. The crux is that
the limiting problem enjoys conservation of charge (Lemma 4.2.6). Note that there is
necessarily a little bit of technicality at that point. Indeed, we make a local assumption
on g (assumption (4.2.4)), we apply a global regularization, and eventually we recover a
local property at the limit. This seems rather unnatural, but there does not seem to be
any obvious way of avoiding that difficulty.
From now on, we consider ϕ ∈ H10 (Ω) and we set M = ‖ϕ‖H1 .
Step 1. Construction of approximate solutions. Given a positive integer m, let
Jm = (I − 1
m4)−1.
56
In other words, for every f ∈ H−1(Ω), Jmu ∈ H10 (Ω) is the unique solution of the equation
u− 1
m4u = f, in H−1(Ω).
We summarize below the main properties of the self-adjoint opeartor Jm (see Section 2.4).
‖Jm‖L(H−1,H10 ) ≤
1
m; (4.2.10)
‖Jm‖L(X,X) ≤ 1, whenever X = H10 (Ω), L2(Ω), H−1(Ω); (4.2.11)
Jmu −→m→∞
u in X, for every u ∈ X, whenever X = H10 (Ω), L2(Ω), H−1(Ω); (4.2.12)
if um if bounded in X, whenever X = H10 (Ω), L2(Ω), H−1(Ω); then
Jmum − um 0 in X, as m→∞;
(4.2.13)
‖Jm‖L(Lp,Lp) ≤ 1, for 1 ≤ p <∞. (4.2.14)
We define
gm(u) = Jm(g(Jmu)), for every u ∈ L2(Ω),
and
Gm(u) = G(Jmu) for every u ∈ H10 (Ω).
It is clear from (4.2.10) that the above definitions make sense. It is easy to verify by
(4.2.10) and (4.2.3) that gm is Lipschitz continuous on bounded sets of L2(Ω), and by
(4.2.10) and (4.2.1) that Gm ∈ C1(H10 (Ω),R) and G′m = gm. In addition, it follows easily
from (4.2.4) that
(gm(u), iu)L2 = (g(Jmu), Jmu)L2 = 0, for every u ∈ L2(Ω).
Therefore, we can apply Theorem 4.1.1; and so there exists a sequence (um)m∈N of func-
tions of C(R, H10(Ω)) ∩ C1(R, H−1(Ω)) such that
ium
t +4um + gm(um) = 0,
um(0) = ϕ.(4.2.15)
Furthermore,
‖um(t)‖L2 = ‖ϕ‖L2 (4.2.16)
57
and1
2
∫
Ω
|∇um(t)|2 dx−Gm(um(t)) =1
2
∫
Ω
|∇ϕ|2 dx−Gm(ϕ) (4.2.17)
for all t ∈ R.
Step 2. Estimates on the sequence um. We denote by C(M) various constants depending
only on M . Let
θm = supτ > 0; ‖um(t)‖H1 ≤ 2M on (−τ, τ). (4.2.18)
Note that by (4.2.11) and (4.2.14),
gm verifies (4.2.2) and (4.2.3) uniformly in m ∈ N. (4.2.19)
Therefore, by (4.2.15)
supm∈N
‖umt ‖L∞(−θm,θm;H−1) ≤ C(M). (4.2.20)
It follows from (4.2.18), (4.2.20) and Lemma 4.2.4 that
‖um(t)− um(s)‖L2 ≤ C(M)|t− s|1/2, for all s, t ∈ (−θm, θm). (4.2.21)
Applying (4.2.16), (4.2.17), (4.2.19), Lemma 4.2.5 (ii), (4.2.18) and (4.2.21), we obtain
‖um(t)‖2H1 ≤ ‖ϕ‖2
L2 + ‖∇ϕ‖2L2 + 2|Gm(um(t))−Gm(ϕ)|
≤ ‖ϕ‖2H1 + C(M)|t|β/2,
(4.2.22)
for all t ∈ (−θm, θm). If we define T (M) by
C(M)T (M)β/2 = 3M2,
it follows from (4.2.22) that
‖um‖L∞(−T,T ;H1) ≤ 2M,
for T = minT (M), θm. This implies that T (M) ≤ θm; and so
‖um‖L∞(−T (M),T (M);H1) ≤ 2M, (4.2.23)
and by (4.2.20)
‖umt ‖L∞(−T (M),T (M);H−1) ≤ C(M). (4.2.24)
58
Step 3. Passage to the limit. It follows from (4.2.23), (4.2.24) and Proposition 2.3.13
that there exists u ∈ L∞(−T (M), T (M);H10(Ω)) ∩W 1,∞(−T (M), T (M);H−1(Ω)) and a
subsequence, which we still denote by (um) such that for all t ∈ [−T (M), T (M)],
um(t) u(t) in H10 (Ω), as m→∞. (4.2.25)
In addition, by (4.2.23), (4.2.25), Lemma 4.2.4, (4.2.19) and Lemma 4.2.5, gm(um) is
bounded in C0,α/2(−T (M), T (M);Lρ′(Ω)). Therefore, it follows from Proposition 2.1.7
that there exists f ∈ C0,α/2(−T (M), T (M);Lρ′(Ω)) and a subsequence, which we still
denote by (gm(um)) such that for all t ∈ [−T (M), T (M)],
gm(um(t)) f(t) in Lρ′(Ω), as m→∞. (4.2.26)
On the other hand, it follows from (4.2.15) that for every w ∈ H10 (Ω) and for every
φ ∈ D(−T (M), T (M)), we have
∫ T (M)
−T (M)
−〈ium, w〉H−1,H10φ′(t) + 〈4um + gm(um), w〉H−1,H1
0φ(t) dt = 0.
Applying (4.2.25), (4.2.26) and the dominated convergence theorem, it follows easily that
∫ T (M)
−T (M)
−〈iu, w〉H−1,H10φ′(t) + 〈4u+ f, w〉H−1,H1
0φ(t) dt = 0.
This means that u satisfiesiut +4u+ f = 0, for almost all t ∈ (−T (M), T (M)),
u(0) = ϕ.(4.2.27)
Now the crux of the proof is the following result.
Lemma 4.2.6. For all t ∈ (−T (M), T (M)), we have Im(f(t)u(t)) = 0 almost everywhere
on Ω.
Proof. It suffices to show that for every bounded subset B of Ω, we have
〈f(t)|B, iu(t)|B〉Lρ′(B),Lρ(B) = 0.
To see this, we omit the time dependence and we write
〈f, iu〉Lρ′(B),Lρ(B) = 〈f − Jmg(Jmum), iu〉+ 〈Jmg(Jmu
m)− g(Jmum), iu〉
+ 〈g(Jmum), i(u− um)〉+ 〈g(Jmu
m), i(um − Jmum)〉
+ 〈g(Jmum), iJmu
m〉 −→m→∞
a+ b+ c+ d+ e.
59
Note first that Jmg(Jmum) = gm(um) f in Lρ′(Ω), hence in Lρ′(B). Therefore, a = 0.
Next, observe that g(Jmum) is bounded in Lρ′(Ω). It follows from (4.2.13) and (4.2.14)
that Jmg(Jmum) − g(Jmu
m) 0 in H−1(Ω), hence in Lρ′(B). Therefore, b = 0. Since
um u in H10 (Ω), we have um → u in Lρ(B). Since g(Jmu
m) is bounded in Lρ′(B),
it follows that c = 0. By (4.2.11) and (4.2.13), um − Jmum is bounded in H1
0 (Ω) and
converges weakly to 0 in H−1(Ω). It follows that um − Jmum → 0 in Lρ(B). Since
g(Jmum) is bounded in Lρ′(B), it follows that d = 0. Finally, e = 0 by (4.2.4). Hence the
result.
End of the proof of Proposition 4.2.3. Taking the H−1 −H10 duality product of the first
equation in (4.2.27) with iu, it follows that
d
dt‖u(t)‖2
L2 = 0, for all t ∈ (−T (M), T (M));
and so
‖u(t)‖L2 = ‖ϕ‖L2 . (4.2.28)
It follows from (4.2.16), (4.2.28) and Proposition 2.3.13 (ii) that
um → u, in C(−T (M), T (M);L2(Ω)). (4.2.29)
Applying (4.2.23), (4.2.29) and Theorem 2.3.7, it follows that
um → u, in C(−T (M), T (M);Lp(Ω)), for every 2 ≤ p <2n
n− 2. (4.2.30)
It follows easily from (4.2.3), (4.2.12) and (4.2.3) that
gm(um(t)) → g(u(t)) in Lρ′(Ω), for all t ∈ (−T (M), T (M)).
Therefore, f = g(u); and so, u satisfies (4.2.6). (4.2.7) follows from (4.2.23) and (4.2.8) from
(4.2.28). It remains to prove (4.2.9). This follows from (4.2.17), weak lower semicontinuity
of the H1-norm, and the fact that Gm(um(t)) → G(u(t)) as m →∞. This completes the
proof.
Before proceeding further, we make the following definition.
60
Definition 4.2.7. In all what follows, we say that we have uniqueness for problem (4.2.6)
if the following hold. For every interval J containing 0 and for every ϕ ∈ H10 (Ω), any two
solutions of (4.2.6) in L∞(J,H10(Ω)) ∩W 1,∞(J,H−1(Ω)) coincide.
The main result of this section is the following.
Theorem 4.2.8. Let g satisfy assumptions (4.2.1) through (4.2.7), and assume that we
have uniqueness for problem (4.2.6). Then, the following properties hold.
(i) For every ϕ ∈ H10 (Ω), there exists T∗(ϕ), T ∗(ϕ) > 0 and there exists a unique, maximal
solution u ∈ C((−T∗(ϕ), T ∗(ϕ)), H10 (Ω)) ∩ C1((−T∗(ϕ), T ∗(ϕ)), H−1(Ω)) of problem
(4.2.6). u is maximal in the sense that if T ∗(ϕ) <∞ (respectively, T∗(ϕ) <∞), then
‖u(t)‖H1 →∞, as t ↑ T ∗(ϕ) (respectively, as t ↓ −T∗(ϕ));
(ii) in addition, we have conservation of charge and energy, that is
‖u(t)‖L2 = ‖ϕ‖L2 , and E(u(t)) = E(ϕ),
for all t ∈ (−T∗(ϕ), T ∗(ϕ));
(iii) we have continuous dependence of the solution on the initial datum in the sense that
both functions T∗(ϕ) and T ∗(ϕ) are lower semicontinuous, and that if ϕm → ϕ in
H10 (Ω) and if [−T1, T2] ⊂ (−T∗(ϕ), T ∗(ϕ)), then um → u in C([−T1, T2], H
10(Ω)),
where um is the maximal solution of (4.2.6) with initial datum ϕm.
Proof. The proof proceeds in three steps. We first show that the solution u given by
Proposition 4.2.3 belongs to C((−T (M), T (M)), H10(Ω)) ∩ C1((−T (M), T (M)), H−1(Ω)),
and that we have conservation of energy. Next, we consider the maximal solutions and show
that T∗ and T ∗ satisfy the above alternative. Finally, we establish continuous dependence.
Step 1. Regularity. Let I be an interval and let u ∈ L∞(I,H10 (Ω)) ∩W 1,∞(I,H−1(Ω))
satisfy
iut +4u+ g(u) = 0, for all t ∈ I.
We claim that u enjoys both conservation of charge and energy, and that u ∈ C(I,H10(Ω))∩
C1(I,H−1(Ω)). To see this, consider
M = sup‖u(t)‖H1 , t ∈ I,
61
and let us first show that ‖u(t)‖L2 and E(u(t)) are constant on every interval J ⊂ I
of lenght at most T (M), where T (M) is given by Proposition 4.2.3. Indeed, let J be
as above and let σ, τ ∈ J . Let ϕ = u(σ) and let v be the solution of (4.2.6) given by
Proposition 4.2.3. v(· − σ) is defined on J and by uniqueness, v(· − σ) = u(·) on J . By
(4.2.8) and (4.2.9), it follows in particular that
‖u(τ)‖L2 = ‖u(σ)‖L2, and E(u(τ)) ≤ E(u(σ)). (4.2.31)
Now, let ϕ = u(τ) and let w be the solution of (4.2.6) given by Proposition 4.2.3. w(· − τ)is defined on J and by uniqueness, w(·− τ) = u(·) on J . By (4.2.9), it follows in particular
that
E(u(σ)) ≤ E(u(τ)).
Comparing with (4.2.31), this implies that both ‖u(t)‖L2 and E(u(t)) are constant on J .
Since J is arbitrary, it follows that
‖u(t)‖L2 = ‖u(s)‖L2 , and E(u(t)) = E(u(s)), for all s, t ∈ I. (4.2.32)
Furthermore, note that by Lemma 4.2.4, u ∈ C0,1/2(I, L2(Ω)); and so, by Lemma 4.2.5,
the function t 7→ G(u(t)) is continuous I → R. In view of (4.2.32), it follows that ‖u(t)‖H1
is continuous I → R. Therefore (Lemma 2.1.5), u ∈ C(I,H10 (Ω)), and by the equation,
u ∈ C1(I,H−1(Ω)).
Step 2. Maximality. Consider ϕ ∈ H10 (Ω) and let
T ∗(ϕ) = supT > 0; there exists a solution of (4.2.6) on [0, T ],
T∗(ϕ) = supT > 0; there exists a solution of (4.2.6) on [−T, 0].
By uniqueness and Step 1, there exists a solution u ∈ C((−T∗(ϕ), T ∗(ϕ)), H10(Ω)) ∩
C1((−T∗(ϕ), T ∗(ϕ)), H−1(Ω)) of (4.2.6). Suppose now that T ∗(ϕ) <∞, and assume that
there exists M < ∞ and a sequence tj ↑ T ∗(ϕ) such that ‖u(tj)‖H1 ≤ M . Let k be such
that tk + T (M) > T ∗(ϕ). By Proposition 4.2.3 and Step 1, and starting from u(tk), one
can extend u up to tk + T (M), which is a contradiction with the maximality. Therefore,
‖u(t)‖H1 → ∞, as t ↑ T ∗(ϕ). One shows by the same argument that if T∗(ϕ) < ∞, then
62
‖u(t)‖H1 → ∞, as t ↓ −T∗(ϕ). Therefore, we have established statements (i) and (ii) of
Theorem 4.2.7.
Step 3. Continuous dependence. We have to show that if ϕm → ϕ in H10 (Ω) and if
[−T1, T2] ⊂ (−T∗(ϕ), T ∗(ϕ)), then [−T1, T2] ⊂ (−T∗(ϕm), T ∗(ϕm)) for m large enough,
and um → u in C([−T1, T2], H10 (Ω)). To do this, consider
M =1
2sup‖u(t)‖H1 ; t ∈ [−T1, T2].
For m large enough, we have ‖ϕm‖H1 ≤M ; and so [−T (M), T (M)] ⊂ (−T∗(ϕm), T ∗(ϕm))
and um is bounded in L∞(−T (M), T (M);H10(Ω)) ∩W 1,∞(−T (M), T (M);H−1(Ω)). Ap-
plying the argument of Step 3 of the proof of Proposition 4.2.3, it follows that um → u
in C([−T (M), T (M)], L2(Ω)). By Lemma 4.2.5 and conservation of energy, this implies
that ‖um‖H1 → ‖u‖H1 uniformly on [−T (M), T (M)]. Applying Proposition 2.3.13 (iii), it
follows that um → u in C([−T (M), T (M)], H10(Ω)). Since T (M) depends only on M , we
can repeat this argument to cover the interval [−T1, T2]. This proves (iii).
Remark 4.2.9. The proof of Theorem 4.2.8 is trivially adapted to the case where g is a
finite sum of terms gj , where each of the gj ’s satisfies assumptions (4.2.1) through (4.2.4)
for some exponents rj , ρj. However, note that one needs the uniqueness assumption for g,
and not for each of the gj’s.
Remark 4.2.10. Assumptions (4.2.2) and (4.2.3) can be weakened, without modifying
the proof of Theorem 4.2.8. One may assume that g verifies the following assumptions
(4.2.2’) and (4.2.3’).
There exists ρ ∈ [2,2n
n− 2] (r, ρ ∈ [2,∞] if n = 1, r, ρ ∈ [2,∞) if n = 2)
such that g is a bounded mapping H10 (Ω) → Lρ′(Ω);
(4.2.2′)
for every M > 0, there exists C(M) <∞ and α > 0 such that
‖g(v)− g(u)‖H−1 ≤ C(M)‖v − u‖αL2 , for every u, v ∈ H1
0 (Ω)
such that ‖u‖H1 + ‖v‖H1 ≤M.
(4.2.3′)
It follows from Lemma 4.2.5 that assumptions (4.2.2’) and (4.2.3’) are actually weaker
than assumptions (4.2.2) and (4.2.3). However, they do not make any difference in the
classical applications.
63
Remark 4.2.11. By Theorem 4.2.8, if g satisfies assumptions (4.2.1) through (4.2.4), we
have well posedness of problem (4.2.6) in H10 (Ω) provided we have uniqueness. Unfortu-
nately, the techniques that are used to prove uniqueness depend on the problem (see the
following sections). However, we give below a general sufficient condition for uniqueness.
Corollary 4.2.12. Let G ∈ C1(H10 (Ω),R) and let g = G′. Assume that 〈g(u), iu〉 = 0,
for every u ∈ H10 (Ω) and that for every M , there exists C(M) such that
‖g(v)− g(u)‖L2 ≤ C(M)‖v − u‖L2 ,
for all u, v ∈ H10 (Ω) such that ‖u‖H1 +‖v‖H1 ≤M . Then, the conclusions of Theorem 4.2.8
hold.
Proof. We only have to show uniqueness. Let I be an interval containing 0, let j ∈ H10 (Ω)
and let u1, u2 ∈ L∞(I,H10(Ω)) ∩W 1,∞(I,H−1(Ω)) be two solutions of (4.2.6). It follows
from Remark 2.5.1 (iii) that
u2(t)− u1(t) = i
∫ t
0
T (t− s)(g(u2(s))− g(u1(s))) ds,
for all t ∈ I. It follows that there exists a constant C such that
‖u2(t)− u1(t)‖L2 ≤ C
∫ t
0
‖u2(s)− u1(s)‖L2 ds,
and the result follows from Gronwall’s lemma.
Remark 4.2.13. Theorem 4.2.8 (and also Corollary 4.2.12) is stated for one equation,
but the method applies as well for systems of the same form. More precisely, consider
an integer µ ≥ 1 and set H10 = (H1
0 (Ω))µ, H−1 = (H−1(Ω))µ and Lp = (Lp(Ω))µ. Let
(αj)1≤j≤µ, (βj)1≤j≤µ be two familes of real numbers such that αj 6= 0 and βj 6= 0, for
every 1 ≤ j ≤ µ. Set AU = (α14u1, · · · , αµ4uµ) for U = (u1, · · · , uµ) ∈ H10, and BU =
(β1u1, · · · , βµuµ), for every U ∈ Cµ. Then, if g satisfies assumptions (4.2.1) through (4.2.4),
but with H10 (Ω), H−1(Ω), Lp(Ω) replaced by H1
0,H−1,Lp, the conclusions of Theorem 4.2.8
(under uniqueness assumption) hold for the system
iBUt +AU + g(U) = 0,
U(0) = Φ,
64
where Φ is a given initial datum in H10.
Remark 4.2.14. Let g be as in Proposition 4.2.3, or more generally, as in Remark 4.2.9.
Assume further that for every A > 0, there exists K(A) <∞ such that
G(u) ≤ K(A) +1
4‖u‖2
H1 , for all u ∈ H10 (Ω) such that ‖u‖L2 ≤ A. (4.2.33)
Then, we can take T (M) = ∞ in the conclusion of Proposition 4.2.3. In particular, it g
verifies the assumptions of Theorem 4.2.8, we have T ∗(ϕ) = T∗(ϕ) = ∞. Indeed, we only
have to modify Step 2 of the proof of Proposition 4.2.3 (i.e. the uniform H1 estimate).
Note that by (4.2.16), (4.2.17) and (4.2.33), we have
‖um(t)‖2H1 = ‖ϕ‖2
H1 − 2Gm(ϕ) + 2Gm(um(t))
≤ ‖ϕ‖2H1 − 2Gm(ϕ) +K(‖ϕ‖L2) +
1
2‖um(t)‖2
H1 ,
from which the result follows. In other words, assumption (4.2.33) ensures that the solu-
tions are global. We will come back to that property in Chapter 6.
Remark 4.2.15. Let g be as in Theorem 4.2.8, or more generally, as in Remark 4.2.9.
Consider ϕ ∈ H10 (Ω), and let u be the maximal solution of (4.2.6). Let um be the approxi-
mate solutions constructed in Step 1 of the proof of Proposition 4.2.3. Then, following the
argument of the proof of Theorem 4.2.8, one shows easily that um → u in C([S, T ], H10(Ω)),
as m→∞, for every interval [S, T ] ⊂ (−T∗(ϕ), T ∗(ϕ)).
4.3. The nonlinear Schrodinger equation in Rn. Throughout this section, we
assume that Ω = Rn. Consider g ∈ C(H1(Rn)), H−1(Rn) and assume that there exists
g1, · · · , gk ∈ C(H1(Rn)), H−1(Rn) such that
g = g1 + · · ·+ gk,
where each of the gj ’s satifies assumptions (4.2.1) through (4.2.4) for some exponents rj , ρj.
Let
G = G1 + · · ·+Gk,
and set
E(u) =1
2
∫
Rn
|∇u(x)|2 dx−G(u), for u ∈ H1(Rn).
65
For convenience, we recall problem (4.2.6).iut +4u+ g(u) = 0, for almost all t ∈ I;
u(0) = ϕ,(4.2.6)
We will apply the results of Section 3.2 to establish the following result.
Theorem 4.3.1. Let g be as above. Then, the following holds.
(i) For every ϕ ∈ H1(Rn), there exists T∗(ϕ), T ∗(ϕ) > 0 and there exists a unique,
maximal solution u ∈ C((−T∗(ϕ), T ∗(ϕ)), H1(Rn)) ∩ C1((−T∗(ϕ), T ∗(ϕ)), H−1(Rn))
of problem (4.2.6). u is maximal in the sense that if T ∗(ϕ) <∞ (respectively, T∗(ϕ) <
∞), then ‖u(t)‖H1 →∞, as t ↑ T ∗(ϕ) (respectively, as t ↓ −T∗(ϕ));
(ii) in addition, we have conservation of charge and energy, that is
‖u(t)‖L2 = ‖ϕ‖L2 , and E(u(t)) = E(ϕ),
for all t ∈ (−T∗(ϕ), T ∗(ϕ));
(iii) we have continuous dependence of the solution on the initial datum in the sense that
both functions T∗(ϕ) and T ∗(ϕ) are lower semicontinuous, and that if ϕm → ϕ in
H1(Rn) and if [−T1, T2] ⊂ (−T∗(ϕ), T ∗(ϕ)), then um → u in C([−T1, T2], H1(Rn)),
where um is the maximal solution of (4.2.6) with initial datum ϕm.
Proof. By Theorem 4.2.8 and Remark 4.2.9, we only have to show uniqueness. Further-
more, iniqueness being a local property, we only have to establish it for possibly small
intervals. Let I be an interval containing 0, and let u, v ∈ L∞(I,H1) ∩W 1,∞(I,H1) be
two solutions of (4.2.6). By Remark 2.5.1 (iii), we have
w(t) = i
k∑
j=1
∫ t
0
T (t− s)[gj(v(s))− gj(u(s))] ds, for all t ∈ I,
where w = v − u. Let (γj)1≤j≤k and (qj)1≤j≤k be defined by
2
γj= n
(1
2− 1
ρj
),
2
qj= n
(1
2− 1
rj
), for 1 ≤ j ≤ k,
and let (q, r) be any admissible pair. By Theorem 3.2.5 (ii), there exists C, depending
only on q, γ1, · · · , γk such that
‖w‖Lq(I,Lr) ≤ Ck∑
j=1
‖gj(v)− gj(u)‖L
γ′j (I,L
ρ′j ).
66
Setting M = supmax‖u(t)‖H1 , ‖u(t)‖H1; t ∈ I, it follows from (4.2.3) and Holder’s
inequality in time that there exists K depending only on M such that
‖gj(v)− gj(u)‖L
γ′j (I,L
ρ′j )≤ K|I|
qj−γ′j
qj γ′j ‖w‖Lqj (I,Lrj ), for 1 ≤ j ≤ k.
Note that every of the numbersqj − γ′jqjγ
′j
is positive. Therefore, there exists α > 0 and
A <∞ such that
|I|qj−γ′
j
qjγ′j ≤ A(|I|+ |I|α) for all 1 ≤ j ≤ k.
Therefore, we have
‖w‖Lq(I,Lr) ≤ CKA(|I|+ |I|α)
k∑
j=1
‖w‖Lqj (I,Lrj ).
Chosing successively r = r1, · · · , r = rk, it follows that there exists a constant B such that
k∑
j=1
‖w‖Lqj (I,Lrj ) ≤ B(|I|+ |I|α)k∑
j=1
‖w‖Lqj (I,Lrj ).
Therefore, if B(|I|+ |I|α) < 1, we obtain w ≡ 0 on I. Hence the result.
Remark 4.3.2. Consider a real valued potential V : Rn → R such that V ∈ Lp(Rn) +
L∞(Rn) for some p ≥ 1, p > n/2.
Consider a function f : Rn × [0,∞) → R. Assume that f(x, u) is measurable in x
and continuous in u, and that f(x, 0) = 0, almost everywhere on Rn. Assume that there
exist constants C and α ∈ [0,4
n− 2) (α ∈ [0,∞) if n = 2) such that
|f(x, v)− f(x, u)| ≤ C(1 + |u|α + |v|α)|v − u|, for almost all x ∈ Rn and all u, v ∈ R.
If n = 1, assume instead that for every M , there exists L(M) such that
|f(x, v)− f(x, u)| ≤ L(M)|v − u|, for almost all x ∈ Rn
and all u, v ∈ R such that |u|+ |v| ≤M.
Extend f to Rn ×C by setting
f(x, z) =z
|z|f(x, |z|), for all z ∈ C, z 6= 0, and almost all x ∈ Rn.
67
Set
F (x, z) =
∫ |z|
0
f(x, s) ds, for all z ∈ C and almost all x ∈ Rn.
Finally, consider an even, real-valued potential W : Rn → R. Assume that W ∈Lp(Rn) + L∞(Rn), for some p ≥ 1, p > n/4.
Then, Theorem 4.2.1 applies to the case where
g(u) = V u+ f(·, u(·)) + (W ∗ |u|2)u,
and the functional G is given by
G(u) =
∫
Rn
1
2V (x)|u(x)|2 + F (x, u(x)) +
1
4(W ∗ |u|2)(x)|u(x)|2
dx.
This follows easily from Remark 4.2.2. In particular, one can apply Theorem 4.3.1 to the
case where g(u) = λ|u|αu, where λ ∈ R and 0 ≤ α ≤ 4
n− 2(or 0 ≤ α <∞, if n = 1, 2).
Remark 4.3.3. Note that the only ingredient that we used for proving uniqueness is
Theorem 3.2.5. In particular, it follows from Remark 3.4.5 that Theorem 4.3.1 still holds
if one replaces Rn by Rn+, or by certain cones of Rn.
Remark 4.3.4. Like Theorem 4.2.8, Theorem 4.3.1 is stated for one equation, but the
method applies as well for systems of the same form (see Remark 4.2.13).
4.4. The nonlinear Schrodinger equation in one dimension. In this section,
we assume that n = 1. Without restricting the generality, we may also assume that
Ω is connected. Therefore, Ω is either R, or a half line, or a bounded interval. The
case Ω is either R or half line falls into the scope of Theorem 4.3.1 or Remark 4.3.3.
Therefore, the local Cauchy problem in H10 (Ω) is well set for example for the type of
nonlinearities considered in Remark 4.3.2. On the other hand, if Ω is a bounded interval,
we know (Remark 3.4.1) that the estimates of Proposition 3.2.1 do not hold. However,
one can obtain a fairly general result for local nonlinearities, by using the embedding
H10 (Ω) → L∞(Ω).
Consider a function f : Ω× [0,∞) → R. Assume that f(x, u) is measurable in x and
continuous in u, and that f(x, 0) = 0, almost everywhere on Ω. Assume that f is uniformly
locally Lipschitz continuous in the sense that for every M , there exists L(M) such that
|f(x, v)− f(x, u)| ≤ L(M)|v − u|, for almost all x ∈ Ω
and all u, v ∈ R such that |u|+ |v| ≤M.
68
Extend f to Ω×C by setting
f(x, z) =z
|z|f(x, |z|), for all z ∈ C, z 6= 0, and almost all x ∈ Ω.
Let g be defined by
g(u)(x) = f(x, u(x)) almost everywhere, for every u ∈ H10 (Ω),
and let
G(u) =
∫
Ω
F (x, u(x)) dx, for every u ∈ H10 (Ω),
where
F (x, z) =
∫ |z|
0
f(x, s) ds, for all z ∈ C and almost all x ∈ Ω.
We have the following result.
Theorem 4.4.1. Let g be as above. Then, the following holds.
(i) For every ϕ ∈ H10 (Ω), there exists T∗(ϕ), T ∗(ϕ) > 0 and there exists a unique, maximal
solution u ∈ C((−T∗(ϕ), T ∗(ϕ)), H10 (Ω)) ∩ C1((−T∗(ϕ), T ∗(ϕ)), H−1(Ω)) of problem
(4.2.6). u is maximal in the sense that if T ∗(ϕ) <∞ (respectively, T∗(ϕ) <∞), then
‖u(t)‖H1 →∞, as t ↑ T ∗(ϕ) (respectively, as t ↓ −T∗(ϕ));
(ii) in addition, we have conservation of charge and energy, that is
‖u(t)‖L2 = ‖ϕ‖L2 , and E(u(t)) = E(ϕ),
for all t ∈ (−T∗(ϕ), T ∗(ϕ));
(iii) we have continuous dependence of the solution on the initial datum in the sense that
both functions T∗(ϕ) and T ∗(ϕ) are lower semicontinuous, and that if ϕm → ϕ in
H10 (Ω) and if [−T1, T2] ⊂ (−T∗(ϕ), T ∗(ϕ)), then um → u in C([−T1, T2], H
10(Ω)),
where um is the maximal solution of (4.2.6) with initial datum ϕm.
Proof. By The result follows from Corollary 4.2.12 and Remark 4.3.2, Example 2
Remark 4.4.2. In particular, one can apply Theorem 4.4.1 to the case g(u) = V u +
λ|u|αu, where V is a real valued potential V ∈ L∞(Ω), λ ∈ R and 0 ≤ α <∞.
69
Remark 4.4.3. Like Theorem 4.2.8, Theorem 4.4.1 is stated for one equation, but the
method applies as well for systems of the same form (see Remark 4.2.13).
4.5. The nonlinear Schrodinger equation in two dimensions. In this section, we
assume that n = 2. The case Ω is either R2 or some special cone fall into the scope of
Theorem 4.3.1 or Remark 4.3.3. Therefore, the local Cauchy problem in H10 (Ω) is well set
for example for the type of nonlinearities considered in Remark 4.3.2. On the other hand,
if Ω is a bounded domain, we know (Remark 3.4.1) that the estimates of Proposition 3.2.1
do not hold. Furthermore, H10 (Ω) 6→ L∞(Ω); and so, one can not apply the method of
Section 4.4. However, one still can do something by using the fact that H10 (Ω) is “almost”
embedded in L∞(Ω), or more precisely by using Trudinger’s inequality (Remark 2.3.6).
Let Ω be an open subset of R2 and consider a function f : Ω× [0,∞) → R. Assume
that f(x, u) is measurable in x and continuous in u, and that f(x, 0) = 0, almost everywhere
on Ω. Assume that there exist a constant C such that
|f(x, v)− f(x, u)| ≤ C(1 + |u|2 + |v|2)|v − u|, for almost all x ∈ Ω and all u, v ∈ R.
Extend f to Ω×C by setting
f(x, z) =z
|z|f(x, |z|), for all z ∈ C, z 6= 0, and almost all x ∈ Ω.
Let g be defined by
g(u)(x) = f(x, u(x))almost everywhere, for every u ∈ H10 (Ω),
and let
G(u) =
∫
Ω
F (x, u(x)) dx,
where
F (x, z) =
∫ |z|
0
f(x, s) ds, for all z ∈ C and almost all x ∈ Ω.
We have the following result, due to Vladimirov (see Vladimirov [1], Ogawa [1], Ogawa
and Ozawa [1]).
Theorem 4.5.1. If g is as above, then the following properties hold.
70
(i) For every ϕ ∈ H10 (Ω), there exists T∗(ϕ), T ∗(ϕ) > 0 and there exists a unique, maximal
solution u ∈ C((−T∗(ϕ), T ∗(ϕ)), H10 (Ω)) ∩ C1((−T∗(ϕ), T ∗(ϕ)), H−1(Ω)) of problem
(4.2.6). u is maximal in the sense that if T ∗(ϕ) <∞ (respectively, T∗(ϕ) <∞), then
‖u(t)‖H1 →∞, as t ↑ T ∗(ϕ) (respectively, as t ↓ −T∗(ϕ));
(ii) in addition, we have conservation of charge and energy, that is
‖u(t)‖L2 = ‖ϕ‖L2 , and E(u(t)) = E(ϕ),
for all t ∈ (−T∗(ϕ), T ∗(ϕ));
(iii) we have continuous dependence of the solution on the initial datum in the sense that
both functions T∗(ϕ) and T ∗(ϕ) are lower semicontinuous, and that if ϕm → ϕ in
H10 (Ω) and if [−T1, T2] ⊂ (−T∗(ϕ), T ∗(ϕ)), then um → u in C([−T1, T2], H
10(Ω)),
where um is the maximal solution of (4.2.6) with initial datum ϕm.
Proof. By Theorem 4.2.8 and Remark 4.3.2, Example 3, we only have to show uniqueness.
Furthermore, iniqueness being a local property, we only have to establish it for possibly
small intervals. Let I be an interval containing 0, and let u, v ∈ L∞(I,H10)∩W 1,∞(I,H1)
be two solutions of (4.2.6). We have
iwt +4w + g(v)− g(u) = 0.
On multiplying the above equation in the H−1 −H10 duality by iw, it follows that
1
2
d
dt‖w(t)‖2
L2 = Im
(∫
Ω
(g(v(t))− g(u(t)))w(t)dx
).
Therefore, if we define the function h ∈ L∞(I,H10 (Ω)) by
h(t) = |u(t)|+ |v(t)|, for all t ∈ I,
we have1
2
d
dt‖w(t)‖2
L2 ≤ C
∫
Ω
(1 + h(s)2)|w(s)|2 dx.
Integrating the above inequality between 0 and t ∈ I, we obtain
‖w(t)‖2L2 ≤ 2C|
∫ t
0
‖w(s)‖2L2 +
∫
Ω
h(s)2|w(s)|2 dx ds|. (4.5.1)
71
Consider any number p ∈ (2,∞). It follows from Holder’s inequality that
∫
Ω
h2|w|2 dx =
∫
Ω
(hp|w|2) 2p |w|
2p−4p dx ≤
(∫
Ω
hp|w|2 dx) 2
p
‖w‖2p−4
p
L2
≤(∫
Ω
h2p dx
) 1p
‖w‖4p
L4‖w‖2p−4
p
L2
.
(4.5.2)
Note first that w(t) is bounded in H10 (Ω), hence in L4(Ω). Furthermore, h(t) is also
bounded in H10 (Ω). Therefore (Remark 2.3.6), there exists two positive constants K,µ
such that ∫
Ω
(eµh(t)2 − 1) dx ≤ K. (4.5.3)
It follows from (4.5.2), (4.5.3) and the elementary inequality x2p ≤(
pµ
)p
(eµx2 − 1) that
∫
Ω
h2|w|2 dx ≤ CpK1p ‖w‖
2p−4p
L2 ,
for some constant C. Since K1p ≤ 1 +K, we have, after possibly modifying C,
∫
Ω
h2|w|2 dx ≤ Cp‖w‖2p−4
p
L2 .
Let now φ(t) = ‖w‖2L2 . It follows from the above inequality and (4.5.1) that
φ(t) ≤ C∣∣∣∫ t
0
φ(s) + pφ(s)p−2
p ds∣∣∣.
Note that φ is bounded, thus for p large enough, we have φ(t) ≤ pφ(t)p−2
p . Therefore,
φ(t) ≤ Cp∣∣∣∫ t
0
φ(s)p−2
p ds∣∣∣,
for all t ∈ I.Let now
Φp(t) =
∫ t
0
φ(s)p−2
p ds.
It follows from (4.5.4) that Φ′p(t) ≤ Cp|Φp(t)p−2
p |, for all t ∈ I. Integrating this inequality
yields Φp(t) ≤ (2C|t|) p2 . Therefore, if 2C|T | < 1, we obtain
lim infp→∞
Φp(T ) = 0,
72
which implies that ∫ T
0
φ(s) ds = 0.
It follows that w ≡ 0 on [−T, T ]. Hence the result.
Remark 4.5.2. A quite similar result, but for strong solutions (i.e. solutions with values
in H2(Ω) ∩H10 (Ω)) was obtained by Brezis and Gallouet [1].
Remark 4.5.3. In particular, one can apply Theorem 4.5.1 to the case g(u) = V u +
λ|u|αu, where V is a real valued potential V ∈ L∞(Ω), λ ∈ R and 0 ≤ α ≤ 2.
Remark 4.5.4. Like Theorem 4.2.8, Theorem 4.5.1 is stated for one equation, but the
method applies as well for systems of the same form (see Remark 4.2.13).
4.6. Comments. Theorem 4.2.8 admits a generalization in the setting of Section 4.1.
More precisely, with the notation of Section 4.1, consider a C-linear, self-adjoint ≤ 0
operator A on X = L2(Ω). Assume that
XA → Lp(Ω) for 2 ≤ p <2n
n− 2. (4.6.1)
Assume further that for every2n
n+ 2≤ p ≤ 2n
n− 2, (I−εA)−1 is continuous Lp(Ω) → Lp(Ω)
and
sup‖(I − εA)−1‖L(Lp,Lp); ε > 0 <∞. (4.6.2)
Consider a function g ∈ C(XA, X′A) such that
there exists G ∈ C1(XA,R) such that g = G′; (4.6.3)
there exists r, ρ ∈ [2,2n
n− 2) (r, ρ ∈ [2,∞] if n = 1, r, ρ ∈ [2,∞) if n = 2)
such that g : XA → Lρ′(Ω) → X ′A;
(4.6.4)
for every M > 0, there exists C(M) <∞ such that
‖g(v)− g(u)‖Lρ′ ≤ C(M)‖v − u‖Lr , for every u, v ∈ XA
such that ‖u‖A + ‖v‖A ≤M ;
(4.6.5)
Im(g(u)u) = 0 almost everywhere on Ω, for every u ∈ XA, (4.6.6)
73
and let E be defined by (4.1.4). We consider the following problem.
iut + Au+ g(u) = 0,
u(0) = x,(4.6.7)
for a given x ∈ XA. Then we have the following result.
Theorem 4.6.1. Let A and g be as above. Assume that we have uniqueness for problem
(4.6.7) in L∞(I,XA) ∩W 1,∞(I,X ′A), for every x ∈ XA and every interval I containing 0.
Then, the following holds.
(i) For every x ∈ XA, there exists T∗(x), T ∗(x) > 0 and there exists a unique, maximal
solution u ∈ C((−T∗(x), T ∗(x)), XA)∩C1((−T∗(x), T ∗(x)), X ′A) of problem (4.6.7). u
is maximal in the sense that if T ∗(x) <∞ (respectively, T∗(x) <∞), then ‖u(t)‖A →∞, as t ↑ T ∗(x) (respectively, as t ↓ −T∗(x));
(ii) in addition, we have conservation of charge and energy, that is
‖u(t)‖L2 = ‖x‖L2 , and E(u(t)) = E(x),
for all t ∈ (−T∗(x), T ∗(x));
(iii) we have continuous dependence of the solution on the initial datum in the sense that
both functions T∗(x) and T ∗(x) are lower semicontinuous, and that if xm → x in XA
and if [−T1, T2] ⊂ (−T∗(x), T ∗(x)), then um → u in C([−T1, T2], XA), where um is
the maximal solution of (4.6.7) with initial datum xm.
Proof. The proof is an adaptation of the proof of Theorem 4.2.8. We only point out
the modifications that are not absolutely trivial. Lemma 4.2.4 is easily adapted with the
duality inequality ‖u‖2X ≤ ‖u‖A‖u‖X′
A. The proof of Lemma 4.2.5 is adapted as follows.
Consider 2 ≤ p ≤ 2n
n− 2, and let p < q <
2n
n− 2. By Holder’s inequality and (4.6.1), there
exists α ∈ (0, 1) such that
‖u‖Lp ≤ ‖u‖αLq‖u‖1−α
L2 ≤ ‖u‖αA‖u‖1−α
L2 ,
and the rest of the proof is unchanged. To adapt the proof of Proposition 4.2.3, we
need inequalities of type (4.2.10)-(4.2.14). They follow easily from the self-adjointness
74
of A, except for (4.2.14), which follows from (4.6.2). The rest of the proof, including
Lemma 4.2.6, is unchanged except that one has to apply Proposition 2.1.7 instead of
Proposition 2.3.13.
Remark 4.6.2. Remarks 4.2.9 and 4.2.10 and Corollary 4.2.12 are easily adapted to the
above situation.
Remark 4.6.3. Like Theorem 4.2.8 (see Remark 4.2.13), Theorem 4.6.1 is stated for one
equation, but the method applies as well for systems of the same form. More precisely,
considering an integer µ ≥ 1, one may assume that A is a self adjoint operator on (L2(Ω))µ
and replace everywhere H10 (Ω), H−1(Ω), Lp(Ω) by (H1
0 (Ω))µ, (H−1(Ω))µ, (Lp(Ω))µ. Then
the conclusions of Theorem 4.6.1 remain valid.
Remark 4.6.4. As noted in Remark 4.3.3, the only ingredient that we used for proving
uniqueness is Theorem 3.2.5. On the other hand, if A is as in Theorem 4.6.1 and if
furthermore T (t) = eitA verifies the estimate
‖T (t)ϕ‖L∞ ≤ C|t|−n2 ‖ϕ‖L1 , for all ϕ ∈ D(Ω), (4.6.8)
then T (t) verifies similar estimates to those of Theorem 3.2.5 (see Theorem 3.4.1). There-
fore, we have the following result (see also Remark 4.6.2).
Theorem 4.6.5. Let A be as in the statement of Theorem 4.6.1, and assume that T (t) =
eitAverifies estimate (4.6.8). Let g ∈ C(XA, X′A), and assume that there exists g1, · · · , gk ∈
C(XA, X′A) such that
g = g1 + · · ·+ gk,
where each of the gj ’s satifies assumptions (4.6.3) through (4.6.6) for some exponents rj , ρj.
Finally, let
G = G1 + · · ·+Gk,
and let E be defined by (4.1.4). Then, the following holds.
(i) For every x ∈ XA, there exists T∗(x), T ∗(x) > 0 and there exists a unique, maximal
solution u ∈ C((−T∗(x), T ∗(x)), XA)∩C1((−T∗(x), T ∗(x)), X ′A) of problem (4.6.7). u
75
is maximal in the sense that if T ∗(x) <∞ (respectively, T∗(x) <∞), then ‖u(t)‖A →∞, as t ↑ T ∗(x) (respectively, as t ↓ −T∗(x));
(ii) in addition, we have conservation of charge and energy, that is
‖u(t)‖L2 = ‖x‖L2 , and E(u(t)) = E(x),
for all t ∈ (−T∗(x), T ∗(x));
(iii) we have continuous dependence of the solution on the initial datum in the sense that
both functions T∗(x) and T ∗(x) are lower semicontinuous, and that if xm → x in XA
and if [−T1, T2] ⊂ (−T∗(x), T ∗(x)), then um → u in C([−T1, T2], XA), where um is
the maximal solution of (4.6.7) with initial datum xm.
Remark 4.6.6. Like Theorem 4.6.1, Theorem 4.6.5 is stated for one equation, but the
method applies as well for systems of the same form (see Remark 4.6.3).
Remark 4.6.7. If we assume further that for every A > 0, there exists K(A) <∞ such
that
G(u) ≤ K(A) +1
4‖u‖2
A, for all u ∈ H10 (Ω) such that ‖u‖L2 ≤ A, (4.6.9)
then, all solutions given by Theorem 4.6.1 or Theorem 4.6.5 are global (see Remark 4.2.14).
76
5. Regularity and smoothing effect. In this chapter, we consider an open subset Ω
of Rn and a mapping g ∈ C(H10 (Ω), H−1(Ω)). Given an interval I containing 0, an initial
datum ϕ ∈ H10 (Ω), and a solution u ∈ C(I,H1
0(Ω)) ∩ C1(I,H−1(Ω)) of the problem
iut +4u+ g(u) = 0, for all t ∈ I;
u(0) = ϕ;(5.1.1)
we study the following problems.
H2 regularity: find sufficient conditions on g and/or Ω, which imply that if ϕ ∈H2(Ω), then u ∈ C(I,H2(Ω))∩C1(I, L2(Ω)). Essentially, we want an analogous of the H2
regularity property of the linear equation.
W 1,p smoothing effect: find sufficient conditions on g, when Ω = Rn, which imply
that if ϕ ∈ H1, then u ∈ Lq(I,W 1,r), for all admissible pairs (q, r). In other words, when
are the properties of Section 3.2 preserved for the nonlinear equation?
C∞ smoothing effect: when Ω = Rn, under what conditions on ϕ and g is the solution
u in C∞((I \ 0)×Rn). In other words, when are the properties of Section 3.3 preserved
for the nonlinear equation?
5.1. A general result. Consider an open domain Ω ⊂ Rn. Let g ∈ C(H10 (Ω), H−1(Ω))
and assume that for every M , there exists C(M) such that
‖g(v)− g(u)‖L2 ≤ C(M)‖v − u‖L2 ,
for all u, v ∈ H10 (Ω) such that ‖u‖H1 + ‖v‖H1 ≤M . Then, we have the following result.
Proposition 5.1.1 . Let g be as above. Consider an open interval I containing 0, an
initial datum ϕ ∈ H10 (Ω), and a solution u ∈ C(I,H1
0 (Ω)) ∩ C1(I,H−1(Ω)) of problem
(5.1.1). Then, if 4ϕ ∈ L2(Ω), we have 4u ∈ C(I, L2(Ω)) and u ∈ C1(I, L2(Ω)).
Proof. Assume that I = (−T1, T2), and let S1 ∈ (0, T1), S2 ∈ (0, T2). For |h| < minT1 −S1, T2−S2, let uh(t) = u(t+h), for t ∈ (−S1, S2). It follows from Remark 2.5.1 (iii) that
u(t) = T (t)ϕ+ i
∫ t
0
T (t− s)g(u(s)) ds,
where T (t) = eit4; and so
uh(t) = T (t)T (h)ϕ+ i
∫ t
0
T (t− s)g(uh(s)) ds+ i
∫ h
0
T (t+ h− s)g(u(s)) ds.
77
On the other hand, we know that
T (h)ϕ = ϕ+ i
∫ h
0
T (s)4ϕds.
It follows that
uh(t)− u(t) = i
∫ h
0
T (s)4ϕds
+ i
∫ t
0
T (t− s)(g(uh(s))− g(u(s))) ds+ i
∫ h
0
T (t+ h− s)(g(u(s)) ds.
Therefore, we have
‖uh(t)− u(t)‖L2 ≤ h‖4ϕ‖L2 + C(M)|∫ t
0
‖uh(s)− u(s)‖L2 ds|
+ h sup‖g(u(s))‖L2; 0 ≤ s ≤ h.
The above inequality holds for every t ∈ (−S1, S2). Since sup‖g(u(s))‖L2; 0 ≤ s ≤ h <∞, it follows from Gronwall’s Lemma that
‖uh(t)− u(t)‖L2 ≤ Ch eC(M)|I|, for all t ∈ (−S1, S2).
In particular, u : (−S1, S2) → L2(Ω) is lipschitz continuous; and so, g(u) : (−S1, S2) →L2(Ω) is also lipschitz continuous. It follows from Remark 2.3.12 (i) that we have g(u) ∈W 1,∞(−S1, S2;L
2(Ω)). It follows (see Section 2.5) that 4u ∈ C((−S1, S2), L2(Ω)) and
that u ∈ C1((−S1, S2), L2(Ω)) Hence the result, since S1 and S2 are arbitrary in (−T1, T2).
Remark 5.1.2. If we assume that Ω has a boundary of class C2, then the assumption
4ϕ ∈ L2(Ω) is equivalent to ϕ ∈ H2(Ω), and the conclusion of Proposition 5.1.1 means
that u ∈ C(I,H2(Ω)) ∩ C1(I, L2(Ω)).
Corollary 5.1.3. Let g verify the assumptions of Corollary 4.2.12 and let ϕ ∈ H10 (Ω). If
u ∈ C((−T∗(ϕ), T ∗(ϕ)), H10 (Ω)) is the solution of (5.1.1) given by Corollary 4.2.12 and if
4ϕ ∈ L2(Ω), then 4u ∈ C((−T∗(ϕ), T ∗(ϕ)), L2(Ω)) and u ∈ C1((−T∗(ϕ), T ∗(ϕ)), L2(Ω)).
Corollary 5.1.4 Let g verify the assumptions of Theorem 4.4.1, let ϕ ∈ H10 (Ω), and let
u ∈ C((−T∗(ϕ), T ∗(ϕ)), H10(Ω)) be the solution of problem (5.1.1) given by Theorem 4.4.1.
If ϕ ∈ H2(Ω), then u ∈ C((−T∗, T ∗), H2(Ω)) ∩ C1((−T∗, T ∗), L2(Ω)).
78
5.2. H2 regularity in Rn. Throughout this section, we assume that Ω = Rn. We have
the following result.
Theorem 5.2.1. Let g satisfy assumptions (4.2.1) through (4.2.4). Assume further that
there exists p > 2 such that for every M > 0, there exists C(M) <∞ such that
‖g(u)‖Lp ≤ C(M)(1 + ‖u‖H2), for all u ∈ H2(Rn) such that ‖u‖H1 ≤M. (5.2.1)
Let ϕ ∈ H1(Rn), and consider a solution u ∈ C(I,H1(Rn)) of problem (5.1.1), where I is
some interval containing 0. If ϕ ∈ H2(Rn), then u ∈ C(I,H2(Rn)) ∩ C1(I, L2(Rn)).
Before proceeding to the proof of Theorem 5.2.1, we establish some preliminary lem-
mas.
Lemma 5.2.2. Let g satisfy the assumptions of Theorem 5.2.1, let r, ρ be defined by
(4.2.2), let γ and q be such that (γ, ρ) and (q, r) are admissible pairs, and let J be an
interval. Then, the following properties hold.
(i) For every u ∈ L∞(J,H2(Rn)) ∩ C(J, Lr(Rn)), we have g(u) ∈ C(J, L2(Rn)). In
addition,
‖g(u)‖L∞(J,L2) ≤1
2‖u‖L∞(J,H2) +K,
where K depends only on ‖u‖L∞(J,H1);
(ii) For every v ∈ L∞(J,H1(Rn)) ∩W 1,q(J, Lr(Rn)), we have g(v) ∈ W 1,γ′(J, Lρ′(Rn)),
and
‖ ddtg(v)‖Lγ′(J,Lρ′ ) ≤ C|J |
q−γ′
qγ′ ‖v′‖Lq(J,Lr),
where C depends only on γ, q and ‖v‖L∞(J,H1).
Proof. Note that
‖w‖L2 ≤ ‖w‖αLρ′‖w‖1−α
Lp ,
where1
2=α
ρ′+
1− α
p. Taking w = g(u(s))− g(u(t)), the L2 continuity follows, by (4.2.3)
and (5.2.1). Furthermore, for every ε > 0, there exists C(ε) such that
‖w‖L2 ≤ ε‖w‖Lp + C(ε)‖w‖Lρ′ .
79
(i) follows, by taking w = g(u) and applying (4.2.3) and (5.2.1). (ii) follows easily from
(4.2.3) and Proposition 2.3.11 (see the proof of Theorem 4.3.1).
Lemma 5.2.3. Let J be an interval, let (a, b) be an admissible pair, and consider f ∈C(J, L2(Rn)) ∩W 1,a′(J, Lb′(Rn)). If
v(t) = i
∫ t
0
T (t− s)f(s) ds, for all t ∈ J,
then, v ∈ C(J,H2(Rn)) ∩ C1(J, L2(Rn)). Furthermore, v ∈ W 1,α(J, Lβ(Rn)), for every
admissible pair (α, β), and
‖v′‖Lα(J,Lβ) ≤ C‖f(0)‖L2 + C‖f ′‖La′(J,Lb′),
where C depends only on b and β.
Proof. Suppose first that f ∈ C1(J, Lb′(Rn)) → C1(J,H−1(Rn)). Then,
dv
dt(t) = iT (t)f(0) + i
∫ t
0
T (s)f ′(t− s) ds = iT (t)f(0) + i
∫ t
0
T (t− s)f ′(s) ds.
It follows from Theorem 3.2.5 that v ∈ C1(J, L2(Rn)) ∩W 1,α(J, Lβ(Rn)), and that
‖v′‖Lα(J,Lβ) + ‖v′‖L∞(J,L2) ≤ C‖f(0)‖L2 + C‖f ′‖La′(J,Lb′).
By density of C1(J, Lb′(Rn))∩C(J, L2(Rn)) inW 1,a′(J, Lb′(Rn))∩C(J, L2(Rn)), it follows
that for f as in the statement of the lemma, we have v ∈ C1(J, L2(Rn))∩W 1,α(J, Lβ(Rn)).
Furthermore, we have iv′ + 4v + f = 0. Therefore, 4v ∈ C(J, L2(Rn)); and so v ∈C(J,H2(Rn)).
Proof of Theorem 5.2.1. We follow the proof of Kato [1,3]. The idea is to use the space-
time estimates of Theorem 3.2.5 in order to obtain estimates of ut in Lq(I, Lr), and to
apply again Theorem 3.2.5 to obtain the H2 regularity. Note also that it is sufficient
to establish the H2 regularity on a time interval [−T, T ], where T depends only on the
norm of ϕ in H1(Rn). Indeed, since ‖u(t)‖H1 is bounded on I, the result can be applied
repeatedly to cover the interval I.
Consider the solutions um ∈ C(R, H1(Rn)) ∩ C(R, H−1(Rn)) of problem (4.2.15).
We know that there exists T > 0, depending only on the norm of ϕ in H1(Rn), such
80
that um → u, in C((−T, T ), Lp(Rn)), for 2 ≤ p <2n
n− 2, and that um is bounded in
L∞(−T, T ;H1(Rn)) (see the proof of Proposition 4.2.3). In addition, since gm is Lip-
schitz continuous on bounded sets of L2, it follows that um ∈ C([−T, T ], H2(Rn)) ∩C1([−T, T ], L2(Rn)) (see Section 2.5). In particular, gm(um) ∈ W 1,∞(−T, T ;L2(Rn)),
and it follows from Lemma 5.2.3 that umt ∈ La(−T, T ;Lb(Rn)), for every admissible pair
(a, b). In particular, umt ∈ Lq(−T, T ;Lr(Rn)), where r is given by assumption (4.2.2) and
2
q= n
(1
2− 1
r
). It follows easily from (4.2.3) and Lemma 5.2.2 (ii) that
d
dtgm(um(t)) ∈ Lγ′(−T, T ;Lρ′(Rn)),
where ρ is given by assumption (4.2.2) and2
γ= n
(1
2− 1
ρ
), and that
‖ ddtgm(um)‖Lγ′ (−T,T ;Lρ′ ) ≤ CT
q−γ′
qγ′ ‖umt ‖Lq(−T,T ;Lr), (5.2.2)
where C depends only on ‖ϕ‖H1 (note that gm verifies the same assumptions as g, uniformly
with respect to m ≥ 1). Applying Theorem 3.2.5, (5.2.2) and Lemma 5.2.3, it follows that
‖umt ‖Lq(−T,T ;Lr) + ‖um
t ‖L∞(−T,T ;L2) ≤ C(‖4ϕ‖L2 + ‖gm(ϕ)‖L2)
+ CTq−γ′
qγ′ ‖umt ‖Lq(−T,T ;Lr),
(5.2.3)
where C does not depend on ϕ. Note that (Jm)m≥1 is a bounded family of L(H2(Rn)). It
follows from assumption (5.2.1) and the definition of gm that gm(ϕ) is bounded in L2(Rn),
as m→∞. Therefore, there exists K <∞ such that
‖gm(ϕ)‖L2 ≤ K, for all m ≥ 1. (5.2.4)
Choosing T possibly smaller, but still depending only on ‖ϕ‖H1 , it follows from (5.2.3) and
(5.2.4) that (see the proof of Theorem 4.3.1) umt is a bounded sequence in L∞(−T, T ;L2)∩
Lq(−T, T ;Lr). In other words,
um is bounded in W 1,∞(−T, T ;L2) ∩W 1,q(−T, T ;Lr). (5.2.5)
It follows from (5.2.5) and Remark 3.2.12 (ii) that
u ∈W 1,∞(−T, T ;L2) ∩W 1,q(−T, T ;Lr). (5.2.6)
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Furthermore, it follows from (5.2.2) and (5.2.5) that the sequence gm(um)) is bounded in
W 1,γ′(−T, T ;Lρ′). Since gm(um)) → g(u) in C((−T, T ), Lρ′) (see the proof of Proposi-
tion 4.2.3), it follows from Theorem 2.2.4 that
g(u) ∈ W 1,γ′(−T, T ;Lρ′). (5.2.7)
On the other hand, it follows from the equation that
‖4um‖L∞(−T,T ;L2) ≤ ‖umt ‖L∞(−T,T ;L2) + ‖gm(um)‖L∞(−T,T ;L2);
and so, by (5.2.5) and conservation of charge,
‖um‖L∞(−T,T ;H2) ≤ C + ‖gm(um)‖L∞(−T,T ;L2).
Applying Lemma 5.2.2 (i) (note that gm verifies the same assumptions as g, uniformly
with respect to m ≥ 1), it follows that um is bounded in L∞(−T, T ;H2); and so,
u ∈ L∞(−T, T ;H2). (5.2.8)
The result follows, by applying (5.2.8), (5.2.6), Lemma 5.2.2, (5.2.7) and Lemma 5.2.3.
Remark 5.2.4. Note that in fact, we have also shown that, under the assumptions of
Theorem 5.2.1, u ∈W 1,aloc (I, Lb(Rn)) for every admissible pair (a, b).
Remark 5.2.5. Theorem 5.2.1 is essentially due to Kato [1,3]. Partial results in that
direction were obtained in particular by Baillon, Cazenave and Figueira [1], Lin and
Strauss [1], Y. Tsutsumi [2]. Note that we obtained the H2 regularity, essentially by
differentiating the equation with respect to t. This is especially economical, concerning
the assumptions on g, since we do not need any regularity on g(u) with respect to x.
Remark 5.2.6. One can continue the same kind of estimates, and differentiate the equa-
tion with respect to t several times, in order to obtain estimates of the solution in higher
order Sobolev spaces. However, the calculations become more and more complicated, as
well as the assumptions that have to be made on g.
Remark 5.2.7. Alternatively, when g(u) is smooth with respect ot both u and x, H2
regularity can be obtained by differentiating the equation with respect to x. One can
82
proceed further, and establish regularity in higher order Sobolev spaces. However, the
calculations become more and more complicated, as well as the assumptions that have to
be made on the growth of the derivatives of g with respect ot both u and x.
Remark 5.2.8. The proof of Theorem 5.2.1 is easily adapted to cover the case where
g = g1 + · · ·+ gk,
where each of the gj ’s satifies assumptions (4.2.1) through (4.2.4) and (5.2.1) for some
exponents rj , ρj, pj (see the proof of Theorem 4.3.1).
Remark 5.2.9. In particular, the conclusions of Theorem 5.2.1 apply to the model case
where
g(u) = V u+ f(·, u(·)) + (W ∗ |u|2)u,
when V , f and W are as follows. V : Rn → R is a real valued potential such that
V ∈ Lp(Rn) +L∞(Rn) for some p ≥ 2, p > n/2. f(x, u) : Rn × [0,∞) → R is measurable
in x and continuous in u, and f(x, 0) = 0, almost everywhere on Rn. Furthermore, there
exist constants C and α ∈ [0,4
n− 2) (α ∈ [0,∞) if n = 2) such that
|f(x, v)− f(x, u)| ≤ C(1 + |u|α + |v|α)|v − u|, for almost all x ∈ Rn and all u, v ∈ R.
If n = 1, we assume instead that for every M , there exists L(M) such that
|f(x, v)− f(x, u)| ≤ L(M)|v − u|, for almost all x ∈ Rn
and all u, v ∈ R such that |u|+ |v| ≤M.
f is extended to Rn ×C by setting
f(x, z) =z
|z|f(x, |z|), for all z ∈ C, z 6= 0, and almost all x ∈ Rn.
Finally, W : Rn → R is an even, real-valued potential such that W ∈ Lp(Rn) + L∞(Rn),
for some p ≥ 1, p > n/4. By Remarks 4.3.2 and 5.2.8, it is sufficient to verify assumption
(5.2.1) for each term of the decomposition of g, and this follows from Sobolev’s inequalities.
Remark 5.2.10. Under the assumptions of Theorem 5.2.1 (see also Remark 5.2.8), one
verifies easily the following property (follow the proof of Theorem 5.2.1, and apply Theo-
rem 4.3.1 (iii) and Remark 2.3.12 (iii)). Let ϕ ∈ H2(Rn), let ϕm → ϕ, in H2(Rn), and let
83
u, um be the corresponding maximal solutions of problem (5.1.1). Then, for every interval
[S, T ] ⊂ (−T∗(ϕ), T ∗(ϕ)), we have
‖u‖L∞(S,T ;H2) ≤ lim infm→∞
‖um‖L∞(S,T ;H2) <∞.
Note that by Theorem 4.3.1 (iii), um is defined on [S, T ] for m large enough. On the other
hand, we do not necessarily have continuous dependence in H2, if g is not smooth enough.
One shows as well (see Remark 5.2.4) that
‖ut‖Lq(S,T ;Lr) ≤ lim infm→∞
‖umt ‖Lq(S,T ;Lr) <∞,
for every admissible pair (q, r).
5.3. The W 1,p smoothing effect in Rn. Throughout this section, we assume that
Ω = Rn. We establish a result that is essentially due to Kato [1,3]. Let us first introduce
some notation.
For i ∈ 1, · · · , n and h ∈ R \ 0, we set
Dhi u =
τhi u− u
h, for all u ∈ L1
loc(Rn),
where
τhi u(x1, · · · , xn) = u(x1, · · · , xi−1, xi + h, xi+1, · · · , xn).
Clearly, Dhi is a finite difference approximation of ∂i. Our main result of this section is
the following.
Theorem 5.3.1. Let g verify assumptions (4.2.1) through (4.2.4). Assume further that
(with the notation of (4.2.2)) for every M > 0, i ∈ 1, · · · , n and h ∈ R \ 0, there exists
C(M) <∞ such that
‖Dhi g(u)‖Lρ′ ≤ C(M)‖Dh
i u‖Lr , (5.3.1)
for every u ∈ H10 (Ω) such that ‖u‖H1 ≤ M . Let ϕ ∈ H1(Rn), and consider a solution
u ∈ C(I,H1(Rn)) of problem (5.1.1), where I is some interval containing 0. Then, u ∈La
loc(I,W1,b(Rn)), for every admissible pair (a, b).
Proof. Note that it is sufficient to establish the result on a time interval [−T, T ], where
T depends only on the norm of ϕ in H1(Rn). Indeed, since ‖u(t)‖H1 is bounded on I, the
result can be applied repeatedly to cover any compact subinterval of I.
84
It is clear that Dhi commutes with T (t). Therefore, we have
Dhi u(t) = T (t)(Dh
i ϕ) + i
∫ t
0
T (t− s)Dhi g(u(s)) ds, (5.3.2)
for every t ∈ I. Note also that
Dhi ϕ(x) =
1
h
∫ h
0
∂iϕ(x1, · · · , xi + s, · · · , xn) ds;
and so,
‖Dhi ϕ‖L2 ≤ ‖∇ϕ‖L2 . (5.3.3)
Applying Theorem 3.2.5, it follows from (5.3.2) and (5.3.3) that for every admissible pair
(a, b), we have (see the proof of Theorem 4.3.1)
‖Dhi u‖La(−T,T ;Lb) ≤ C‖∇ϕ‖L2 + CT
q−γ′
qγ′ ‖Dhi u‖Lq(−T,T ;Lr),
for every T > 0 such that (−T, T ) ⊂ I. Here, r is defined by (4.2.2) and C depends only
on ‖u‖L∞(I,H1). This implies easily that if T is small enough (depending on ‖u‖L∞(I,H1)),
then (see the proof of Theorem 4.3.1) Dhi u is bounded in La(−T, T ;Lb(Rn)). Therefore,
∇u ∈ La(−T, T ;Lb(Rn)). Hence the result.
Remark 5.3.2. The conclusion of Theorem 5.3.1 holds as well if we replace assumption
(5.3.1) by the weaker assumption
‖Dhi g(u)‖Lρ′ ≤ C(M)(1 + ‖Dh
i u‖Lr ), (5.3.1′)
Remark 5.3.3. The proof of Theorem 5.3.1 is easily adapted to cover the case where
g = g1 + · · ·+ gk,
where each of the gj ’s satifies assumptions (4.2.1) through (4.2.4) and (5.3.1) (or (5.3.1’))
for some exponents rj and ρj (see the proof of Theorem 4.3.1).
Remark 5.3.4. Note that Theorem 5.3.1 contains some nontrivial information. To see
this clearly, consider the cases n = 2 and n = 3.
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In the case n = 2, we have (assuming that I is bounded) u ∈ La(I,W 1,b(R2)),
for every b ∈ [2,∞) and with1
a=
1
2− 1
b. Since W 1,b(R2) → L∞(R2) for b > 0, it
follows in particular that u ∈ La(I, L∞(R2)), for every a ∈ [2,∞). Note that in general,
u 6∈ L∞(I, L∞(R2)), since we assume only that ϕ ∈ H1(R2) and that H1(R2) 6→ L∞(R2).
In the case n = 3, we have (assuming that I is bounded) u ∈ La(I,W 1,b(R3)), for
every b ∈ [2, 6) and with2
a=
3
2− 3
b. In particular, ∇u ∈ L8/3(I, L4(R3)). On the
other hand, since u ∈ L∞(I,H1(R3)), it follows in particular that u ∈ L∞(I, L6(R3)).
Considering the inequality
‖u‖4L∞ ≤ C‖∇u‖8/3
L4 ‖u‖4/3L6 ,
it follows that u ∈ L4(I, L∞(R3)). Note that in general, u 6∈ L∞(I, L∞(R3)), since we
assume only that ϕ ∈ H1(R3) and that H1(R3) 6→ L∞(R3).
Remark 5.3.5. Let us give some examples of nonlinearities to which Theorem 5.3.1
applies. Consider a real valued potential V : Rn → R such that V ∈ Lp(Rn) + L∞(Rn),
and ∇V ∈ Lp(Rn) + L∞(Rn) for some p ≥ 1, p > n/2.
Consider a function f : [0,∞) → R. Assume that f(0) = 0, and that there exist
constants C and α ∈ [0,4
n− 2) (α ∈ [0,∞) if n = 2) such that
|f(v)− f(u)| ≤ C(1 + |u|α + |v|α)|v − u|, for all u, v ∈ R.
If n = 1, assume instead that for every M , there exists L(M) such that
|f(v)− f(u)| ≤ L(M)|v − u|, for all |u|+ |v| ≤M.
Extend f to C by setting
f(z) =z
|z|f(|z|), for all z ∈ C, z 6= 0.
Finally, consider an even, real-valued potential W : Rn → R. Assume that W ∈Lp(Rn) + L∞(Rn), for some p ≥ 1, p > n/4.
Then, Theorem 5.3.1 applies to the case where
g(u) = V u+ f(u(·)) + (W ∗ |u|2)u.
86
This follows easily from Remarks 5.3.2 and 5.3.3, and Sobolev’s inequalities (see Re-
mark 4.2.2), since
Dhi g(u) = uDh
i V + V Dhi u+
f(τhi u)− f(u)
h+ (W ∗ |u|2)Dh
i u+ (W ∗Dhi (|u|2))u.
Remark 5.3.6. Under the assumptions of Theorem 5.3.1, one verifies easily the fol-
lowing property (follow the proof of Theorem 5.3.1, and apply Theorem 4.3.1 (iii) and
Remark 2.3.12 (iii)). Let ϕ ∈ H1(Rn), let ϕm → ϕ, in H1(Rn), and let u, um be the
corresponding maximal solutions of problem (5.1.1). Then, for every interval [S, T ] ⊂(−T∗(ϕ), T ∗(ϕ)) and every admissible pair (q, r), we have
‖u‖Lq(S,T ;W 1,r) ≤ lim infm→∞
‖um‖Lq(S,T ;W 1,r) <∞.
Note that by Theorem 4.3.1 (iii), um is defined on [S, T ] for m large enough. On the
other hand, we do not necessarily have continuous dependence in Lq(S, T ;W 1,r), if g is
not smooth enough.
5.4. The C∞ smoothing effect in Rn. Throughout this section, we assume that
Ω = Rn. We present here a result of Hayashi, Nakamitsu and Tsutsumi [1,2,3], describing
a C∞ smoothing effect similar to the one observed for the linear equation (see Section 3.3).
More precisely, and under suitable assumptions on the nonlinearity, if the initial datum
ϕ decays fast enough as |x| → ∞, then the corresponding solution of (5.1.1) is smooth in
both t and x for t 6= 0, even if ϕ is not smooth.
There are many results in that direction, depending on what are the assumptions
on g(u) and on the initial data. Most of these results, however, are quite complicated
technically. Therefore, and for the sake of simplicity, we only give a simple, typical result
in order to illustrate the idea, and we refer to the papers of Hayashi, Nakamitsu and
Tsutsumi [1,2,3] for a more complete study.
Theorem 5.4.1. Assume that n = 1. Let T > 0, let ϕ ∈ H1(R), and let u ∈C([0, T ), H1(R)) ∩ C1([0, T ), H−1(R)) satisfy the equation
iut + uxx + |u|2u = 0;
u(0) = ϕ.(5.4.1)
87
If ϕ has compact support, then u ∈ C∞((0, T )×R).
Proof. Let us first do a formal calculation, in order to make clear the idea, which is quite
simple. We use the operators Pα defined in Section 3.3. More precisely, for any positive
integer m, let
um(t, x) = (x+ 2it∂x)mu(t, x). (5.4.2)
It follows from formula (3.3.4) that
um(t, x) = (2it)mei x2
4t ∂mx (e−i x2
4t u(t, x)). (5.4.3)
It follows from (5.4.2), (3.3.5), (5.4.1) and (5.4.3) that
iumt + um
xx + (2it)mei x2
4t ∂mx (|e−i x2
4t u|2e−i x2
4t u) = 0. (5.4.4)
Note that |v|2v = vvv; and so,
∂mx |v|2v =
∑
i+j+k=m
∂ixv∂
jxv∂
kxv.
Therefore, setting
v(t, x) = e−i x2
4t u(t, x), (5.4.5)
it follows from (5.4.4) that
iumt + um
xx + (2it)mei x2
4t
∑
i+j+k=m
∂ixv∂
jxv∂
kxv = 0.
Since um(0) = xmϕ, it follows that
um(t) = T (t)(xmϕ) + i
∫ t
0
T (t− s)
(2is)mei x2
4s
∑
i+j+k=m
∂ixv(s)∂
jxv(s)∂
kxv(s)
ds;
and so, by (5.4.3) and (5.4.5),
‖um(t)‖L2 ≤ ‖xmϕ‖L2 +
∫ t
0
(2s)m‖∑
i+j+k=m
∂ixv(s)∂
jxv(s)∂
kxv(s)‖L2 ds. (5.4.6)
Finally, it follows from Holder’s inequality that
‖∑
i+j+k=m
∂ixv(s)∂
jxv(s)∂
kxv(s))‖L2 ≤
∑
i+j+k=m
‖∂ixv(s)‖L
2mi‖∂j
xv(s)‖L
2mj‖∂k
xv(s)‖L2mk.
88
Furthermore, it follows from Theorem 2.3.7 that
‖∂ixv(s)‖L
2mi≤ C‖∂m
x v(s)‖i
m
L2 ‖v(s)‖m−i
m
L∞ ,
for every i ∈ 0, · · · ,m. Therefore,
∑
i+j+k=m
‖∂ixv(s)‖L
2mi‖∂j
xv(s)‖L
2mj‖∂k
xv(s)‖L2mk
≤ C‖∂mx v(s)‖L2 ‖v(s)‖2
L∞ ≤ C
sm‖um(s)‖L2 ‖u(s)‖2
L∞ ≤ C
sm‖um(s)‖L2 .
Applying (5.4.6), it follows from Gronwall’s inequality that
‖um(t)‖L2 ≤ C‖xmϕ‖L2 , (5.4.7)
for all t ∈ [0, T ), where C depends only on T , m and ‖u‖L∞(0,T ;H1). In particular, given
0 < ε < T , we have v ∈ L∞(ε, T ;Hm(R)), for every positive integer m. Since the
mapping v 7→ |v|2v is continuous Hm → Hm (see above), it follows from (5.4.4) that umt ∈
L∞(ε, T ;L2loc(R)); and so, vt ∈ L∞(ε, T ;Hm
loc(R)), for every positive integer m. It follows
in particular that v ∈ C([ε, T ), Hmloc(R)), for every positive integer m. Applying again
(5.4.4), it follows that v ∈ C1([ε, T ), Hmloc(R)), for every positive integer m. Differentiating
the equation with respect to t, k times, we obtain eventually, with the same argument, that
v ∈ Ck([ε, T ), Hmloc(R)), for every positive integers m and k. Therefore, v ∈ C∞([ε, T ]×R),
which means that u ∈ C∞([ε, T ]×R). Hence the result, since ε > 0 is arbitrary.
Now, we want to make that argument rigorous. In order to do that, we need the
following result.
Lemma 5.4.2. If ϕ and u are as above and if ϕ ∈ S(R), then u ∈ C∞([0, T ),S(R)).
Proof. We proceed in three steps.
Step 1. u ∈ L∞(0, T ;Hp(R)), for every positive integer p. We argue by iteration. The
result is certainely true for p = 1. Assuming it is true up to some p ≥ 1, let us show that it is
true for p+1. To do this, consider the approximate solution um constructed in Step 1 of the
proof of Proposition 4.2.3. It follows from Remark 4.2.15 that um → u in C([0, T ], H1(R)),
89
as m→∞. Therefore, it is sufficient to estimate um in L∞(0, T ;Hp+1(R)), independently
of m. Note that, with the notation of the proof of Proposition 4.2.3, we have
um(t) = T (t)ϕ+ i
∫ t
0
T (t− s)Jm(|Jmum|2Jmu
m(s)) ds. (5.4.8)
Since by assumption um ∈ Hp(R), we have Jmum ∈ Hp+2(R) (see Section 2.4). It
follows that |Jmum|2Jmu
m ∈ Hp+2(R) (this is due essentially to the fact that Hp+2(R) →L∞(R), see the proof of (5.4.7)). In particular, Jm(|Jmu
m|2Jmum) ∈ C([0, T ], Hp+1(R)),
and it follows from (5.4.8) that um ∈ C([0, T ], Hp+1(R)). Applying again (5.4.8), we
obtain
‖∂p+1x um(t)‖L2 ≤ ‖∂p+1
x ϕ‖L2 +
∫ t
0
T (t− s)‖∂p+1x (|Jmu
m|2Jmum(s))‖L2 ds. (5.4.9)
Finally, the integrand in the right hand side of (5.4.9) is estimated by (see the proof of
(5.4.7))
C‖Jmum(s)‖2
L∞‖∂p+1x Jmu
m(s)‖L2 ≤ C‖um(s)‖2L∞‖∂p+1
x um(s)‖L2.
This completes the proof of the first step.
Step 2. For every nonnegative integer p, xpu ∈ L∞(0, T ;Hm(R)), for every nonnegative
integer m. We argue by induction on p. We have already established the result for p = 0
(Step 1). Assuming it is true up to some p ≥ 0, let us show that it is true for p+ 1. Let
uk(t) = ∂kxu(t), for every nonnegative integer k. Given a positive integer m, we have
iumt + um
xx +∑
i+j+k=m
uiujuk = 0. (5.4.10)
Taking the L2 scalar product of (5.4.10) with ie−2εx2
x2p+2um, where ε ∈ (0, 1), it follows
that
1
2
d
dt‖e−εx2
xp+1um(t)‖2L2 = Im
∫um
xxe−2εx2
x2p+2um
+Im
∫ e−2εx2
x2p+2um∑
i+j+k=m
uiujuk
= α+ β.
(5.4.11)
We integrate the term α by parts, and we note that Imumx u
mx = 0. It follows that
α = −Im
∫ ((2p+ 2)− 4εx2)e−εx2
xpum+1e−εx2
xp+1um)
≤ C(p)‖xpum+1‖L2 ‖e−εx2
xp+1um‖L2 ≤ C(p,m)‖e−εx2
xp+1um‖L2 ,
(5.4.12)
90
by assumption. On the other hand, we have
β ≤ ‖e−εx2
xp+1um‖L2
∑
i+j+k=m
‖e−εx2
xp+1uiujuk‖L2
≤ C(m)‖e−εx2
xp+1um‖L2
m∑
k=0
‖e−εx2
xp+1uk‖L2 ,
(5.4.13)
since uj is bounded in L∞ for every j, by Step 1. It follows from (5.4.11), (5.4.12) and
(5.4.13) that
1
2
d
dt‖e−εx2
xp+1um(t)‖2L2 ≤ C(p,m)‖e−εx2
xp+1um‖L2
+C(m)‖e−εx2
xp+1um‖L2
m∑
k=0
‖e−εx2
xp+1uk‖L2 ,
for every nonnegative integer m. It follows that
1
2
d
dt
m∑
k=0
‖e−εx2
xp+1uk(t)‖2L2 ≤ C(p,m)
m∑
k=0
‖e−εx2
xp+1uk‖L2
+C(m)
m∑
k=0
‖e−εx2
xp+1uk‖2L2 .
The result follows, by integrating the above differential inequality and letting ε ↓ 0.
Step 3. Conclusion. It follows easily from Step 2 and equation (5.4.10) that xpumt ∈
L∞(0, T ;L2), for every nonnegative integers m and p. In particular, we have xpum ∈C([0, T ], L2), for every nonnegative integers m and p. Considering again equation (5.4.10),
it follows that xpum ∈ C1([0, T ], L2), for every nonnegative integers m and p. Iterating
that argument, we obtain that xpum ∈ C∞([0, T ], L2), for every nonnegative integers m
and p. This completes the proof.
End of the proof of Theorem 5.4.1. Consider a sequence ϕk ∈ S(R), such that ϕk → ϕ
in H1(R) as k →∞, and such that ‖xmϕk‖L2 ≤ 2‖xmϕ‖L2 for all positive integers k and
m. Let uk be the solution of (5.4.1) with initial datum ϕk, given by Theorem 4.4.1. It
follows from Theorem 4.4.1 that uk → u as k → ∞, in C([0, T ], H1(R)). On the other
hand, it follows from Lemma 5.4.2 that the calculations of the formal argument above are
rigorous for the solutions uk. Therefore, we have estimate (5.4.7) for the solution uk, and
it is uniform in k. It follows that ‖um(t)‖L2 ≤ C(m) for all t ∈ [0, T ) and all m ≥ 1. The
result follows, by arguing as in the formal argument that we described before.
91
Remark 5.4.3. Note that we have shown in fact that the function v defined by (5.4.5)
belongs to C∞((0, T ), Hm(Rn)), for all m ≥ 0, whenever (1 + x2)m/2ϕ ∈ L2(R), for
every positive integer m. Evidently, we also have partial results if we assume only that
(1 + x2)m0/2ϕ ∈ L2(R), for some given positive integer m0.
Remark 5.4.4. Note that we did not use fully the assumption u ∈ C([0, T ), H1(R)).
What we used precisely is that u ∈ L2(0, T, L∞(R)), and that the solution depends con-
tinuously on ϕ in L2(0, T, L∞(R)). This can be used to show a C∞ smoothing effect for
initial data in L2(R) (see Section 6.3 below).
Remark 5.4.5. In Theorem 5.4.1, we chosed the nonlinearity g(u) = |u|2u in order that
the calculation of ∂mx g(u) easy. With exactly the same method, one can establish the same
result for g(u) = λ|u|2ku, where k is a nonnegative integer and λ ∈ R. More generaly,
the result holds true when g(u) = f(|u|2)u, where f : [0,∞) → R is in C∞, but the
calculations are technically a little bit more complicated.
Remark 5.4.6. The C∞ smoothing effect of Theorem 5.4.1 holds as well (with an obvious
adaptation of the proof) for the nonlinearity g(u) = (W ∗ |u|2)u, where W ∈ L1(R) +
L∞(R). The proof makes as well use of the property that the solution depends continuously
on ϕ in L2(0, T, L∞(R)).
5.5. Comments.
Theorem 5.2.1 can be generalized in the framework of Theorem 4.6.5, with the same
proof. More precisely, one has the following result.
Theorem 5.5.1. Let A and g be as in the statement of Theorem 4.6.5. Assume further
that there exists p > 2 such that D(A) → Lp(Ω) and such that for every M > 0, there
exists C(M) <∞ such that
‖g(u)‖Lp ≤ C(M)(1 + ‖u‖D(A)), for all u ∈ D(A) such that ‖u‖A ≤M.
Let ϕ ∈ XA, and consider a solution u ∈ C(I,XA) of problem (4.6.7), where I is some
interval containing 0. Then, if ϕ ∈ D(A), we have u ∈ C(I,D(A)) ∩ C1(I, L2(Ω)).
92
Remark 5.5.2. Let Ω be a smooth, open subset of R2, and let g satisfy the assumptions
of Theorem 4.5.1. Consider ϕ ∈ H10 (Ω), and let u be the maximal solution of (4.2.6),
given by Theorem 4.5.1. Then, if ϕ ∈ H2(Ω), we have u ∈ C((−T∗(ϕ), T ∗(ϕ)), H2(Ω)) ∩C1((−T∗(ϕ), T ∗(ϕ)), L2(Ω)). This follows from Brezis and Gallouet [1].
Theorem 5.5.3. Assume that n = 2 or n = 3, and let k be any positive integer if n = 2,
and k = 1, if n = 3. Let T > 0, λ ∈ R, let ϕ ∈ H1(R), and let u ∈ C([0, T ), H1(R)) ∩C1([0, T ), H−1(R)) satisfy the equation
iut +4u+ λ|u|2ku = 0;
u(0) = ϕ.
If ϕ has compact support, then u ∈ C∞((0, T )×R).
Proof. The proof is an obvious adaptation of the proof of Theorem 5.4.1 (se also Re-
mark 5.4.5), making use of Remark 5.3.4.
Remark 5.5.4. The smoothing effect described in Section 5.4 can be proved for other
types of nonlinearities (with the same method), and in particular for Hartree type nonlin-
earities (see Hayashi [3], Hayashi and Ozawa [2]).
Remark 5.5.5. A smoothing effect of analytic type was established for equations of
the type iut + 4u = F (u, u) in Rn, where F is a polynomial in (u, u) (for example,
F (u, u) = |u|2mu, where m is a nonnegative integer). Under some decay and smoothness
assumptions on the initial value u(0) (that do not imply that u(0) is analytic), it is shown
that the corresponding solution is (real) analytic in space (see Hayashi [4,5], Hayashi and
Saitoh [1,2]), and in time (see Hayashi and Kato [1]).
93
6. Global existence and blow up. So far, we have studied the local properties of
solutions of nonlinear Schrodinger equations: local existence, regularity and smoothing
effect. In this chapter, we begin the study of the global properties of the solutions. We
establish criteria on the nonlinearity and/or the initial data to determine whether the
solutions exist for all times, or blow up in finite time. Of course, at that stage, it is
fundamental that we solved the local Cauchy problem in the energy space, since we rely
heavily on the conservation of energy to establish either global existence or blow up.
6.1. Energy estimates and global existence. We have established the local solv-
ability of the Cauchy problem in the space H10 (Ω) (see Theorems 4.2.8, 4.3.1, 4.4.1, 4.5.1,
Corollary 4.2.12, Remark 4.2.9). Therefore, in order to show that a solution u is global, it
is sufficient to establish a priori estimates on ‖u(t)‖H1 . Under some appropriate assump-
tions on the nonlinearity, this can be done easily by using the conservation laws (charge
and energy).
Let Ω be an open domain of Rn, and let g ∈ C(H10 (Ω), H−1(Ω)). Assume that
g = g1 + · · ·+ gk,
where each of the gj ’s satisfies assumptions (4.2.1) through (4.2.4), for some exponents
rj , ρj, and let
E(u) =1
2
∫
Ω
|∇u|2 dx−G(u),
where
G(u) = G1(u) + · · ·+Gk(u).
Assume further that for every ϕ ∈ H10 (Ω) and every interval I containing 0, we have
uniqueness for problem iut +4u+ g(u) = 0;
u(0) = ϕ.(6.1.1)
The main result of this section is the following.
Theorem 6.1.1. Let g be as above, and assume that there exists M > 0, C(M) > 0 and
ε ∈ (0, 1) such that
G(u) ≤ 1− ε
2‖u‖2
H1 + C(M), (6.1.2)
94
for all u ∈ H10 (Ω) such that ‖u‖L2 ≤ M . Let ϕ ∈ H1
0 (Ω) be such that ‖ϕ‖L2 ≤ M ,
and let u ∈ C((−T∗(ϕ), T ∗(ϕ)), H10 (Ω)) ∩ C1((−T∗(ϕ), T ∗(ϕ)), H−1(Ω)) be the maximal
solution of problem (6.1.1) given by Theorem 4.2.8 and Remark 4.2.9. Then, u is global
(i.e. T∗(ϕ) = T ∗(ϕ) = ∞), and sup‖u(t)‖H1 ; t ∈ R <∞.
Proof. By conservation of charge and energy, we have
‖u(t)‖2H1 ≤ ‖ϕ‖2
H1 − 2G(ϕ) + 2G(u(t)),
for all t ∈ (−T∗(ϕ), T ∗(ϕ)). Applying (6.1.2), it follows that
‖u(t)‖2H1 ≤ ‖ϕ‖2
H1 − 2G(ϕ) + (1− ε)‖u‖2H1 + 2C(‖ϕ‖L2);
and so,
ε‖u(t)‖2H1 ≤ ‖ϕ‖2
H1 − 2G(ϕ) + 2C(‖ϕ‖L2),
for all t ∈ (−T∗(ϕ), T ∗(ϕ)). Hence the result, by Theorem 4.2.8 (i) and Remark 4.2.9.
Corollary 6.1.2. Let g be as above, and assume that for every M > 0, there exists
C(M) > 0 and ε ∈ (0, 1) such that
G(u) ≤ 1− ε
2‖u‖2
H1 + C(M), (6.1.3)
for all u ∈ H10 (Ω) such that ‖u‖L2 ≤ M . Then, for every ϕ ∈ H1
0 (Ω), the maximal
solution u ∈ C((−T∗(ϕ), T ∗(ϕ)), H10(Ω))∩C1((−T∗(ϕ), T ∗(ϕ)), H−1(Ω)) of problem (6.1.1)
given by Theorem 4.2.8 and Remark 4.2.9 is global (i.e. T∗(ϕ) = T ∗(ϕ) = ∞) and
sup‖u(t)‖H1 ; t ∈ R <∞.
Proof. Apply Theorem 6.1.1.
We now give examples of applications of Theorem 6.1.1 and Corollary 6.1.2.
Example 1. Assume Ω = Rn, and let
g(u) = V u+ f(·, u(·)) + (W ∗ |u|2)u,
95
where V , f and W are as follows. V is a real valued potential V : Rn → R such that
V ∈ Lp(Rn) + L∞(Rn) for some p ≥ 1, p > n/2.
f : Rn × [0,∞) → R is measurable in x and continuous in u, and f(x, 0) = 0, almost
everywhere on Rn. If n ≥ 2, assume that there exist constants C and α ∈ [0,4
n− 2)
(α ∈ [0,∞) if n = 2) such that
|f(x, v)− f(x, u)| ≤ C(1 + |u|α + |v|α)|v − u|, for almost all x ∈ Rn and all u, v ∈ R.
If n = 1, assume instead that for every M , there exists L(M) such that
|f(x, v)− f(x, u)| ≤ L(M)|v − u|, for almost all x ∈ Rn
and all u, v ∈ R such that |u|+ |v| ≤M.
Extend f to Rn ×C by setting
f(x, z) =z
|z|f(x, |z|), for all z ∈ C, z 6= 0, and almost all x ∈ Rn.
Set
F (x, z) =
∫ |z|
0
f(x, s) ds, for all z ∈ C and almost all x ∈ Rn,
and assume that
F (x, u) ≤ A|u|2(1 + |u|δ), (6.1.4)
where 0 ≤ δ <4
n.
Finally, W : Rn → R is an even, real-valued potential such that W ∈ Lq(Rn) +
L∞(Rn), for some q ≥ 1, q > n/4. Assume further that
W+ ∈ Lσ(Rn) + L∞(Rn),
for some σ ≥ 1, σ ≥ n/2 (and σ > 1, if n = 2).
We know (see Section 4.2, Examples 1, 2, 3, 4) that g is the gradient of the potential
G defined by
G(u) =
∫
Rn
1
2V (x)|u(x)|2 + F (x, u(x)) +
1
4(W ∗ |u|2)(x)|u(x)|2
dx.
We claim that
G(u) ≤ 1
2‖u‖2
H1 + C(‖u‖L2), for all u ∈ H1(Rn). (6.1.5)
96
To see this, let V = V1 + V2, where V1 ∈ L∞ and V2 ∈ Lp, and let W+ = W1 +W2, where
W1 ∈ L∞ and W2 ∈ Lσ. It follows that
4G(u) ≤ 2‖V1‖L∞‖u‖2L2 + 2‖V2‖Lp‖u‖2
L2p
p−1+ 4A‖u‖2
L2 + 4A‖u‖δ+2Lδ+2
+‖W1‖L∞‖u‖4L2 + ‖W2‖Lσ‖u‖4
L4σ
σ−1
≤ C(1 + ‖u‖4L2) + C‖u‖2
L2p
p−1+ C‖u‖δ+2
Lδ+2 + C‖u‖4
L4σ
σ−1.
On the other hand, it follows from Theorem 2.3.7 that
‖u‖2
L2p
p−1≤ C‖u‖
np
H1‖u‖2p−n
p
L2 ;
‖u‖δ+2Lδ+2 ≤ C‖u‖
nδ2
H1‖u‖δ+2−nδ2
L2 ;
‖u‖4
L4σ
σ−1≤ C‖u‖
nσ
H1‖u‖4σ−n
σ
L2 .
Sincen
p,nδ
2,n
σ< 1, (6.1.5) follows from the inequality ab ≤ εar + C(ε)br
′
.
In particular, for g as above, the maximal solution of
iut +4u+ g(u) = 0,
u(0) = ϕ,
given by Remark 4.3.2 is global, and sup‖u(t)‖H1 ; t ∈ R <∞ (apply (6.1.5) and Corol-
lary 6.1.2).
In the case where F satisfies (6.1.4) with δ =4
n, then instead of (6.1.5), we obtain
the following inequality
G(u) ≤(
1
2+ C‖u‖
4n
L2
)‖u‖2
H1 + C(‖u‖L2), for all u ∈ H1(Rn). (6.1.6)
In this case, all solutions of (6.1.1) are global and uniformly bounded in H1, provided
‖ϕ‖L2 is small enough. This follows from (6.1.6) and Theorem 6.1.1.
Example 2. Assume that Ω is an open subset of R. Consider a function f : Ω×[0,∞) →R. Assume that f(x, u) is measurable in x and continuous in u, and that f(x, 0) = 0,
almost everywhere on Ω. Assume that f is uniformly locally Lipschitz continuous in the
sense that for every M , there exists L(M) such that
|f(x, v)− f(x, u)| ≤ L(M)|v − u|,
97
for almost all x ∈ Ω and all u, v ∈ R such that |u| + |v| ≤ M . Extend f to Ω × C by
setting
f(x, z) =z
|z|f(x, |z|), for all z ∈ C, z 6= 0, and almost all x ∈ Ω.
Let g be defined by
g(u)(x) = f(x, u(x)) almost everywhere, for every u ∈ H10 (Ω),
and let
G(u) =
∫
Ω
F (x, u(x)) dx, for every u ∈ H10 (Ω),
where
F (x, z) =
∫ |z|
0
f(x, s) ds, for all z ∈ C and almost all x ∈ Ω.
We know that the local Cauchy problem for (6.1.1) is well set inH10 (Ω) (see Theorem 4.4.1).
Furthermore, if we assume that
F (x, u) ≤ C(1 + |u|s)|u|2,
for some s < 4, then all solutions of (6.1.1) are global and uniformly bounded in H1. If
s = 4, the same holds true, provided ‖u‖L2 is small enough. The calculations are the same
as in Example 1.
Example 3. Assume Ω is an open subset of R2. Consider a function f : Ω× [0,∞) → R.
Assume that f(x, u) is measurable in x and continuous in u, and that f(x, 0) = 0, almost
everywhere on Ω. Assume that there exist a constant C such that
|f(x, v)− f(x, u)| ≤ C(1 + |u|2 + |v|2)|v − u|, for almost all x ∈ Ω and all u, v ∈ R.
Extend f to Ω×C by setting
f(x, z) =z
|z|f(x, |z|), for all z ∈ C, z 6= 0, and almost all x ∈ Ω.
Let g be defined by
g(u)(x) = f(x, u(x))almost everywhere, for every u ∈ H10 (Ω),
and let
G(u) =
∫
Ω
F (x, u(x)) dx,
98
where
F (x, z) =
∫ |z|
0
f(x, s) ds, for all z ∈ C and almost all x ∈ Ω.
We know that the local Cauchy problem for (6.1.1) is well set inH10 (Ω) (see Theorem 4.5.1).
Furthermore, it follows from Corollary 6.1.1 that all solutions of (6.1.1) are global and
uniformly bounded in H1, provided ‖u‖L2 is small enough. In addition, if we assume that
F (x, u) ≤ C(1 + |u|s)|u|2,
for some s < 2, then all solutions of (6.1.1) are global and uniformly bounded in H1. The
calculations are the same as in Example 1.
6.2. Global existence for small data. In the preceding section, we gave sufficient
conditions on the nonlinearity in order that all solutions of (6.1.1) are global. In this
section, we show that, under a different type of assumption on g, all solutions of (6.1.1)
are global, provided the initial data are small enough in H10 (Ω). Our result is the following.
Theorem 6.2.1. Let g be as in Section 6.1. Assume further that there exists ε > 0 and
a nonnegative function θ ∈ C([0, ε),R+), with θ(0) = 0, such that
G(u) ≤ 1− ε
2‖u‖2
H1 + θ(‖u‖L2), (6.2.1)
for all u ∈ H10 (Ω) such that ‖u‖H1 ≤ ε. Then, there exists δ > 0 such that for every
ϕ ∈ H10 (Ω) with ‖ϕ‖H1 ≤ δ, the maximal solution u ∈ C((−T∗(ϕ), T ∗(ϕ)), H1
0(Ω)) ∩C1((−T∗(ϕ), T ∗(ϕ)), H−1(Ω)) of problem (6.1.1) given by Theorem 4.2.8 and Remark 4.2.9
is global (i.e. T∗(ϕ) = T ∗(ϕ) = ∞) and sup‖u(t)‖H1 ; t ∈ R <∞.
Proof. First of all, note that we may assume that G(0) = 0 without restricting the
generality. Consider ϕ ∈ H10 (Ω), and suppose that on some interval [−T1, T2] 3 0, we have
‖u(t)‖H1 ≤ ε. It follows from the conservation laws (charge and energy) and inequality
(6.2.1) that
‖u(t)‖2H1 ≤ ‖ϕ‖2
H1 − 2G(ϕ) + 2θ(‖ϕ‖L2) + (1− ε)‖u(t)‖2H1 ;
and so,
‖u(t)‖2H1 ≤ 1
ε(‖ϕ‖2
H1 − 2G(ϕ) + 2θ(‖ϕ‖L2)), (6.2.2)
99
for all t ∈ [−T1, T2]. Note that the right hand side of (6.2.2) is a continuous function of ϕ
(in H10 (Ω)), which is 0 for ϕ = 0. It follows that there exists δ > 0, such that
1
ε(‖ϕ‖2
H1 − 2G(ϕ) + 2θ(‖ϕ‖L2)) ≤ ε2
4,
if ‖ϕ‖H1 ≤ δ. Therefore, if we assume that ‖ϕ‖H1 ≤ δ, it follows from (6.2.2) that
‖u(t)‖H1 ≤ ε
2,
on every interval [−T1, T2] 3 0 on which ‖u(t)‖H1 ≤ ε. An elementary continuity argument
shows that ‖u(t)‖H1 ≤ ε/2, for every t ∈ (−T∗(ϕ), T ∗(ϕ)). Therefore, T∗(ϕ) = T ∗(ϕ) = ∞.
Hence the result.
We now give examples of applications of Theorem 6.2.1.
Example 1. Assume Ω = Rn, and let
g(u) = V u+ f(·, u(·)) + (W ∗ |u|2)u,
where V , f and W are as follows. V is a real valued potential V : Rn → R such that
V ∈ Lp(Rn) + L∞(Rn) for some p ≥ 1, p > n/2.
f : Rn × [0,∞) → R is measurable in x and continuous in u, and f(x, 0) = 0, almost
everywhere on Rn. If n ≥ 2, assume that there exist constants C and α ∈ [0,4
n− 2)
(α ∈ [0,∞) if n = 2) such that
|f(x, v)− f(x, u)| ≤ C(1 + |u|α + |v|α)|v − u|, for almost all x ∈ Rn and all u, v ∈ R.
If n = 1, assume instead that for every M , there exists L(M) such that
|f(x, v)− f(x, u)| ≤ L(M)|v − u|, for almost all x ∈ Rn
and all u, v ∈ R such that |u|+ |v| ≤M.
Extend f to Rn ×C by setting
f(x, z) =z
|z|f(x, |z|), for all z ∈ C, z 6= 0, and almost all x ∈ Rn.
Set
F (x, z) =
∫ |z|
0
f(x, s) ds, for all z ∈ C and almost all x ∈ Rn.
100
Finally, W : Rn → R is an even, real-valued potential such that W ∈ Lq(Rn) +
L∞(Rn), for some q ≥ 1, q > n/4.
We know (see Section 4.2, Examples 1, 2, 3, 4) that g is the gradient of the potential
G defined by
G(u) =
∫
Rn
1
2V (x)|u(x)|2 + F (x, u(x)) +
1
4(W ∗ |u|2)(x)|u(x)|2
dx.
It follows from the same type of estimates as in Example 1 of Section 6.1 that
G(u) ≤ ‖V2‖Lp ‖u‖2H1 + C(‖u‖α+2
H1 + ‖u‖4H1) + ‖V1‖L∞ ‖u‖2
H1 , for all u ∈ H1(Rn),
where V = V1 + V2. Note that we can choose ‖V2‖Lp arbitrarily small, and for example
less than 1/2.
In particular, we can apply Theorem 6.2.1, and the maximal solution of (6.1.1) given
by Remark 4.3.2 is global, and sup‖u(t)‖H1 ; t ∈ R <∞, provided ‖ϕ‖H1 is small enough.
Example 2. Assume that Ω is an open subset of R. Consider a function f : Ω×[0,∞) →R. Assume that f(x, u) is measurable in x and continuous in u, and that f(x, 0) = 0,
almost everywhere on Ω. Assume that f is uniformly locally Lipschitz continuous in the
sense that for every M , there exists L(M) such that
|f(x, v)− f(x, u)| ≤ L(M)|v − u|,
for almost all x ∈ Ω and all u, v ∈ R such that |u| + |v| ≤ M . Extend f to Ω × C by
setting
f(x, z) =z
|z|f(x, |z|), for all z ∈ C, z 6= 0, and almost all x ∈ Ω.
Let g be defined by
g(u)(x) = f(x, u(x)) almost everywhere, for every u ∈ H10 (Ω),
and let
G(u) =
∫
Ω
F (x, u(x)) dx, for every u ∈ H10 (Ω),
where
F (x, z) =
∫ |z|
0
f(x, s) ds, for all z ∈ C and almost all x ∈ Ω.
101
As in Example 1, we can apply Theorem 6.2.1, and the maximal solution of (6.1.1) given by
Theorem 4.4.1 is global, and sup‖u(t)‖H1 ; t ∈ R <∞, provided ‖ϕ‖H1 is small enough.
6.3. L2 solutions in Rn. In this section, we assume that Ω = Rn. We apply the
results of Section 6.1 and the estimates of Section 3.2 to construct solutions of some
nonlinear Schrodinger equations, for initial data in L2(Rn). Such results were first ob-
tained by Y. Tsutsumi [4] (see also Cazenave and Weissler [2,4]). We assume that g ∈C(H1(Rn), H−1(Rn)) is the gradient of a functional G ∈ C1(H1(Rn),R). We assume
that there exists r ∈ [2,2n
n− 2) (r ∈ [2,∞], if n = 1, r ∈ [2,∞), if n = 2) such that g maps
L2(Rn) ∩ Lr(Rn) to Lr′(Rn). We define q by
2
q= n
(1
2− 1
r
), (6.3.1)
so that (q, r) is an admissible pair. Furthermore, we assume that for every M > 0, there
exists C(M) <∞ and α > 0 such that
‖g(v)− g(u)‖Lr′ ≤ C(M)(‖u‖αLr + ‖v‖α
Lr)‖v − u‖Lr , (6.3.2)
for all u, v ∈ L2(Rn) ∩ Lr(Rn) such that ‖u‖L2 + ‖v‖L2 ≤M . Finally, we suppose that g
verifies assumption (4.2.4), and that
g(0) = 0. (6.3.3)
We have the following result.
Theorem 6.3.1. Let g be as above, and let q be defined by (6.3.1). If α + 2 < q, then
for every ϕ ∈ L2(Rn), there exists a unique function u ∈ C(R, L2(Rn))∩Lqloc(R, L
r(Rn))
with ut ∈ Lqloc(R, H
−2(Rn)), solution of the following problem
iut +4u+ g(u) = 0;
u(0) = ϕ.(6.3.4)
In addition, we have ‖u(t)‖L2 = ‖ϕ‖L2 , for all t ∈ R, and u ∈ Laloc(R, L
b(Rn)) for every
admissible pair (a, b). Furthermore, if ϕm → ϕ in L2(Rn) and if um denotes the solution
of (6.3.4) with initial datum ϕ, then um → u in u ∈ Laloc(R, L
b(Rn)) for every admissible
pair (a, b).
102
Remark. Note that equation (6.3.4) makes sense for almost all t ∈ R, in H−1(Rn).
Proof. We claim that g verifies the assumptions of Corollary 6.1.2. Indeed, by Theo-
rem 4.3.1, all we have to check is inequality (6.1.3). Take u ∈ H1(Rn). We have by (6.3.3)
and (6.3.2),
G(u) =
∫ 1
0
d
dsG(u(s)) ds =
∫ 1
0
〈g(su), u〉H−1,H1 ds
=
∫ t
0
〈g(su), u〉Lr′,Lr ds ≤ C(‖ϕ‖L2)‖u‖rLr .
Since
‖u‖α+2Lr ≤ C‖u‖
α+2q
H1 ‖u‖(α+2)(q−1)
q
L2 ,
by Theorem 2.3.7, (6.1.3) follows. Furthermore, note that by (6.3.2) and Holder’s inequality
(in time), we have
‖g(v)− g(u)‖Lq′(−T,T ;Lr′ ) ≤ (C(M))1/q′T 1−α+2q (‖u‖r−2
Lq(−T,T ;Lr)
+‖v‖r−2Lq(−T,T ;Lr))‖v − u‖Lq(−T,T ;Lr),
(6.3.5)
for all u, v ∈ Lq(−T, T ;Lr(Rn))∩L∞(−T, T ;L2(Rn)) such that ‖u‖L∞(−T,T ;L2) ≤M and
‖v‖L∞(−T,T ;L2) ≤M .
Consider now ϕ ∈ L2(Rn), and let ϕm ∈ H1(Rn) converge to ϕ in L2(Rn), asm→∞.
Let um be the solution of (6.3.4). It follows from Corollary 3.1.2 that um ∈ C(R, H1(Rn)).
Furthermore, we have conservation of charge, that is
‖um(t)‖L2 = ‖ϕ‖L2 , (6.3.6)
for all t ∈ R. It follows from (6.3.5) and Theorem 3.2.5 that for every T > 0,
‖um‖Lq(−T,T ;Lr) ≤ C‖ϕm‖L2 + CT 1−α+2q ‖um‖α+1
Lq(−T,T ;Lr),
where C is independent of ϕ and m. It follows that for T small enough (depending on
‖ϕ‖L2), we have
‖um‖Lq(−T,T ;Lr) ≤ 2C‖ϕ‖L2 , (6.3.7)
for all m ≥ 1. Consider now two positive integers m, j. It follows from (6.3.5), (6.3.7) and
Theorem 3.2.5 that for every T > 0,
‖um − uj‖Lq(−T,T ;Lr) ≤ C‖ϕm − ϕj‖L2 +KT 1−α+2q ‖um − uj‖Lq(−T,T ;Lr),
103
where K depends on ‖ϕ‖L2 . Choosing T possibly smaller, but still depending on ‖ϕ‖L2 ,
it follows that um is a Cauchy sequence in Lq(−T, T ;Lr(Rn)). Applying (6.3.5) and
Theorem 3.2.5, it follows that um is a Cauchy sequence in La(−T, T ;Lb(Rn)), for every
admissible pair (a, b), and in particular in C([−T, T ], L2(Rn)). Therefore, if we denote by
u the limit of um, we have in particular, by (6.3.6),
‖um(t)‖L2 = ‖ϕ‖L2 , (6.3.8)
for all t ∈ [−T, T ]. One shows easily that u solves problem (6.3.4) (or rather its integral
formulation, which is equivalent). Since T depends on ‖ϕ‖L2 , it follows from (6.3.8) that
one can iterate the argument in order to cover the whole real line. Uniqueness in the class
C(R, L2(Rn))∩Lqloc(R, L
r(Rn)) and continuous dependence on the initial data are shown
by the same argument.
Theorem 6.3.2. Let g = g1 + · · ·+ gk, where each of the gj’s verifies the assumptions
of Theorem 6.3.1 for some exponents rj and αj . Let r = maxr1, · · · , rk, and let q
be defined by (6.3.1). Then, for every ϕ ∈ L2(Rn), there exists a unique function u ∈C(R, L2(Rn))∩Lq
loc(R, Lr(Rn)) with ut ∈ Lq
loc(R, H−2(Rn)), solution of problem (6.3.4).
In addition, we have ‖u(t)‖L2 = ‖ϕ‖L2 , for all t ∈ R, and u ∈ Laloc(R, L
b(Rn)), for every
admissible pair (a, b). Furthermore, if ϕm → ϕ in L2(Rn) and if um denotes the solution
of (6.3.4) with initial datum ϕ, then um → u in u ∈ Laloc(R, L
b(Rn)), for every admissible
pair (a, b).
Proof. The proof of Theorem 6.3.1 is easily adapted (see the proof of Theorem 4.3.1).
Remark 6.3.3. It follows from uniqueness and from Corollary 6.1.2 (see also Example 1
of Section 6.1) that under the assumptions of Theorem 6.3.2, if ϕ ∈ H1(Rn), then u ∈C(R, H1(Rn)) ∩ C1(R, H−1(Rn)).
Remark 6.3.4. Let us now give an example of application of Theorem 6.3.2. Let
g(u) = V u+ f(·, u(·)) + (W ∗ |u|2)u,
where V , f and W are as follows. V is a real valued potential V : Rn → R such that
V = V1 + V2, with V1 ∈ L∞(Rn) and V2 ∈ Lp(Rn) for some p ≥ 1, p > n/2.
104
f : Rn × [0,∞) → R is measurable in x and continuous in u, and f(x, 0) = 0, almost
everywhere on Rn. Assume that there exist constants C and β ∈ [0,4
n) such that
|f(x, v)− f(x, u)| ≤ C(1 + |u|β + |v|β)|v − u|, for almost all x ∈ Rn and all u, v ∈ R.
Extend f to Rn ×C by setting
f(x, z) =z
|z|f(x, |z|), for all z ∈ C, z 6= 0, and almost all x ∈ Rn.
Finally, W : Rn → R is an even, real-valued potential such that W = W1 +W2, with
W1 ∈ L∞(Rn) and W2 ∈ Lσ(Rn) for some σ ≥ 1, σ > n/2 (σ ≥ 1, if n = 2).
Then, g verifies the assumptions of Theorem 6.3.2. Indeed, it is sufficient to show that
each of the terms V1u, V2u, f(·, u(·)), (W1 ∗ |u|2)u, (W2 ∗ |u|2)u verifies the assumptions
of the theorem. It is immediate that V1u (respectively V2u) verify the assumptions, with
C(M) ≡ C, α = 0 and r = 2 (respectively r =2p
p− 1). Applying Holder’s and Young’s
inequalities, one verifies easily that (W1 ∗ |u|2)u (respectively (W2 ∗ |u|2)u) satisfy the
assumptions with C(M) = M2, α = 0 and r = 2 (respectively r =2σ
σ − 1). Finally, one
can write
f(x, s) = f1(x, s) + f2(x, s),
where f1 is Lipschitz continuous in s, uniformly in x, and
|f2(x, v)− f2(x, u)| ≤ C(|u|β + |v|β)|v − u|, for almost all x ∈ Rn and all u, v ∈ R.
One extends f1 and f2 to the complex plane, and one verifies easily that f1(u)(respectively
f2(u)) verify the assumptions, with C(M) ≡ C, α = 0 and r = 2 (respectively α = β and
r = β + 2).
Corollary 6.3.5. Assume n = 1, and let ϕ ∈ R and W ∈ L1(R) + L∞(R). Consider
ϕ ∈ L2(R) and let u ∈ C(R, L2(R)) ∩ L4loc(R, L
∞(R)) be the solution of the problem
iut +4u+ λ|u|2u+ (W ∗ |u|2)u = 0,
u(0) = ϕ,
given by Theorem 6.3.1 and Remark 6.3.4. If ϕ has compact support, then u ∈ C∞((R \0)×R).
105
Proof. Note that (4,∞) is an admissible pair in dimension 1. Therefore, the result follows
from Remarks 5.4.4. and 5.4.6
6.4. Blow up. Throughout this section, we assume that Ω = Rn. We show that,
under suitable assumptions on the nonlinearity, some solutions of the nonlinear Schrodinger
equation blow up in finite time. We follow the method of Glassey [1]. This is essentially a
convexity method, but not purely energetic. It is based on the calculation of
∫
Rn
|x|2|u(t, x)|2 dx.
That calculation is technically complicated. Therefore, for the sake of simplicity, we con-
sider a specific type of nonlinearity. More precisely, we assume that
g(u) = V u+ f(u(·)) + (W ∗ |u|2)u,
where V , f and W are as follows.
V is a real valued potential V : Rn → R such that V ∈ Lp(Rn) + L∞(Rn) for some
p ≥ 1, p > n/2 and x · ∇V (x) ∈ Lσ(Rn) + L∞(Rn), for some σ ≥ 1, σ > n/2.
f : [0,∞) → R is continuous and f(0) = 0. If n ≥ 2, assume that there exist constants
C and α ∈ [0,4
n− 2) (α ∈ [0,∞) if n = 2) such that
|f(v)− f(u)| ≤ C(1 + |u|α + |v|α)|v − u|, for all u, v ∈ R.
If n = 1, assume instead that for every M , there exists L(M) such that
|f(v)− f(u)| ≤ L(M)|v − u|, for all u, v ∈ R such that |u|+ |v| ≤M.
Extend f to C by setting
f(z) =z
|z|f(|z|), for all z ∈ C, z 6= 0.
Set
F (z) =
∫ |z|
0
f(s) ds, for all z ∈ C.
Finally, W : Rn → R is an even, real-valued potential such that W ∈ Lq(Rn) +
L∞(Rn), for some q ≥ 1, q > n/4 and x · ∇W (x) ∈ Lδ(Rn) + L∞(Rn), for some δ ≥ 1,
δ > n/4.
106
We know (see Section 4.2, Examples 1, 2, 3, 4) that g is the gradient of the potential
G defined by
G(u) =
∫
Rn
1
2V (x)|u(x)|2 + F (x, u(x)) +
1
4(W ∗ |u|2)(x)|u(x)|2
dx.
Finally, we set
E(u) =1
2
∫
Ω
|∇u(x)|2 dx−G(u), for all u ∈ H1(Rn).
We summarize below the local existence properties for the problem
iut +4u+ g(u) = 0;
u(0) = ϕ;(6.4.1)
(see Remarks 4.3.2 and 5.2.9)
Proposition 6.4.1. If g is as above, then the following properties hold.
(i) For every ϕ ∈ H1(Rn), there exists T∗(ϕ), T ∗(ϕ) > 0 and there exists a unique,
maximal solution u ∈ C((−T∗(ϕ), T ∗(ϕ)), H1(Rn)) ∩ C1((−T∗(ϕ), T ∗(ϕ)), H−1(Rn))
of problem (6.4.1). u is maximal in the sense that if T ∗(ϕ) <∞ (respectively, T∗(ϕ) <
∞), then ‖u(t)‖H1 →∞, as t ↑ T ∗(ϕ) (respectively, as t ↓ −T∗(ϕ));
(ii) in addition, we have conservation of charge and energy, that is
‖u(t)‖L2 = ‖ϕ‖L2 , and E(u(t)) = E(ϕ),
for all t ∈ (−T∗(ϕ), T ∗(ϕ));
(iii) we have continuous dependence of the solution on the initial datum in the sense that
both functions T∗(ϕ) and T ∗(ϕ) are lower semicontinuous, and that if ϕm → ϕ in
H1(Rn) and if [−T1, T2] ⊂ (−T∗(ϕ), T ∗(ϕ)), then um → u in C([−T1, T2], H1(Rn)),
where um is the maximal solution of (4.2.6) with initial datum ϕm;
(iv) if in addition, we assume that ϕ ∈ H2(Rn), then u ∈ C((−T∗(ϕ), T ∗(ϕ)), H2(Rn)) ∩C1((−T∗(ϕ), T ∗(ϕ)), L2(Rn)).
Our blow up result is based on the following identities, which will also be essential in
the next chapter, to establish the pseudo-conformal conservation law.
107
Proposition 6.4.2. Let g be as above. Consider ϕ ∈ H1(Rn) such that | · |ϕ(·) ∈L2(Rn). Then, with the notation of Proposition 6.4.1, the function t 7→ | · |u(t, ·) belongs
to C((−T∗(ϕ), T ∗(ϕ)), L2(Rn)). Furthermore, the function
t 7→ f(t) ≡∫
Rn
|x|2|u(t, x)|2 dx (6.4.2)
is in C2(−T∗(ϕ), T ∗(ϕ)), and we have
f ′(t) = 4Im
∫
Rn
u(x)x · ∇u(x) dx, (6.4.3)
and
f ′′(t) = 16E(ϕ) +
∫
Rn
(8(n+ 2)F (u)− 4nRe(f(u)u)) dx
+ 8
∫
Rn
(V +1
2x · ∇V )|u|2 dx+ 4
∫
Rn
((W +1
2x · ∇W ) ∗ |u|2)|u|2 dx,
(6.4.4)
for all t ∈ (−T∗(ϕ), T ∗(ϕ)).
Before proceeding to the proof, we establish some preliminary results.
Lemma 6.4.3. Let g and ϕ be as in Proposition 6.4.2. Then, the function t 7→ | · |u(t, ·)belongs to C((−T∗(ϕ), T ∗(ϕ)), L2(Rn)). Furthermore, the function f defined by (6.4.2)
belongs to C1(−T∗(ϕ), T ∗(ϕ)), and identity (6.4.3) holds.
Proof. Let ε > 0, and take the H−1 − H1 duality product of equation (6.4.1) with
ie−2ε|x|2 |x|2u(t, x) ∈ H1(Rn). It follows easily that
1
2
d
dt‖e−ε|x|2 |x|u(t)‖2
L2 = −〈4u, ie−2ε|x|2 |x|2u〉 − 〈g(u), ie−2ε|x|2|x|2u〉
= Im
∫
Rn
∇u · ∇(e−2ε|x|2 |x|2u)− e−2ε|x|2 |x|2g(u)u
dx.
Since Im(∇u · ∇u) = Im(g(u)u) = 0 almost everywhere, it follows that
1
2
d
dt‖e−ε|x|2 |x|u(t)‖2
L2 = Im
∫
Rn
u∇u · ∇(e−2ε|x|2 |x|2) dx
= 2Im
∫
Rn
e−ε|x|2ux · ∇ue−ε|x|2(1− 2ε|x|2) dx
.
108
It follows that
‖e−ε|x|2 |x|u(t)‖2L2 = ‖e−ε|x|2 |x|ϕ‖2
L2
+ 4
∫ t
0
Im
∫
Rn
e−ε|x|2ux · ∇ue−ε|x|2(1− 2ε|x|2) dx dt.(6.4.5)
Note that e−ε|x|2(1 − 2ε|x|2) is bounded in both x and ε and that ‖e−ε|x|2 |x|ϕ‖L2 ≤‖xϕ‖L2 . Furthermore, ∇u is bounded in L2(Rn), on compact intervals of (−T∗(ϕ), T ∗(ϕ)).
Therefore, it follows from (6.4.5) that for every interval [S, T ] ⊂ (−T∗(ϕ), T ∗(ϕ)), there
exists a constant C such that
‖e−ε|x|2 |x|u(t)‖L2 ≤ C,
for all t ∈ [S, T ] and all ε > 0. It follows easily that the function t 7→ | · |u(t, ·) is weakly
continuous (−T∗(ϕ), T ∗(ϕ)) → L2(Rn). In particular, we can let ε ↓ 0 in (6.4.5), and we
obtain
‖xu(t)‖2L2 = ‖xϕ‖2
L2 + 4
∫ t
0
Im
∫
Rn
ux · ∇u dx dt. (6.4.6)
Note that the right hand side is a continuous function of t; and so, the function t 7→ |·|u(t, ·)is continuous (−T∗(ϕ), T ∗(ϕ)) → L2(Rn). It follows that the right hand side of (6.4.6) is
a C1 function. Therefore, f is C1, and identity (6.4.3) holds.
Corollary 6.4.4. Let ϕ ∈ H1(Rn), and let u be the corresponding maximal solution of
(6.4.1). Let (ϕm)m∈N ⊂ H1(Rn) be such that ϕm → ϕ in H1(Rn) and xϕm → xϕ in
L2(Rn) as m →∞, and let um be the corrseponding maximal solutions of (6.4.1). Then,
for every interval [S, T ] ⊂ (−T∗(ϕ), T ∗(ϕ)), we have xum → xu in C([S, T ], L2(Rn)) as
m→∞.
Proof. Assume [S, T ] ⊂ (−T∗(ϕ), T ∗(ϕ)), and consider m large enough so that [S, T ] ⊂(−T∗(ϕm), T ∗(ϕm)) (see Proposition 6.4.1). It follows from Lemma 6.4.3 that for every
t ∈ [S, T ],
‖xum(t)‖2L2 = ‖xϕm‖2
L2 + 4
∫ t
0
Im
∫
Rn
um x · ∇um dx dt, (6.4.7)
and
‖xu(t)‖2L2 = ‖xϕ‖2
L2 + 4
∫ t
0
Im
∫
Rn
ux · ∇u dx dt. (6.4.8)
109
On the other hand, we have ∇um → ∇u, in C([S, T ], L2(Rn)), and it follows easily from
Lemma 6.4.3 that xum(t) xu(t) in L2(Rn), for all t ∈ [S, T ]. Applying (6.4.7) and
(6.4.8), it follows that ‖xum(t)‖L2 → ‖xu(t)‖L2, as m → ∞, for all t ∈ [S, T ]; and so,
xum(t) → xu(t) in L2(Rn), as m → ∞, for all t ∈ [S, T ]. It remains to show that the
convergence is uniform. We argue by contradiction, and we assume that there exists tm
and ε > 0 such that ‖um(tm)− u(tm)‖L2 ≥ ε. We may assume that there exists t ∈ [S, T ]
such that tm → t, as m→∞. We have
‖xum(tm)− xu(tm)‖2L2 = ‖xum(tm)‖2
L2 + ‖xu(tm)‖2L2 − 2〈xum(tm), xu(tm)〉L2 .
It follows easily from (6.4.7) and (6.4.8) that ‖xum(tm)‖2L2 + ‖xu(tm)‖2
L2 → 2‖xu(t)‖2L2.
Furthermore, xu(tm) → xu(t) and xum(tm) xu(t), in L2(Rn); and so, ‖xum(tm) −xu(tm)‖2
L2 → 0, which is a contradiction. This completes the proof.
Remark 6.4.5. Note that for proving Lemma 6.4.3 and Corollary 6.4.4, we did not use
the specific assumptions that we made on the nonlinearity. These results hold whenever g
is as in Theorem 5.2.1.
Lemma 6.4.6. Let g and ϕ be as in Proposition 6.4.2. Assume furthermore that ϕ ∈H2(Rn). Then, the function f defined by (6.4.2) belongs to C2(−T∗(ϕ), T ∗(ϕ)), and
identity (6.4.4) holds.
Proof. Note first that by Proposition 6.4.1, we have u ∈ C((−T∗(ϕ), T ∗(ϕ)), H2(Rn)) ∩C1((−T∗(ϕ), T ∗(ϕ)), L2(Rn)). Given ε > 0, let θε(x) = e−ε|x|2 , and let
hε(t) = Im
∫
Rn
θεux · ∇u dx = −〈rθεu, i∂ru〉,
for every t ∈ (−T∗(ϕ), T ∗(ϕ)) Let us first show that hε is C1, and that
h′ε(t) = 〈ut, i2θεr∂ru+ (nθε + r∂rθε)u〉, (6.4.9)
where ∂r =∂
∂r. Let us first observe that (6.4.9) is equivalent to
hε(t) = hε(0) +
∫ t
0
〈ut, i2θεr∂ru+ (nθε + r∂rθε)u〉 ds. (6.4.10)
110
By density, it is sufficient to establish (6.4.10) for u ∈ C1((−T∗(ϕ), T ∗(ϕ)), H1(Rn)). In
this case, we have
h′ε(t) = −〈rθεut, i∂ru〉 − 〈rθεu, i∂rut〉 = −〈ut, irθε∂ru〉 − 〈θεu, ix · ∇ut〉,
and (6.4.10) follows after integration by parts, since
θεux · ∇ut = ∇ · (xθεuut)− nθεuut − θεutx · ∇u− r∂rθεuut.
Since the right hand side of (6.4.10) is clearly a C1 function, it follows that h is C1, and
that (6.4.9) holds. Making use of equation (6.4.1), it follows that
h′ε(t) = −〈4u+ g(u), 2θεr∂ru+ (nθε + r∂rθε)u〉. (6.4.11)
On the other hand, an elementary calculation based on the identity
Re(2θε∇u · ∇(r∂ru)) = −((n− 2)θε + r∂rθε)|∇u|2 +∇ · (xθε|∇u|2),
shows that for every u ∈ H2(Rn), we have
〈4u,2θεr∂ru+ (nθε + r∂rθε)u〉 = −2
∫
Rn
θε|∇u|2 dx
−∫
Rn
2r∂rθε|∂ru|2 + r((n+ 1)∂rθε + r∂2rθε)Re(u∂ru) dx.
(6.4.12)
Furthermore, note that
〈g(u), (nθε+r∂rθε)u〉 =
∫
Rn
(nθε + r∂rθε)Re(g(u)u) dx
=
∫
Rn
(nθε + r∂rθε)V |u|2 + Re(f(u)u) + (W ∗ |u|2)|u|2 dx.(6.4.13)
On the other hand, we have
〈g(u), 2θεr∂ru〉 = −∫
Rn
(nθε + r∂rθε)V |u|2 + 2F (u) + (W ∗ |u|2)|u|2 dx
−∫
Rn
rθε|u|2(W ∗ ∂r|u|2) + θε|u|2 x · ∇V dx;
and so, by applying (6.4.11), (6.4.12) and (6.4.13),
h′ε(t) = 2
∫
Rn
θε|∇u|2 dx+
∫
Rn
(nθε + r∂rθε)(2F (u)− Re(f(u)u)) dx
+
∫
Rn
θε|u|2 x · ∇V dx+
∫
Rn
rθε|u|2(W ∗ ∂r|u|2) dx
+
∫
Rn
2r∂rθε|∂ru|2 + r((n+ 1)∂rθε + r∂2rθε)Re(u∂ru) dx.
(6.4.12)
111
Note that θε, r∂rθε and r2∂2rθε are bounded with respect to both x and ε. Furthermore, θε ↑
1, r∂rθε → 0, and r2∂2rθε → 0, as ε ↓ 0. On the other hand, for every t ∈ (−T∗(ϕ), T ∗(ϕ)),
we have u ∈ H2(Rn) by assumption, and |x|u ∈ L2(Rn) by Lemma 6.4.3; and so, it follows
from Sobolev’s, Young’s and Holder’s inequalities that we can pass to the limit in the right
hand side of (6.4.12), as ε ↓ 0. We obtain
limε↓0
h′ε(t) = 2
∫
Rn
|∇u|2 dx+ n
∫
Rn
(2F (u)− Re(f(u)u)) dx
+
∫
Rn
|u|2 x · ∇V dx+
∫
Rn
r|u|2(W ∗ ∂r|u|2) dx.
Since also
limε↓0
hε(t) = Im
∫
Rn
ux · ∇u dx ≡ h(t),
it follows that h is of class C1, and that
h′(t) = 2
∫
Rn
|∇u|2 dx+ n
∫
Rn
(2F (u)− Re(f(u)u)) dx
+
∫
Rn
|u|2 x · ∇V dx+
∫
Rn
r|u|2(W ∗ ∂r|u|2) dx.(6.4.15)
Finally, an integration by parts yields
∫
Rn
r|u|2(W ∗ ∂r|u|2) dx =1
2
∫
Rn
((x · ∇W ) ∗ |u|2)|u|2 dx; (6.4.16)
and so, (6.4.4) follows from (6.4.15), (6.4.16) and conservation of energy.
Proof of Proposition 6.4.2. The first part of the statement follows from Lemma 6.4.3.
Let now (ϕm)m∈N ⊂ H2(Rn) be such that ϕm → ϕ in H1(Rn) and xϕm → xϕ in
L2(Rn), as m → ∞, and let um be the corresponding maximal solutions of (6.4.1). Let
t ∈ (−T∗(ϕ), T ∗(ϕ)). It follows from Lemma 6.4.5 that
‖xum(t)‖2L2 = ‖xϕm‖2
L2 + 4tIm
∫
Rn
ϕm x · ∇ϕm dx+
∫ t
0
∫ s
0
hm(s) ds dt,
where hm is the right hand side of (6.4.4) corresponding to the solution um. By continuous
dependence, hm converges to the right hand side of (6.4.4), uniformly on [0, t], and by
Corollary 6.4.4, ‖xum(t)‖2L2 → ‖xu(t)‖2
L2 . (6.4.4) follows easily.
112
Theorem 6.4.7. Let g be as in Proposition 6.4.2. Assume further that
2(n+ 2)F (s)− nsf(s) ≤ 0, for all s ≥ 0; (6.4.17)
V +1
2x · ∇V ≤ 0, almost everywhere; (6.4.18)
W +1
2x · ∇W ≤ 0, almost everywhere. (6.4.19)
Let ϕ ∈ H1(Rn) be such that | · |ϕ(·) ∈ L2(Rn), and assume that E(ϕ) < 0. Then,
T∗(ϕ) < ∞ and T ∗(ϕ) < ∞.. In other words, the solution u blows up in finite time, for
both t > 0 and t < 0.
Proof. It follows from (6.4.17), (6.4.18), (6.4.19) and Proposition 6.4.2 that for every
t ∈ (−T∗(ϕ), T ∗(ϕ)), we have
‖xu(t)‖2L2 ≤ θ(t), (6.4.20)
where
θ(t) = ‖xϕ‖2L2 + 4tIm
∫
Rn
ux · ∇u dx+ 8t2E(ϕ).
Observe that θ(t) is second degree polynomial, and that the coefficient of t2 is negative.
Therefore θ(t) < 0, for |t| large enough. Since ‖xu(t)‖2L2 ≥ 0, it follows from (6.4.20) that
both T∗(ϕ) and T ∗(ϕ) are finite.
Remark 6.4.8. Note that the proof does not show that ‖xu(t)‖L2 → 0, as t ↑ T ∗(ϕ) or
t ↓ −T ∗(ϕ). See Ball [1,2] for an interesting discussion of the methods for proving blow
up.
Remark 6.4.9. Note that 2(n + 2)F (s) − nsf(s) = −ns3+ 4nd
ds(s−(2+ 4
n )F (s)); and so,
assumption (6.4.17) is equivalent to the property that s−(2+ 4n )F (s) is a nondecreasing
function of s.
On the other hand, V +1
2x · ∇V = V +
1
2r∂rV =
1
2r∂r(r
2V ). Therefore, assumption
(6.4.18) (respectively (6.4.19)) is equivalent to the property that |x|2V (x) (respectively
|x|2W (x)) is a nonincreasing function of |x|.
6.5. Comments. We begin with some examples of applications of the results of the
preceding sections.
113
Remark 6.5.1. Take g(u) = λ|u|αu, where λ ∈ R, and 0 < α <4
n− 2(0 < α < ∞, if
n = 1, 2).
(i) If λ < 0, then all solutions of (6.1.1) are global;
(ii) if λ > 0 and α <4
n, then all solutions of (6.1.1) are global;
(iii) if λ > 0 and α ≥ 4
n, then the solution of (6.1.1) is global if ‖ϕ‖H1 is small enough.
On the other hand, given ψ ∈ H1(Rn), ψ 6= 0, the solution of (6.1.1) with ϕ = kψ
blows up in finite time, provided |k| is large enough.
(i) and (ii) follow from Example 1 of Section 6.1. The first part of (iii) follows from
Example 1 of Section 6.2. Finally, the last part of (iii) follows from Theorem 6.4.7. Indeed,
it is clear that (6.4.17) is verified, and that E(kψ) < 0 for |k| large enough.
Remark 6.5.2. Take g(u) = λ(|x|−ν ∗ |u|2)u, where λ ∈ R, and 0 < ν < minn, 4.
(i) If λ < 0, then all solutions of (6.1.1) are global;
(ii) if λ > 0 and 0 < ν < 2, then all solutions of (6.1.1) are global;
(iii) if λ > 0 and ν ≥ 2, then the solution of (6.1.1) is global if ‖ϕ‖H1 is small enough. On
the other hand, given ψ ∈ H1(Rn), ψ 6= 0, the solution of (6.1.1) with ϕ = kψ blows
up in finite time, provided |k| is large enough.
(i) and (ii) follow from Example 1 of Section 6.1. The first part of (iii) follows from
Example 1 of Section 6.2. Finally, the last part of (iii) follows from Theorem 6.4.7. Indeed,
it is clear that (6.4.19) is verified, and that E(kψ) < 0 for |k| large enough.
Remark 6.5.3. Take g(u) = λ|u|αu + β(|x|−ν ∗ |u|2)u, λ, β ∈ R, 0 < α <4
n− 2, and
0 < ν < minn, 4.
(i) The solution of (6.1.1) is global if ‖ϕ‖H1 is small enough;
(ii) If λ, β < 0, then all solutions of (6.1.1) are global;
(iii) if λ < 0, β > 0 and 0 < ν < 2, then all solutions of (6.1.1) are global;
(iv) if λ > 0, β < 0 and α <4
n, then all solutions of (6.1.1) are global;
(v) if λ, β > 0, α <4
nand 0 < ν < 2, then all solutions of (6.1.1) are global;
114
(vi) if λ, β > 0, α ≥ 4
nand ν ≥ 2, then given ψ ∈ H1(Rn), ψ 6= 0, the solution of (6.1.1)
with ϕ = kψ blows up in finite time, provided |k| is large enough;
(vii) if λ > 0, β < 0, α ≥ 4
n, α > 2 and ν ≤ 2, then given ψ ∈ H1(Rn), ψ 6= 0, the solution
of (6.1.1) with ϕ = kψ blows up in finite time, provided |k| is large enough;
(viii) if λ < 0, β > 0, α ≤ 4
n, α < 2 and ν ≥ 2, then given ψ ∈ H1(Rn), ψ 6= 0, the solution
of (6.1.1) with ϕ = kψ blows up in finite time, provided |k| is large enough.
(i) follows from Example 1 of Section 6.2. (ii), (iii), (iv) and (v) follow from Example 1 of
Section 6.1. Finally, (vi), (vii) and (viii) follow from Theorem 6.4.7.
Remark 6.5.4. In the case g(u) = |u|4/nu, there is a sharp condition on the L2 norm
of the initial data for global existence. More precisely, there exists R > 0, such that if
‖ϕ‖L2 ≤ R, then the solution of (6.1.1) is global (and converges weakly to 0, as t→ ±∞).
Furthermore, given R′ ≥ R, there exists ϕ ∈ H1 such that ‖ϕ‖L2 = R′, and such that the
solution of (6.1.1) blows up in finite time (see Weinstein [1,4]).
Remark 6.5.5. Assume Ω is a smooth, open subset of Rn, and let g(u) = −|u|αu,for some α ∈ [0, 3]. Then, given ϕ ∈ H2(Ω) ∩ H1
0 (Ω) such that 4ϕ ∈ H10 (Ω), there
exists a unique global solution of (6.1.1). This completes the results of Section 4.6 (see
M. Tsutsumi [2]).
Remark 6.5.6. The nature of the singularities developping at the blow up time is not
quite well known. Here are however some results in this direction.
Assume that g(u) = |u|αu, for some 0 < α <4
n− 2, and let u be a solution of (6.1.1)
that blows up as t ↑ T ∗ > 0. Then, there exists C > 0 such that
‖∇u(t)‖L2 ≥ C(T ∗ − t)− 1α−n−2
4 ,
for all 0 ≤ t < T ∗ (see Cazenave and Weissler [4], Theorem 1.1). It follows from conserva-
tion of energy that
‖u(t)‖Lα+2 ≥ C(T ∗ − t)−2
α+2 1α−n−2
4 ,
for all 0 ≤ t < T ∗. Applying conservation of charge and Holder’s inequality ‖u‖Lα+2 ≤‖u‖a
Lp‖u‖1−aL2 , it follows that for every α+ 2 ≤ p ≤ ∞,
‖u(t)‖Lp ≥ C(T ∗ − t)−2α 1
α−n−2
4 1− 2p,
115
for all 0 ≤ t < T ∗. Finally, forαn
2< p ≤ α + 2 one can write, by using conservation of
energy and Theorem 2.3.7,
‖∇u(t)‖2
α+2
L2 ≤ C‖u‖Lα+2 ≤ C‖∇u(t)‖aL2‖u(t)‖1−a
Lp ,
with1
α+ 2= a
(1
2− 1
n− 1
p
)+
1
p. Since a <
2
α+ 2it follows that ‖u(t)‖Lp is also
estimated from below as a power of (T ∗ − t)−1. On the other hand, these lower bounds
are not upper bounds in general. For example, in the case α =4
n, there exist solutions for
which ‖∇u(t)‖L2 = C(T ∗ − t)−2 1α−n−2
4 (see Weinstein [4]).
In the special case g(u) = |u|4/nu and ‖ϕ‖L2 = R, where R is the number appearing
in Remark 6.5.4, the self-similar nature of blow up can be described precisely as follows
(see Weinstein [4]). Consider the positive ground state solution ψ of equation −4ψ +
ψ = |ψ|4/nψ (see Chapter 8), and let R = ‖ψ‖L2. Consider ϕ ∈ H1(Rn) such that
‖ϕ‖L2 = ‖ψ‖L2 , and assume that the corresponding solution of (6.1.1) blows up at the
finite time T . Let
λ(t) =‖∇ψ‖L2
‖∇u(t)‖L2
−→t→T
0.
Then, there exist functions y : [0, T ) → Rn and γ : [0, T ) → R such that
eiγ(t)Sλ(t)u(t, ·+ y(t))−→t→T
ψ, in H1(Rn),
where Sλ is the space dilation defined by Sλw(x) = λn/2w(λx).
Given k points y1, · · · , xk of R2, there exists ϕ ∈ H1(R2) such that the corrseponding
solution blows up exactly at the points y1, · · · , xk (see Merle [1]).
The solution concentrates near the origin (in L2) for spherically symmetric solutions
(see Merle and Tsutsumi [1], Merle [2,3], Nawa [1,2], Nawa and Tsutsumi [1], Tsutsumi [5]).
Many numerical experiments were performed in order to determine the nature of
singularities at blow up. See Frisch, Sulem and Sulem [1], Le Mesurier, Papanicolaou,
Sulem and Sulem [1], Mac Laughlin, Papanicolaou, Sulem and Sulem [1], Patera, Sulem
and Sulem [1].
Remark 6.5.7. There are some blow up results in strict subdomains Ω ⊂ Rn. For exam-
ple, assume Ω ⊂ Rn is smooth, bounded and starshaped about the origin, and g(u) = |u|αu,for some
4
n≤ α <
4
n− 2. Then, a solution u ∈ C1([0, T ], L2(Ω))∩C([0, T ], H2(Ω)∩H1
0 (Ω))
of (6.1.1) blows up in finite time, provided E(ϕ) < 0 (see Kavian [1], Proposition 1.2).
116
Remark 6.5.8. See M. Tsutsumi [1] for a blow up result for a nonlinear Schrodinger
equation with a damping term.
Remark 6.5.9. Theorem 6.4.7 shows the existence of solutions for which both T ∗ <∞and T∗ <∞. As a matter of fact, there exist solutions for which T ∗ = ∞ and T∗ <∞ and
solutions for which T ∗ <∞ and T∗ = ∞ (see Cazenave and Weissler [6], Remark 2.6).
Remark 6.5.10. Note that the blowup assumption in Theorem 6.4.7 is E(ϕ) < 0. How-
ever, we also assume that | · |ϕ(·) ∈ L2(Rn), and this is essential for our proof. However, it
is natural to ask whether or not this condition is purely technical. As a matter of fact, it
was shown recently that the conclusion of Theorem 6.4.7 holds without the assumption
| · |ϕ(·) ∈ L2(Rn) in two cases: if n = 1, and if n ≥ 2 and ϕ is spherically symmetric (see
Ogawa and Tsutsumi [1,2]).
Remark 6.5.11. The global existence results established in this chapter are either for
all solutions under a certain global unilateral growth condition on g or else for small
initial data under a certain logal unilateral growth condition on g near 0. One can also
obtain global existence for “large” nonlinearities and certain “large” initial data. More
precisely, assume that g(u) = λ|u|αu, with λ > 0 and a(n) < α <4
n− 2(a(n) given
by (7.2.10)). For every ψ ∈ H1(Rn), there exists b0 < ∞ such that for every b > b0
we have T ∗(eib|x|2ψ) = +∞ (see Cazenave and Weissler [6]). Note that one may have
T∗(eib|x|2ψ) <∞ (see Remark 6.5.9).
117
7. Asymptotic behavior in the repulsive case. We continue the study of the global
properties of the solutions. Our purpose in this chapter is to show, for an appropriate
class of nonlinearities, that all solutions converge weakly to 0, as t → ∞. Under stronger
assumptions on the nonlinearity, we show that all solutions are asymptotically free, and we
develop a scattering theory. Such results can be proved in two settings. The simplest one
is based on the pseudo-conformal conservation law, and applies to solutions with initial
data ϕ ∈ H1, such that | · |ϕ(·) ∈ L2. But a scattering theory can also be developped in
H1, based on Morawetz’ estimate. Throughout this chapter, we assume that Ω = Rn.
7.1. The pseudo-conformal conservation law. Throughout this section, we consider
a nonlinearity g as in Section 6.4. More precisely, we assume that
g(u) = V u+ f(u(·)) + (W ∗ |u|2)u,
where V , f and W are as follows.
V is a real valued potential V : Rn → R such that V ∈ Lp(Rn) + L∞(Rn) for some
p ≥ 1, p > n/2 and x · ∇V (x) ∈ Lσ(Rn) + L∞(Rn), for some σ ≥ 1, σ > n/2.
f : [0,∞) → R is continuous and f(0) = 0. If n ≥ 2, assume that there exist constants
C and α ∈ [0,4
n− 2) (α ∈ [0,∞) if n = 2) such that
|f(v)− f(u)| ≤ C(1 + |u|α + |v|α)|v − u|, for allu, v ∈ R.
If n = 1, assume instead that for every M , there exists L(M) such that
|f(v)− f(u)| ≤ L(M)|v − u|, for all u, v ∈ R such that |u|+ |v| ≤M.
Extend f to C by setting
f(z) =z
|z|f(|z|), for all z ∈ C, z 6= 0.
Set
F (z) =
∫ |z|
0
f(s) ds, for all z ∈ C.
Finally, W : Rn → R is an even, real-valued potential such that W ∈ Lq(Rn) +
L∞(Rn), for some q ≥ 1, q > n/4 and x · ∇W (x) ∈ Lδ(Rn) + L∞(Rn), for some δ ≥ 1,
δ > n/4.
118
We know (see Section 4.2, Examples 1, 2, 3, 4) that g is the gradient of the potential
G defined by
G(u) =
∫
Rn
1
2V (x)|u(x)|2 + F (x, u(x)) +
1
4(W ∗ |u|2)(x)|u(x)|2
dx.
Finally, we set
E(u) =1
2
∫
Ω
|∇u(x)|2 dx−G(u), for all u ∈ H1(Rn).
The local existence properties for the problem
iut +4u+ g(u) = 0;
u(0) = ϕ;(7.1.1)
are summarized in Proposition 6.4.1. The following “pseudo-conformal conservation law”,
discovered by Ginibre and Velo [1,3], is essential for the study of the asymptotic behavior
of solutions.
Theorem 7.1.1. Let g be as above. If ϕ ∈ H1(Rn) and | · |ϕ(·) ∈ L2(Rn), then
‖(x+ 2it∇)u(t)‖2L2 − 8t2G(u(t)) = ‖xϕ‖2
L2 −∫ t
0
sθ(s) ds, (7.1.2)
where
θ(t) =
∫
Rn
(8(n+ 2)F (u)− 4nRe(f(u)u)) dx+ 8
∫
Rn
(V +1
2x · ∇V )|u|2 dx
+4
∫
Rn
((W +1
2x · ∇W ) ∗ |u|2)|u|2 dx
,
(7.1.3)
for all t ∈ (−T∗(ϕ), T ∗(ϕ)).
Remark. Note that by Proposition 6.4.2, the left hand side of (7.1.2) makes sense.
Proof. Let
h(t) = ‖(x+ 2it∇)u(t)‖2L2 − 8t2G(u(t)).
We have
h(t) = ‖xu(t)‖2L2 + 4t2‖∇u(t)‖2
L2 − 4tIm
∫
Rn
ux · ∇u dx− 8t2G(u(t))
= ‖xu(t)‖2L2 − 4tIm
∫
Rn
ux · ∇u dx+ 8t2E(ϕ).
119
Applying (6.4.2) and (6.4.3), it follows that h ∈ C1(−T∗(ϕ), T ∗(ϕ)) and that
h′(t) =d
dt‖xu(t)‖2
L2 − 4Im
∫
Rn
ux · ∇u dx− 4td
dtIm
∫
Rn
ux · ∇u dx+ 16tE(ϕ)
= −t(d2
dt2‖xu(t)‖2
L2 − 16E(ϕ)
).
Applying (6.4.4), we obtain
h′(t) = −t∫
Rn
(8(n+ 2)F (u)− 4nRe(f(u)u)) dx+ 8
∫
Rn
(V +1
2x · ∇V )|u|2 dx
+4
∫
Rn
((W +1
2x · ∇W ) ∗ |u|2)|u|2 dx
.
(7.1.2) follows, after integration on (0, t).
Remark 7.1.3. Note that when V ≡ 0, f(s) = λ|s|4/ns for some λ ∈ R and W (x) =
µ|x|−2 for some µ ∈ R (µ = 0, if n = 1, 2), (7.1.2) is an exact conservation law, which is
‖(x+ 2it∇)u(t)‖2L2 − 4t2
∫
Rn
λn
n+ 2|u|2+ 4
n +µ
2(|x|−2 ∗ |u|2)|u|2
dx = ‖xϕ‖2
L2 . (7.1.4)
It corresponds to the invariance of the equation under a group of transforms (see Ginibre
and Velo [8], Olver [1]).
Remark 7.1.4. Observe that (x+ 2it∇)w = 2itei|x|2
4t ∇(e−i|x|2
4t w) for a smooth function
w(t, x); and so,
‖(x+ 2it∇)w‖2L2 = 4t2‖∇(e−i
|x|2
4t w)‖2L2 .
Therefore, if we set
v(t, x) = e−i|x|2
4t u(t, x), (7.1.5)
then (7.1.2) is equivalent to
8t2E(v(t)) = ‖xϕ‖2L2 −
∫ t
0
sθ(s) ds. (7.1.6)
That formulation of (7.1.2) will be helpful in the sequel.
Remark 7.1.5. For f as in the beginning of Section 7.1, but with |f(v) − f(u)| ≤C(|u|α + |v|α)|v−u|, we have e−
i|x|2
4t f(u) = f(e−i|x|2
4t u). It follows (see Remark 7.1.4) that
|(x+ 2it∇)f(u)| = 2|t| |∇f(v)|,
120
where v is defined by (7.1.5). It follows from the above identity and Remark 2.3.1 (vii)
that
‖(x+ 2it∇)f(u)‖L
α+2α+1
≤ C|t| ‖v‖αLα+2‖∇v‖Lα+2 .
Since |v| = |u| and 2|t| |∇v| = |(x+ 2it∇)u|, we obtain
‖(x+ 2it∇)f(u)‖L
α+2α+1
≤ C‖u‖αLα+2‖(x+ 2it∇)u‖Lα+2 . (7.1.7)
Note that when α = 0, i.e. when f is globally Lipschitz continuous, we have
‖(x+ 2it∇)f(u)‖L2 ≤ C‖(x+ 2it∇)u‖L2 . (7.1.8)
The constants C in the above inequalities are independent of u and t.
7.2. Decay of solutions in the weighted L2 space. Throughout this section, we
assume that Ω = Rn. We apply the pseudo-conformal conservation law to the study of
the asymptotic behavior of solutions. For simplicity, we restrict our attention to the model
case
g(u) = −λ|u|αu, (7.2.1)
where 0 < α <4
n− 2(0 < α <∞, if n = 1, 2) and
λ > 0, (7.2.2)
and we refer to Section 7.7 and Ginibre and Velo [1,2,3,8] for more general results. Note
that in this case, we have
T∗(ϕ) = T ∗(ϕ) = ∞, (7.2.3)
for all ϕ ∈ H1(Rn) (see Remark 6.5.1 (i)).
Furthermore, it follows from (7.1.6) that (7.1.2) is equivalent to
8t2
1
2
∫
Rn
|∇v(t)|2 dx +λ
α+ 2
∫
Rn
|v(t)|α+2 dx
=
‖xϕ‖2L2 + 4λ
4− αn
α+ 2
∫ t
0
s
∫
Rn
|v(s)|α+2 dx ds,
(7.2.4)
for all t ∈ R, where v is defined by (7.1.5). We have the following result.
121
Theorem 7.2.1. Let g be as above, consider ϕ ∈ H1(Rn) such that | · |ϕ(·) ∈ L2(Rn),
and let u be the maximal solution of (7.1.1) (see Proposition 6.4.1). Then, the following
holds.
(i) If α ≥ 4
n, then for every 2 ≤ r ≤ 2n
n− 2(2 ≤ r ≤ ∞, if n = 1; 2 ≤ r < ∞, if n = 2),
there exists C such that
‖u(t)‖Lr ≤ C|t|−n( 12− 1
r ), (7.2.5)
for all t ∈ R;
(ii) If α <4
n, then for every 2 ≤ r ≤ 2n
n− 2(2 ≤ r ≤ ∞, if n = 1; 2 ≤ r < ∞, if n = 2),
there exists C such that
‖u(t)‖Lr ≤ C|t|−n( 12− 1
r )(1−θ(r)), (7.2.6)
for all t ∈ R, where
θ(r) =
0, if 2 ≤ r ≤ α+ 2;
(r − (α+ 2))(4− nα)
(r − 2)(2α+ 4− nα)if r > α+ 2.
Proof. If α ≥ 4
n, it follows from (7.2.4) that
8t2∫
Rn
|∇v(t)|2 dx ≤ ‖xϕ‖2L2 ,
for all t ∈ R. Applying conservation of charge and Theorem 2.3.7, it follows that
‖u(t)‖Lr = ‖v(t)‖Lr ≤ C‖∇v(t)‖n( 12− 1
r )L2 ‖v(t)‖1−n( 1
2− 1r )
L2
≤ C|t|−n( 12− 1
r )‖ϕ‖1−n( 12− 1
r )L2
≤ C|t|−n( 12− 1
r ).
Hence (i). Assume now α <4
n. We consider the case t ≥ 1, the argument being the same
for t < −1. It follows from (7.1.4) that
8t2E(v(t)) = 8E(v(1)) + 4λ4− αn
α+ 2
∫ t
1
s
∫
Rn
|v(s)|α+2 dx ds.
This implies that
h(t) ≤ C +4− nα
2
∫ t
1
1
sh(s),
122
where
h(t) = t2∫
Rn
|v(t)|α+2 dx.
Applying Gronwall’s Lemma, it follows that
h(t) ≤ Ct4−nα
2 ,
from which we deduce
‖v(t)‖Lα+2 ≤ Ct−n( 12− 1
α+2 ). (7.2.7)
Applying (7.2.4) and (7.2.7), we obtain
8t2∫
Rn
|∇v(t)|2 dx ≤ C + C
∫ t
0
s1−nα2 ds ≤ C + Ct2t−
nα2 ;
and so,
‖∇v(t)‖L2 ≤ Ct−nα4 . (7.2.8)
Applying (7.2.7), Holder’s inequality and conservation of charge, it follows that for every
2 ≤ r ≤ α+ 2, we have
‖u(t)‖Lr = ‖v(t)‖Lr ≤ C‖v(t)‖2(α+2)
α ( 12− 1
r )Lα+2 ‖v(t)‖1− 2(α+2)
α ( 12− 1
r )L2
≤ C|t|−n( 12− 1
r )‖ϕ‖1− 2(α+2)α ( 1
2− 1r )
L2
≤ C|t|−n( 12− 1
r ).
This implies (7.2.6), for 2 ≤ r ≤ α + 2. For α + 2 < r ≤ 2n
n− 2, it follows from (7.2.7),
(7.2.8) and Theorem 2.3.7 that
‖u(t)‖Lr = ‖v(t)‖Lr ≤ C‖∇v(t)‖2n(r−α−2)
r(2α+4−nα)
L2 ‖v(t)‖1− 2n(r−α−2)r(2α+4−nα)
Lα+2
≤ Ct−n( 12− 1
r )(1−θ(r)).
Hence (7.2.6), for r > α+ 2. This completes the proof.
Remark 7.2.2. Theorem 7.2.1 implies in particular that all solutions of (7.1.1) for which
| · |ϕ(·) ∈ L2(Rn) converge weakly to 0 in L2(Rn), as |t| → ∞. Indeed, they are bounded
in L2, and they converge to 0 strongly in Lα+2.
123
Remark 7.2.3. Note that for α ≥ 4
n, u has the same decay properties in Lr as the
solutions of the linear Schrodinger equation (see Proposition 3.2.1), for every 2 ≤ r <2n
n− 2. When α <
4
n, the decay properties are the same for r ≤ α+ 2.
Corollary 7.2.4. Let g be as in Theorem 7.2.1, and assume that α > a(n), where
a(n) =2− n+
√n2 + 12n+ 4
2n. (7.2.10)
If ϕ ∈ H1(Rn) and | · |ϕ(·) ∈ L2(Rn), and if u is the maximal solution of (7.1.1) (see
Proposition 6.4.1), then u ∈ Lq(R,W 1,r(Rn)) for every admissible pair (q, r). In addition,
if
v(t) = (x+ 2it∇)u(t), for t ∈ R, (7.2.11)
then v ∈ Lq(R, Lr(Rn)) for every admissible pair (q, r).
Remark. Note that2
n< a(n) <
4
n. On the other hand, it follows from Proposition 6.4.2
that v is well defined, and that v ∈ C(R, L2(Rn)).
Before proceeding to the proof, we need to establish the following lemma.
Lemma 7.2.5. Let g(u)(·) = f(u(·)) with f is as in Example 2 or Example 3 of
Remark 4.2.2. Let ϕ ∈ H1(Rn) be such that | · |ϕ(·) ∈ L2(Rn) and let u be the
maximal solution of (7.1.1) (see Remark 4.3.2). If v is defined by (7.2.11), then v ∈Lq
loc(−T∗(ϕ), T ∗(ϕ);Lr(Rn)) for every admissible pair (q, r).
Proof. Given ε > 0, let
fε(s) = f(s√
1 + εs2),
for s ≥ 0. Let gε be defined accordingly, and let uε be the maximal solution of the
corresponding nonlinear Schrodinger equation, with initial datum ϕ. Note that for every
ε > 0, fε is Lipschitz continuous [0,∞) → R. It follows (see Example 1 of Section 6.1) that
the solutions uε are global. Furthermore, a straightforward adaptation of Theorems 4.2.3
and 4.2.8 shows that uε → u in C([s, t], H1(Rn), for every interval [s, T ] ⊂ (−T∗(ϕ), T ∗(ϕ))
(see also Remark 4.2.15). Therefore, it is sufficient to obtain estimates on the solutions
124
uε, independently of ε. It follows from Lemma 6.4.3 that vε(t) ≡ (x + 2it∇)uε(t) ∈C([s, t], L2(Rn)). Furthermore, it follows from formula (3.3.5) that
vε(t) = T (t)xϕ+ i
∫ t
0
T (t− s)(x+ 2is∇)gε(uε(s)) ds. (7.2.12)
Since fε is Lipschitz continuous, it follows from (7.1.8) that there exists Cε such that
‖(x+ 2it∇)gε(w)‖L2 ≤ Cε‖(x+ 2it∇)w‖L2 .
Therefore, (x + 2it∇)gε(uε) ∈ L∞(s, t, L2(Rn)). Applying Theorem 3.2.5, it follows that
vε ∈ Lq(s, t;Lr(Rn)), for every admissible pair (q, r). Next, one verifies easily that there
exists C, independent of ε, such that
|gε(v)− gε(u)| ≤ C(1 + |u|α + |v|α)|v − u|,
for all u, v ∈ C. Therefore, it follows from Theorem 3.2.5 and (7.2.12) that (see the proof
of Theorem 5.3.1)
‖vε‖Lγ(s,t;Lρ) ≤ C‖xϕ‖L2 + C|t− s| ‖vε‖L1(s,t;L2)
+ C|t− s|q−q′
qq′ ‖vε‖Lq(s,t;Lr),
where (γ, ρ) is any admissible pair and (q, r) is the admissible pair such that r = α + 2.
One concludes like for Theorem 5.3.1.
Proof of Corollary 7.2.4. We proceed in two steps.
Step 1. u ∈ Lq(R,W 1,r(Rn)), for every admissible pair (q, r). Note first that by Re-
mark 5.3.5, we have u ∈ Lqloc(R,W
1,r(Rn)). Consider now r = α + 2, and let q be such
that (q, r) is an admissible pair. We have
u(t) = T (t)ϕ− iλ
∫ t
0
T (t− s)|u|αu(s) ds.
Therefore, it follows from Theorem 3.2.5 that for every t ≥ T ≥ 0, we have
‖u‖Lq(0,t;W 1,r) ≤ C‖ϕ‖H1 + C‖ |u|αu‖Lq′(0,T ;W 1,r′) + C‖ |u|αu‖Lq′(T,t;W 1,r′ ),
where C is independent of t and T . Since
‖ |u|αu‖W 1,r′ ≤ C‖u‖αLr‖u‖W 1,r ,
125
it follows, by Holder’s inequality that for every T ∈ [0, t],
‖u‖Lq(0,t;W 1,r) ≤ C‖ϕ‖H1 + C
(∫ T
0
‖u(s)‖αq
q−2
Lr ds
) q−2q
‖u‖Lq(0,T ;W 1,r)
+ C
(∫ t
T
‖u(s)‖αq
q−2
Lr ds
) q−2q
‖u‖Lq(T,t;W 1,r).
(7.2.13)
By Theorem 7.2.1, we have ‖u(s)‖Lr ≤ Cs−2/q, and so
‖u(s)‖αq
q−2
Lr ≤ Cs−2 αq−2 .
Note that since α > a(n), we have 2α > q − 2. Therefore, for T large enough, we have
C
(∫ t
T
‖u(s)‖αq
q−2
Lr ds
) q−2q
≤ 1
2, (7.2.14)
On the other hand, u ∈ L∞(0, T ;H1(Rn))∩Lq(0, T ;W 1,r(Rn)). Therefore, it follows from
(7.2.13) and (7.2.14) that
‖u‖Lq(0,t;W 1,r) ≤ C +1
2‖u‖Lq(0,t;W 1,r).
Letting t ↑ ∞, we obtain
‖u‖Lq(0,∞;W 1,r) ≤ 2C.
One proves as well that u ∈ Lq(−∞, 0;W 1,r). Applying again Theorem 3.2.5, one obtains
the result for every admissible pair.
Step 2. v ∈ Lq(R, Lr(Rn)), for every admissible pair (q, r). Note that by Lemma 7.2.5,
we have v ∈ Lqloc(R, L
r(Rn)). Consider now r = α+ 2, and let q be such that (q, r) is an
admissible pair. It follows from formula (3.3.5) that
v(t) = T (t)xϕ− iλ
∫ t
0
T (t− s)(x+ 2is∇)|u|αu(s) ds.
Therefore, it follows from Theorem 3.2.5 that for every t > 0, we have
‖v‖Lq(0,t;Lr) ≤ C‖xϕ‖L2 + C‖(x+ 2is∇)|u|αu‖Lq′(0,t;Lr′),
where C is independent of t. By applying (7.1.7), we obtain
‖v‖Lq(0,t;Lr) ≤ C‖xϕ‖L2 + C
(∫ T
0
‖u(s)‖αq
q−2
Lr ds
) q−2q
‖v‖Lq(0,T ;Lr)
+ C
(∫ t
T
‖u(s)‖αq
q−2
Lr ds
) q−2q
‖v‖Lq(T,t;Lr),
(7.2.15)
126
for every T ∈ [0, t]. By Theorem 7.2.1, we have ‖u(s)‖Lr ≤ Cs−2/q, and so
‖u(s)‖αq
q−2
Lr ≤ Cs−2 αq−2 .
Note that since α > a(n), we have 2α > q − 2. Therefore, for T large enough, we have
C
(∫ t
T
‖u(s)‖αq
q−2
Lr ds
) q−2q
≤ 1
2, (7.2.16)
On the other hand, we have v ∈ Lq(0, T, Lr(Rn)) by Lemma 7.2.5. Therefore, it follows
from (7.2.15) and (7.2.16) that
‖v‖Lq(0,t;Lr) ≤ C +1
2‖v‖Lq(0,t;Lr).
Letting t ↑ ∞, we obtain
‖v‖Lq(0,∞;Lr) ≤ 2C.
One proves as well that v ∈ Lq(−∞, 0;Lr). Applying again Theorem 3.2.5, one obtains
the result for every admissible pair.
7.3. Scattering theory in the weighted L2 space. We study with more detail the
asymptotic behavior of the solutions of (7.1.1). We still assume that Ω = Rn, and we
assume that g is as in the preceding section, that is
g(u) = −λ|u|αu,
for some λ > 0 and 0 < α <4
n− 2(0 < α < ∞, if n = 1, 2). We refer to Section 7.7 and
to Ginibre and Velo [1,2,3,8] for more general results. The results of this section are due
to Y. Tsutsumi [1] (see also Tsutsumi and Yajima [1]).
We define the Hilbert space
Σ = u ∈ H1(Rn); | · |u(·) ∈ L2(Rn),
equipped with the norm ‖u‖2Σ = ‖u‖2
H1 + ‖xu‖2L2 , and we begin with the following result.
Theorem 7.3.1. Let g be as above, and assume that α > a(n), where a(n) is defined by
(7.2.10). If ϕ ∈ Σ and if u is the maximal solution of (7.1.1) (see Proposition 6.4.1), then
there exists u+, u− ∈ Σ such that
‖T (−t)u(t)− u±‖Σ −→t→±∞
0.
127
In addition,
‖u+‖L2 = ‖u−‖L2 = ‖ϕ‖L2 ,
and1
2
∫
Rn
|∇u+|2 =1
2
∫
Rn
|∇u−|2 = E(ϕ).
Proof. Let v(t) = T (−t)u(t). We have
v(t) = ϕ− iλ
∫ t
0
T (−s)|u|αu(s) ds.
Therefore, for 0 < t < τ ,
v(t)− v(τ) = −iλ∫ t
τ
T (−s)|u|αu(s) ds. (7.3.1)
It follows from Theorem 3.2.5 that
‖v(t)− v(τ)‖H1 = ‖T (t)(v(t)− v(τ))‖H1 ≤ C‖ |u|αu‖Lq(t,τ ;W 1,r),
where (q, r) is the admissible pair such that r = α+ 2. It follows that
‖v(t)− v(τ)‖H1 −→t,τ→∞
0,
(see the proof of Corollary 7.2.4). Therefore, there exists u+ ∈ H1(Rn) such that v(t) →u+ inH1, as t→∞. One shows as well that there exists u− ∈ H1(Rn) such that v(t) → u−
in H1, as t→ −∞. Finally, it follows from formulas (7.3.1) and (3.3.5) that
x(v(t)− v(τ)) = −iλ∫ t
τ
T (−s)(x+ 2is∇)|u|αu(s) ds.
It follows from Theorem 3.2.5 that
‖x(v(t)− v(τ))‖L2 = ‖T (t)x(v(t)− v(τ))‖L2 ≤ C‖(x+ 2is∇)|u|αu‖Lq(t,τ ;Lr),
where (q, r) is the admissible pair such that r = α+ 2; and so,
‖x(v(t)− v(τ))‖L2 −→t,τ→∞
0,
128
(See the proof of Corollary 7.2.4.) Therefore, x(v(t)−u+) → 0 in L2, as t→∞. One shows
as well that x(v(t)− u−) → 0 in L2, as t→ −∞. The other properties follow immediately
from conservation of charge and energy.
Remark 7.3.2. Theorem 7.3.1 means that u(t) is asymptotic, as t→ ±∞, to a solution of
the linear Schrodinger equation. In other words, the effect of the nonlinearity is neglectable
for large times.
Remark 7.3.3. The mappings U+ : ϕ 7→ u+ and U− : ϕ 7→ u− defined by Theorem 7.3.1
map Σ → Σ. In fact, one can show with the same estimates that U+ and U− are continuous.
Note that U+ and U− are nonlinear operators.
Remark 7.3.4. It follows from Corollary 3.2.7 that one has the following formula
u± = ϕ− iλ
∫ ±∞
0
T (−s)|u|αu(s) ds.
In particular, we have
u(t) = T (t)u± + iλ
∫ ±∞
t
T (t− s)|u|αu(s) ds, (7.3.2)
for all t ∈ R.
Theorem 7.3.5. If g is as in Theorem 7.3.1, then the following properties hold.
(i) For every u+ ∈ Σ, there exists a unique ϕ ∈ Σ such that the maximal solution
u ∈ C(R, H1(Rn)) of (7.1.1) verifies ‖T (−t)u(t)− u+‖Σ → 0, as t→ +∞;
(ii) for every u− ∈ Σ, there exists a unique ϕ ∈ Σ such that the maximal solution
u ∈ C(R, H1(Rn)) of (7.1.1) verifies ‖T (−t)u(t)− u−‖Σ → 0, as t→ −∞;
Proof. We prove (i), the proof of (ii) being similar. The idea of the proof is to solve
equation (7.3.2) by a fixed point argument. To that end, we introduce the functions
ω(t) = T (t)u+, and z(t) = (x + 2it∇)ω(t, x). Let (q, r) be the admissible pair such that
r = α+ 2. It follows from Theorem 3.2.5 and Corollary 3.3.4 that ω ∈ Lq(R,W 1,r(Rn)),
z ∈ Lq(R, Lr(Rn)), and that ‖ω(t)‖Lr ≤ C|t|−2/q. Let
K = ‖ω‖Lq(R,W 1,r) + ‖z‖Lq(R,Lr) + sup|t|2/q‖ω(t)‖Lr ; t ∈ R. (7.3.3)
129
Given S > 0, let
X =u ∈ Lq(S,∞;W 1,r(Rn)); (x+ 2it∇)u(t, x) ∈ Lq(S,∞;Lr(Rn)) and
‖u‖Lq(S,∞;W 1,r) + ‖(x+ 2it∇)u(t)‖Lq(S,∞;Lr)+
sup|t|2/q‖ω(t)‖Lr ; t ∈ [S,∞) ≤ 2K,
and let
d(u, v) = ‖v − u‖Lq(S,∞;Lr), for u, v ∈ X.
It is easily checked that (X, d) is a complete metric space. Given u ∈ X, we have
‖g(u(t))‖W 1,r′ ≤ ‖u(t)‖αLr‖u(t)‖W 1,r ≤ λ(2K)αt−2α/q‖u(t)‖W 1,r ;
and so, by Holder’s inequality,
‖g(u)‖Lq′(S,∞;W 1,r′) ≤ λ(2K)α
(∫ ∞
S
s−2αq−2 ds
) q−2q
‖u‖Lq(S,∞;W 1,r)
≤ C(2K)α+1S1− 2αq−2 ,
since 2α > q − 2. It follows from Corollary 3.2.7 that the function
Fu(t) = −i∫ ∞
t
T (t− s)g(u(s)) ds, (7.3.4)
is well defined, that
Fu ∈ C([S,∞), H1(Rn)) ∩ Lq(S,∞;W 1,r(Rn)), (7.3.5)
and that
‖Fu‖Lq(S,∞;W 1,r) ≤ K/3, for S large enough. (7.3.6)
Furthermore, we have
(x+ 2it∇)Fu(t) = −i∫ ∞
t
T (t− s)[(x+ 2is∇)g(u(s))] ds,
by formula (3.3.5). Since ‖(x + 2is∇)g(u(s))‖Lr′ ≤ C‖u(s)‖αLr‖(x + 2is∇)u(s)‖Lr , one
concludes as above that
(x+ 2it∇)Fu ∈ C([S,∞), L2(Rn)) ∩ Lq(S,∞;Lr(Rn)), (7.3.7)
130
and that
‖(x+ 2it∇)Fu‖Lq(S,∞;Lr) ≤ K/3, for S large enough. (7.3.8)
Finally, it follows from Proposition 3.2.1 that
‖Fu(t)‖Lr ≤ λ
∫ ∞
t
|t− s|−2/q‖u(s)‖α+1Lr ds ≤ C(2K)α+1S1− 2(α+1)
q t−2/q,
since 2(α+ 1) > q. It follows that for S large enough, we have
supt2/q‖Fu(t)‖Lr ; t ∈ [S,∞) ≤ K/3. (7.3.9)
Putting together (7.3.3), (7.3.6), (7.3.8) and (7.3.9), it follows that A defined by
Au(t) = T (t)u+ + Fu(t), for t ≥ S,
maps X to itself if S is large enough. One verifies easily with the same type of estimates
that if S is large enough, one has
d(Au,Av) ≤1
2d(u, v), for all u, v ∈ X. (7.3.10)
It follows from Banach’s fixed point theorem that A has a fixed point u ∈ X, which
satisfies equation (7.3.2) on [S,∞). It follows from (7.3.5), (7.3.7), Theorem 3.2.5 and
Corollary 3.3.4 that u ∈ C([S,∞), H1(Rn)) and that (x+2it∇)u ∈ C([S,∞), L2(Rn)). In
particular, ψ = T (−S)u(S) ∈ Σ. Note also that
u(t) = T (t)ψ − i
∫ t
S
T (t− s)g(u(s)) ds.
Therefore, u is the solution of problem
iut +4u+ g(u) = 0;
u(S) = ψ.
Note that by Remark 6.5.1, the solution u is global. In particular, u(0) is well defined,
and by Proposition 6.4.1, u(0) ∈ Σ. It is not difficult to show with the above arguments
that ϕ = u(0) verifies the conclusions of the theorem. Uniqueness follows easily from
Remark 7.3.4 and (7.3.10).
Remark 7.3.6. The mappings Ω+ : u+ 7→ ϕ and Ω− : u− 7→ ϕ defined by Theorem 7.3.5
map Σ → Σ. In fact, one can show with the same estimates that Ω+ and Ω− are continuous.
131
These operators are called the wave operators. It follows from Theorems 7.3.1 and 7.3.5
that U±Ω± = Ω±U± = I on Σ, where U± is defined by Remark 7.3.3. In particular,
Ω± : Σ → Σ is one to one, with continuous inverse (Ω±)−1 = U±.
Theorem 7.3.7. If g is as in Theorem 7.3.1, then for every u− ∈ Σ, there exists a unique
u+ ∈ Σ with the following property. There exists a unique ϕ ∈ Σ, such that the maximal
solution u ∈ C(R,Σ) of (7.1.1) verifies T (−t)u(t) → u± in Σ, as t→ ±∞. The scattering
operator
S :Σ → Σ
u− 7→ u+
is continuous, one to one, and its inverse is continuous Σ → Σ. In addition, ‖u+‖L2 =
‖u−‖L2 and ‖∇u+‖L2 = ‖∇u−‖L2 , for every u− ∈ Σ.
Proof. The result follows from Theorems 7.3.1 and 7.3.5, and Remark 7.3.6, by setting
S = U+Ω−. Note that S−1 = U−Ω+.
Remark 7.3.8. The scattering operator S is defined on Σ. One naturally wonders
whether S can be defined on the whole energy space H1(Rn). This turns out to be the
case, under more restrictive assumptions on α, and that result is the subject of the following
three sections.
7.4. Morawetz’ estimate. This section is devoted to the proof of Morawetz’ estimate,
which is the essential for construction the scattering operator on the energy space. See
Lin and Strauss [1], and Ginibre and Velo [6,7]. We begin with the following generalized
Sobolev’s estimates.
Lemma 7.4.1. Let 1 ≤ p < ∞. If q < n is such that 0 ≤ q ≤ p, then|u(·)|p| · |q ∈ L1(Rn)
for every u ∈W 1,p(Rn). Furthermore,∫
Rn
|u(x)|p|x|q dx ≤
(p
n− q
)q
‖u‖p−qLp ‖∇u‖q
Lp , (7.4.1)
for every u ∈W 1,p(Rn).
Proof. By density and Fatou’s lemma, it is sufficient to establish (7.4.1) for u ∈ D(Rn).
Let z(x) =x
|x|q . We have ∇ · z =n− q
|x|q . Integrating formula
|u|p∇ · z = ∇ · (|u|pz)− p|u|p−1∇|u|,
132
on the set x ∈ Rn; |x| ≥ r > 0, it follows that
(n− q)
∫
|x|≥r
|u(x)|p|x|q dx ≤ p
∫
|x|≥r
|u(x)|p−1|∇u(x)||x|q−1
dx. (7.4.2)
Applying Holder’s inequality, it follows that
(n− q)
∫
|x|≥r
|u(x)|p|x|q dx ≤ p
(∫
|x|≥r
|u(x)|p|x|q dx
) q−1q
‖u‖p−q
q
Lp ‖∇u‖Lp .
(7.4.1) follows, by letting r ↓ 0.
Corollary 7.4.2. If n ≥ 4, then|u(·)|2| · |3 ∈ L1(Rn) for every u ∈ H2(Rn). Furthermore,
there exists C such that ∫
Rn
|u(x)|2|x|3 dx ≤ C‖u‖2
H2 , (7.4.3)
for every u ∈ H2(Rn).
Proof. Note that it suffices to establish (7.4.3) for u ∈ D(Rn). Applying (7.4.2) with
q = 3 and p = 2, we obtain
(n− 3)
∫
|x|≥r
|u(x)|2|x|3 dx ≤ 2
∫
|x|≥r
|u(x)||x|
|∇u(x)||x| dx
≤(∫
|x|≥r
|u(x)|2|x|2 dx
)1/2(∫
|x|≥r
|∇u(x)|2|x|2 dx
)1/2
.
Applying (7.4.1) with p = q = 2, to both u and ∇u, it follows that
(n− 3)
∫
|x|≥r
|u(x)|2|x|3 dx ≤ C‖u‖H1‖∇u‖H1 ≤ C‖u‖2
H2 .
Hence the result, by letting r ↓ 0.
We now assume that n ≥ 3, and we consider
g(u) = V u+ f(u(·)) + (W ∗ |u|2)u,
where V , f and W are as follows.
V is a real valued potential V : Rn → R such that V ∈ Lp(Rn) + L∞(Rn) and
∇V ∈ Lp(Rn) + L∞(Rn) for some p > n/2.
133
f : [0,∞) → R is continuous and f(0) = 0. We assume that there exist constants C
and α ∈ [0,4
n− 2) such that
|f(v)− f(u)| ≤ C(1 + |u|α + |v|α)|v − u|, for allu, v ∈ R.
Extend f to C by setting
f(z) =z
|z|f(|z|), for all z ∈ C, z 6= 0.
Set
F (z) =
∫ |z|
0
f(s) ds, for all z ∈ C.
Finally, W : Rn → R is an even, real-valued potential such that W ∈ Lq(Rn) +
L∞(Rn) and ∇W ∈ Lq(Rn) + L∞(Rn) for some q ≥ 1, q > n/4.
We know (see Section 4.2, Examples 1, 2, 3, 4) that g is the gradient of the potential
G defined by
G(u) =
∫
Rn
1
2V (x)|u(x)|2 + F (x, u(x)) +
1
4(W ∗ |u|2)(x)|u(x)|2
dx.
Finally, we set
E(u) =1
2
∫
Ω
|∇u(x)|2 dx−G(u), for all u ∈ H1(Rn).
For u ∈ H1(Rn), we set
h(u) =1
2Vr|u|2 +
n− 1
2r2F (u)− uf(u)+
1
2|u|2 x|x| · (∇W ∗ |u|2). (7.4.4)
If u is such that h(u) ∈ L1(Rn), we set
H(u) =
∫
Rn
h(u) dx. (7.4.5)
We have the following regularity result.
Lemma 7.4.3. Let n ≥ 3, let g be as above and set ρ = maxα + 2,4q
2q − 1. Then
h(u) ∈ L1(Rn) for every u ∈ H1(Rn) ∩W 1,ρ(Rn). Furthermore, there exists C such that
|H(v)−H(u)| ≤C(1 + ‖u‖H1 + ‖v‖H1)α+2(‖u‖H1 + ‖u‖W 1,ρ+
‖v‖H1 + ‖v‖W 1,ρ)‖v − u‖H1 ,(7.4.6)
134
for all u, v ∈ H1(Rn) ∩W 1,ρ(Rn).
Proof. Let us write Vr = V1+V2, where V1 ∈ Ln2 (Rn) and V2 ∈ L∞(Rn), ∇W = Z1+Z2,
where Z1 ∈ Lq(Rn) and Z2 ∈ L∞(Rn), f = f1 + f2, where f1 is globally Lipschitz
continuous and |f2(v)− f2(u)| ≤ C(|u|+ |v|)α|v − u| for all u, v ∈ C. Set
φi(u) = |u|2 x|x| · (Zi ∗ |u|2),
and
ψi(u) =1
r2Fi(u)− ufi(u),
for i = 1, 2. Consider u, v ∈ D(Rn). We have
∫
Rn
|V1(|v|2 − |u|2)| ≤∫
Rn
|V1|(|v|+ |u|)|v − u|
≤ C‖V1‖Ln2(‖v‖
L2n
n−2+ ‖u‖
L2n
n−2)‖v − u‖
L2n
n−2
≤ C(‖u‖H1 + ‖v‖H1)‖v − u‖H1 .
As well, ∫
Rn
|V2(|v|2 − |u|2)| ≤ C(‖u‖L2 + ‖v‖L2)‖v − u‖L2 .
Finally, we have ∫
Rn
|ψ1(v)− ψ1(u)| ≤ C
∫
Rn
|v|+ |u|r
|v − u|,
and ∫
Rn
|ψ2(v)− ψ2(u)| ≤ C
∫
Rn
(|v|+ |u|)α+1
r|v − u|.
Applying Holder’s inequality and (7.4.1), it follows that
∫
Rn
|ψ2(v)− ψ2(u)| ≤ C(‖v‖Lα+2 + ‖u‖Lα+2)α(‖∇v‖Lα+2 + ‖∇u‖Lα+2)‖v − u‖Lα+2
≤ C(1 + ‖u‖H1 + ‖v‖H1)α+2(‖u‖H1 + ‖u‖W 1,ρ+
‖v‖H1 + ‖v‖W 1,ρ)‖v − u‖H1 .
As well, ∫
Rn
|ψ1(v)− ψ1(u)| ≤ C(‖∇v‖L2 + ‖∇u‖L2)‖v − u‖L2 .
One obtains the same inequalities for φ1 and φ2 by applying Young’s and Holder’s inequal-
ities. It follows that (7.4.6) holds for all u, v ∈ D(Rn). The result now follows easily by
density.
135
We are now in a position to state and prove the main result of this section.
Theorem 7.4.4. (Morawetz’ estimate) Let n ≥ 3 and let g be as above. If ϕ ∈ H1(Rn)
and if u ∈ C((−T∗(ϕ), T ∗(ϕ)), H1(Rn)) is the maximal solution of problem (7.1.1), then∫ t
s
H(u(τ)) dτ ≤ 1
2〈iu(t), ur(t)〉L2 − 〈iu(s), ur(s)〉L2, (7.4.7)
for all −T∗(ϕ) < s < t < T ∗(ϕ), where H(u) is defined by (7.4.5).
Remark 7.4.5. Note that inequality (7.4.7) makes sense. Indeed, it follows from Theo-
rem 5.3.1 and Remark 5.3.5 that u ∈ Lq(s, t;W 1,r(Rn)), for every adimssible pair (q, r).
Applying Lemma 7.4.3, it follows easily that H(u) ∈ L1(s, t).
Proof of Theorem 7.4.4. We proceed in two steps.
Step 1. (7.4.7) holds, when ϕ ∈ H2(Rn). Note that by Theorem 5.2.1 and Remark 5.2.9,
we have u ∈ C((−T∗(ϕ), T ∗(ϕ)), H2(Rn)) ∩ C1((−T∗(ϕ), T ∗(ϕ)), L2(Rn)). Therefore,
equation (7.1.1) makes sense in L2(Rn); and so, we can take the L2-scalar product of
it with ur +n− 1
2ru ∈ C((−T∗(ϕ), T ∗(ϕ)), L2(Rn)) (by Lemma 7.4.1). Therefore, if we
denote by 〈·, ·〉 the L2-scalar product, we have
〈iut +4u+ g(u), ur +n− 1
2ru〉 = 0, on (−T∗(ϕ), T ∗(ϕ)). (7.4.8)
We claim that
〈iut, ur +n− 1
2ru〉 =
1
2
d
dt〈iu, ur〉. (7.4.9)
Indeed, by density, it is sufficient to establish identity (7.4.9) for smooth functions u. In
this case, it follows from integrating identity
Re
iut
(ur +
n− 1
2ru
)=
1
2∂tRe(iuur) +
1
2∇ ·(xrRe(iutu)
).
We also claim that
〈4u, ur +n− 1
2ru〉 ≤ 0. (7.4.10)
Again by density, it is sufficient to establish (7.4.10) for u ∈ D(Rn). Note that in this case
we have
Re
4u
(ur +
n− 1
2ru
)=∇ · Re
∇u(ur +
n− 1
2ru
)− x
2r|∇u|2
+
∇ ·(n− 1
4r3x|u|2
)− 1
r|∇u|2 − |ur|2 −
(n− 1)(n− 3)
4r3|u|2;
136
and so,
〈4u, ur +n− 1
2ru〉 = −
∫
Rn
1
r|∇u|2 − |ur|2 − a− b,
where
a =
2π|u(0)|2, if n = 3,
0, if n ≥ 4;b =
0, if n = 3,
(n− 1)(n− 3)
4
∫
Rn
|u|2r3
, if n ≥ 4.
Note that b is well defined by Corollary 7.4.2. (7.4.10) follows immediately. Furthermore,
we have
〈V u, ur +n− 1
2ru〉 = −1
2
∫
Rn
Vr|u|2. (7.4.11)
As well, it is sufficient to establish (7.4.11) for u ∈ D(Rn). In this case, it follows from
integrating identity
Re
V u
(ur +
n− 1
2ru
)= ∇ ·
(xV |u|2
2r
)− 1
2Vr|u|2.
Note also that
〈f(u), ur +n− 1
2ru〉 = −
∫
Rn
n− 1
2r2F (u)− uf(u). (7.4.12)
As well, it is sufficient to establish (7.4.12) for u ∈ D(Rn). In this case, it follows from
integrating the identity
Re
f(u)
(ur +
n− 1
2ru
)= ∇ ·
(xF (u)
r
)− n− 1
2r2F (u)− uf(u).
Finally, we claim that
〈(W ∗ |u|2)u, ur +n− 1
2ru〉 = −1
2
∫
Rn
|u|2 x|x| · (∇W ∗ |u|2). (7.4.13)
As well, it is sufficient to establish (7.4.13) for u ∈ D(Rn). In this case, we have
Re
(W ∗ |u|2)u
(ur +
n− 1
2ru
)= ∇ ·
( x2r
(W ∗ |u|2)|u|2)− 1
2|u|2 x|x| · ∇(W ∗ |u|2).
(7.4.13) follows from integrating the above inequality. (7.4.7) follows from putting together
formulas (7.4.8) through (7.4.13).
Step 2. Conclusion. Let ϕ ∈ H1(Rn), let u be the corresponding maximal solution of
(7.1.1), and let −T∗(ϕ) < s < t < T ∗(ϕ). Consider a sequence ϕm ∈ H2(Rn) such
137
that ϕm → ϕ is H1(Rn). Let um be the corresponding solutions of (7.1.1). It follows
from Theorem 4.2.8 that um → u, in C([s, t], H1(Rn)). Furthermore, it follows from
Theorem 5.3.1 and Remark 5.3.6 that for every admissible pair (q, r), um is bounded in
Lq(s, t;W 1,r(Rn)), uniformly with respect to m. In particular, we have
〈ium(τ), umr (τ)〉 −→
m→∞〈iu(τ), ur(τ)〉,
uniformly on [s, t], and by Lemma 7.4.3,
∫ t
s
H(um(τ)) dτ −→m→∞
∫ t
s
H(u(τ)) dτ.
The result now follows by applying Step 1.
Corollary 7.4.6. Assume n ≥ 3, and let g(u) = −λ|u|αu, for some λ > 0 and α ∈(0,
4
n− 2
). For every ϕ ∈ H1(Rn), the maximal solution u ∈ C(R, H1(Rn)) of (7.1.1)
verifies ∫ +∞
−∞
∫
Rn
|u(t, x)||x|
α+2
dxdt <∞. (7.4.14)
In addition, u(t) 0 in H1(Rn) as t→ ±∞.
Proof. It follows from Remark 6.5.1 (i) that the solution u is global and bounded in
H1(Rn). Applying (7.4.7), we obtain
∫ t
−t
∫
Rn
|u(t, x)||x|
α+2
dxds ≤ C(‖u(t)‖2H1 + ‖u(−t)‖2
H1) ≤ C.
(7.4.14) follows, by letting t ↑ ∞. In order to show the weak convergence to 0, we have to
verify that for every ψ ∈ D(Rn), 〈u(t), ψ〉 → 0, as t→ ±∞. Note that
|〈u(t), ψ〉| ≤∫
Rn
|u(t)| |ψ| ≤∫
Rn
|u(t)||x| 1
α+2
|x| 1α+2 |ψ| ≤ C
(∫
Rn
|u(t, x)||x|
α+2) 1
α+2
;
and so, ∫ +∞
−∞|〈u(t), ψ〉|α+2 dt <∞.
Finally, u is bounded in H1(Rn); and so, by the equation, ut is bounded in H−1(Rn). It
follows in particular that the function t 7→ |〈u(t), ψ〉| is (uniformly) Lipschitz continuous
R → R. Hence the result.
138
Remark 7.4.7. Note that Corollary 7.4.6 is more general than Remark 7.2.2, in the
sense that it applies to any H1 solution of (7.1.1), while Remark 7.2.2 applies only when
ϕ ∈ Σ.
7.5. Decay of solutions in the energy space. Throughout this section, we assume
that Ω = Rn and that n ≥ 3. We apply Morawetz’ estimate to the study of the asymptotic
behavior of solutions. For simplicity, we restrict our attention to the model case
g(u) = −λ|u|αu, (7.5.1)
where 0 < α <4
n− 2(0 < α <∞, if n = 1, 2) and
λ > 0, (7.5.2)
and we refer to Section 7.7 and Ginibre and Velo [6,7] for more general results. Note that
in this case, we have
T∗(ϕ) = T ∗(ϕ) = ∞, (7.5.3)
for all ϕ ∈ H1(Rn) (see Remark 6.5.1 (i)). Note also that we can apply Corollary 7.4.6.
Our main result of this section is the following.
Theorem 7.5.1. Let n ≥ 3 and let g be as above. If
α >4
n, (7.5.4)
then for every ϕ ∈ H1(Rn), the maximal solution u ∈ C(R, H1(Rn)) of (7.1.1) verifies
‖u(t)‖Lr −→t→±∞
0, (7.5.5)
for every 2 < r <2n
n− 2.
Remark 7.5.2. The above result is due to Ginibre and Velo [6], however the proof that
we present below follows closely an idea of Lin and Strauss [1].
Proof of Theorem 7.5.1. The proof that we give below does not cover the case n = 3
and α ∈(
43 ,
1+√
172
]. (The proof in that case is slightly more complicated, and the result
follows from Theorem 7.7.2 below.) We only prove the result for t → +∞, since the case
139
t→ −∞ follows from the same argument. Note also that we need only to establish (7.5.5)
for r = α+2, the general case following immediately from the boundedness of the solution
in H1(Rn) and Holder’s inequality. We proceed in several steps.
Step 1. We have the following estimate.
∫
|x|≥tLogt|u(t, x)|α+2 dx −→
t→+∞0. (7.5.6)
Let M > 0, and let
θM (x) =
|x|M, if |x| ≤M ;
1, if |x| ≥M.
We have θM ∈W 1,∞(Rn) and ‖∇θM‖L∞ ≤ 1
M. In particular, θMu ∈ C(R, H1(Rn)) and
〈iut +4u+ g(u), iθMu〉H−1,H1 = 0.
Note that
〈iut, iθMu〉H−1,H1 =1
2
d
dt
∫
Rn
θM |u|2,
that
〈g(u), iθMu〉H−1,H1 = 0,
and that
−〈4u, iθMu〉H−1,H1 = 〈∇u, i∇θMu〉L2 = 〈∇u, iu∇θM 〉L2
= −Re
∫
Rn
iu∇u · ∇θM ≤ 1
M‖u(t)‖2
H1 ≤ C
M.
It follows that ∫
Rn
θM |u(t, x)|2 dx ≤ Ct
M+
∫
Rn
θM |ϕ|2 dx,
for every t ∈ R. Taking M = tLogt, we obtain
∫
|x|≥tLogt|u(t, x)|2 dx ≤ C
Logt+
∫
Rn
θtLogt|ϕ|2 dx.
Applying the dominated convergence theorem to the last term in the right hand side of
the above estimate, it follows that
∫
|x|≥tLogt|u(t, x)|2 dx −→
t→+∞0.
140
The result now follows from Holder’s inequality and boundedness of u in H1(Rn).
Step 2. For every ε > 0, t > 1, τ > 0, there exists t0 > mint, 2τ such that
∫ t0
t0−2τ
∫
|x|≤tLogt|u(t, x)|α+2 dxdt ≤ ε. (7.5.7)
Indeed, by Morawetz’ estimate (7.4.14), we have
+∞ >
∫ ∞
1
1
sLogs
∫
|x|≤sLogs|u(t, x)|α+2
≥∞∑
k=0
∫ t+2(k+1)τ
t+2kτ
1
sLogs
∫
|x|≤sLogs|u(t, x)|α+2
≥∞∑
k=0
1
(t+ 2(k + 1)τ)Log(t+ 2(k + 1)τ)
∫ t+2(k+1)τ
t+2kτ
∫
|x|≤sLogs|u(t, x)|α+2.
Since∞∑
k=0
1
(t+ 2(k + 1)τ)Log(t+ 2(k + 1)τ)= ∞,
it follows that there exists k > 0 for which
∫ t+2(k+1)τ
t+2kτ
∫
|x|≤sLogs|u(t, x)|α+2 ≤ ε.
Hence the result, with t0 = t+ 2(k + 1)τ
Step 3. For every ε > 0, a, b > 0, there exists t0 > mina, b such that
sups∈[t0−b,t0]
‖u(s)‖Lα+2 ≤ ε. (7.5.8)
Consider t > τ > 0, and write
u(t) = T (t)ϕ+ i
∫ t−τ
0
T (t− s)g(u(s)) ds+ i
∫ t
t−τ
T (t− s)g(u(s)) ds
= v(t) + w(t, τ) + z(t, τ).
(7.5.9)
It follows from Corollary 3.2.8 that
‖v(t)‖Lα+2 −→t→∞
0. (7.5.10)
Let now
p =
∞, if α ≥ 1;
2
1− α, if α < 1.
141
Observe that n
(1
2− 1
p
)=n
2minα, 1 > 1. Therefore, it follows from Proposition 3.2.1
that
‖w(t, τ)‖Lp ≤∫ t−τ
0
(t− s)−n( 12− 1
p )‖u(s)‖α+1L(α+1)p′ ds
≤ Cτ−( n2 minα,1−1) sup
s∈R
‖u(s)‖α+1L(α+1)p′ .
Since 2 ≤ (α+1)p′ ≤ 2n
n− 2, and since u is bounded in H1(Rn), it follows that there exists
C such that
‖w(t, τ)‖Lp ≤ Cτ−(n2 minα,1−1), (7.5.11)
for all t > τ > 0. On the other hand, note that
w(t, τ) = T (τ)u(t− τ)− T (t)ϕ;
and so, it follows from conservation of charge that
‖w(t, τ)‖L2 ≤ 2‖ϕ‖L2 . (7.5.12)
Applying (7.5.11), (7.5.12) and Holder’s inequality, it follows that there exists K such that
‖w(t, τ)‖Lα+2 ≤ Kτ−nα−2 maxα,1
2(α+2) , (7.5.13)
for all t > τ > 0. Finally, it follows from Proposition 3.2.1 that
‖z(t, τ)‖Lα+2 ≤∫ t
t−τ
(t− s)−nα
2(α+2) ‖u(s)‖α+1Lα+2 ds. (7.5.14)
Note that nα < 2(α+2), and let p ∈(
1,2(α+ 2)
nα
). It follows in particular that (α+1)p′ >
α+ 2. Applying (7.5.14), Holder’s inequality and the boundedness of u in Lα+2(Rn), we
obtain
‖z(t, τ)‖Lα+2 ≤(∫ t
t−τ
(t− s)−nαp
2(α+2) ds
)1/p(∫ t
t−τ
‖u(s)‖(α+1)p′
Lα+2
)1/p′
≤ Cτ δ
(∫ t
t−τ
‖u(s)‖α+2Lα+2
)µ
,
for some δ, µ > 0. In particular, there exists L such that
‖z(t, τ)‖Lα+2 ≤Lτµ+δ
(sup
s≥t−τ
∫
|x|≥sLogs|u(s, x)|α+2 dx
)µ
+ Lτ δ
(∫ t
t−τ
∫
|x|≤sLogs|u(s, x)|α+2 dxds
)µ
.
(7.5.15)
142
It follows from (7.5.10) that there exists t1 ≥ mina, b such that
‖v(t)‖Lα+2 ≤ ε
4, for t ≥ t1. (7.5.16)
Next, take τ1 > b such that
‖w(t, τ1)‖Lα+2 ≤ ε
4, for t > 0, (7.5.17)
which exists by (7.5.13). By Step 1, there exists t2 ≥ t1 such that
Lτµ+δ1
(sup
s≥t−τ1
∫
|x|≥sLogs|u(s, x)|α+2 dx
)µ
≤ ε
4, for t ≥ t2. (7.5.18)
Finally, by Step 2, there exists t0 ≥ t2 such that
Lτ δ1
(∫ t0
t0−2τ1
∫
|x|≤sLogs|u(s, x)|α+2 dxds
)µ
≤ ε
4. (7.5.19)
Note that for t ∈ [t0 − b, t0], we have [t − τ1, t] ⊂ [t0 − 2τ1, t0]. Therefore, it follows from
(7.5.19) that
Lτ δ1
(∫ t
t−b
∫
|x|≤sLogs|u(s, x)|α+2 dxds
)µ
≤ ε
4. (7.5.20)
Applying (7.5.9), (7.5.16), (7.5.17), (7.5.15), (7.5.18) and (7.5.20), it follows that for every
t ∈ [t0 − b, t0], we have ‖u(t)‖Lα+2 ≤ ε. Hence the result.
Step 4. Conclusion. We need to show that for every ε > 0, we have ‖u(t)‖Lα+2 ≤ ε for
t large. Let t > τ > 0. It follows from (7.5.9) and (7.5.13) that
‖u(t)‖Lα+2 ≤ ‖v(t)‖Lα+2 +Kτ−nα−2 maxα,1
2(α+2) + ‖z(t, τ)‖Lα+2 . (7.5.21)
Consider ε > 0, and let τε be defined by
Kτ−nα−2 maxα,1
2(α+2)ε =
ε
4. (7.5.22)
It follows from (7.5.10) that there exists t1 > 0 such that
‖v(t)‖Lα+2 ≤ ε
4, for t ≥ t1. (7.5.23)
Applying (7.5.21), (7.5.22) and (7.5.23), we obtain
‖u(t)‖Lα+2 ≤ ε
2+ ‖z(t, τε)‖Lα+2 , (7.5.24)
143
for t ≥ t1. Note also that by Proposition 3.2.1, we have for t ≥ τε,
‖z(t, τε)‖Lα+2 ≤∫ t
t−τε
(t− s)−nα
2(α+2) ‖u(s)‖α+1Lα+2 ds
≤Mτ1− nα
2(α+2)ε sup
t−τε,t‖u(s)‖α+1
Lα+2 .
(7.5.25)
By Step 3, there exists t0 ≥ minτε, t1 such that ‖u(t)‖Lα+2 ≤ ε, for t ∈ [t0 − τε, t0].
Therefore, we can define
tε = supt ≥ t0; ‖u(s)‖Lα+2 ≤ ε for all s ∈ [t0 − τε, t]
Assume that tε <∞. Then,
‖u(tε)‖Lα+2 = ε. (7.5.26)
Applying (7.5.24) and (7.5.25) with t = tε, it follows that
ε ≤ ε
2+Mτ
1− nα2(α+2)
ε εα+2,
which implies
τ1− nα
2(α+2)ε εα ≥ 1
2M.
Applying (7.5.22), it follows that
τ−γε ≥ 1
2M(4K)α, (7.5.27)
where
γ =α(nα− 2− 2 maxα, 1) + (nα− 4)
2(α+ 2).
Observe that when α ≤ 1, we have γ > 0 (remember that nα > 4). Therefore, (7.5.27)
implies that τε is bounbed a positive number. This is a contradiction when ε is small,
since τε → ∞, as ε ↓ 0. When α > 1, one verifies easily that γ > 0 when n ≥ 4, or when
n = 3 and α >1 +
√17
2; in which case we obtain the same contradiction. It follows that
tε = ∞, which is the desired estimate.
Theorem 7.5.3. Assume n ≥ 3, and let g be as in Theorem 7.5.1, with α verifying
(7.5.4). For every ϕ ∈ H1(Rn), the maximal solution u ∈ C(R, H1(Rn)) of (7.1.1) verifies
u ∈ Lq(R,W 1,r(Rn)), (7.5.28)
144
for every admissible pair (q, r).
Before proceeding to the proof, we need the following elementary lemma.
Lemma 7.5.4. Let a, b > 0 and p > 1. Assume that b is small enough so that the
function f(x) ≡ a− x+ bxp is negative for some x > 0, and let x0 be the first (positive)
zero of f . Let I ⊂ R be an interval and let φ ∈ C(I,R+) verify
φ(t) ≤ a+ bφ(t)p,
for all t ∈ I. If φ(t0) = 0 (or more generally φ(t0) ≤ x0) for some t0 ∈ I, then φ(t) ≤ x0,
for all t ∈ I.
Proof. By assumption, the set J = x ≥ 0; f(x) ≥ 0 is of the form J = [0, y] ∪ [z,∞),
for some 0 < y < x0 < z. Since φ(t); t ∈ I is a connected set and f(φ(t)) ≥ 0, we must
have either φ(t); t ∈ I ⊂ [0, y], or else φ(t); t ∈ I ⊂ [z,∞). Hence the result.
Proof of Theorem 7.5.3. Let (γ, ρ) be the admissible pair such that ρ = α+ 2. For every
S, t > 0, we have
u(t+ S) = T (t)u(S) + i
∫ t
0
T (t− s)g(u(S + s)) ds.
It follows from Theorem 3.2.5, Remark 2.3.1 (vii) and Holder’s inequality that for every
t > S > 0, we have
‖u‖Lγ(S,t;W 1,ρ) ≤ C‖u(S)‖H1 + C
(∫ t
S
‖u(s)‖αγ′
Lρ ‖u(s)‖γ′
W 1,ρ
)1/γ′
≤ C‖u(S)‖H1 + C
(∫ t
S
‖u(s)‖(α+1)γ′−γLρ ‖u(s)‖γ−γ′
Lρ ‖u(s)‖γ′
W 1,ρ
)1/γ′
.
where C is independent of t, S. Note that (α+ 1)γ ′ > γ; and so,
(∫ t
S
‖u(s)‖(α+1)γ′−γLρ ‖u(s)‖γ−γ′
Lρ ‖u(s)‖γ′
W 1,ρ
)1/γ′
≤ sup‖u(s)‖Lρ; s ≥ Sα+1− γ
γ′ ‖u‖γ
γ′
Lγ(S,t;W 1,ρ).
145
It follows from Theorem 7.5.1 that
(∫ t
S
‖u(s)‖(α+1)γ′−γLρ ‖u(s)‖γ−γ′
Lρ ‖u(s)‖γ′
W 1,ρ
)1/γ′
≤ ε(S)‖u‖γ
γ′
Lγ(S,t;W 1,ρ)
= ε(S)‖u‖γ−1Lγ(S,t;W 1,ρ),
where ε(S) → 0, as S →∞. Therefore,
‖u‖Lγ(S,t;W 1,ρ) ≤ C‖ϕ‖L2 + ε(S)‖u‖γ−1Lγ(S,t;W 1,ρ).
Applying Lemma 7.5.4, it follows that if we fix S large enough, we have ‖u‖Lγ(S,t;W 1,ρ) ≤ K,
for some K independent of t. Therefore, u ∈ Lγ(S,∞;W 1,ρ(Rn)). One shows as well that
for S large enough, u ∈ Lγ(−∞,−S;W 1,ρ(Rn)); and so, u ∈ Lγ(R,W 1,ρ(Rn)). This
implies that g(u) ∈ Lγ′(R,W 1,ρ′(Rn)), and the result follows from Theorem 3.2.5.
Remark 7.5.5. One can add to the statement of Theorem 7.5.3, the following property.
If ϕ ∈ H2(Rn), then ut ∈ Lq(R, Lr(Rn)), for every admissible pair (q, r). These estimates
follow essentially from the same argument, by applying Theorem 5.2.1 and Remark 5.2.4.
Note that this implies in particular, via Sobolev’s inequalities and an iteration process, that
u ∈ Lq(R,W 2,r(Rn)), for every admissible pair (q, r). In particular, u ∈ L∞(R, H2(Rn)).
7.6. Scattering theory in the energy space. We apply the results of Section 7.5 in
order to construct the scattering operator in the energy space H1(Rn). The results below
are due to Ginibre and Velo [6]. We still assume that Ω = Rn, n ≥ 3 and we assume that
g is as in the preceding section, that is
g(u) = −λ|u|αu,
for some λ > 0 and 0 < α <4
n− 2. We refer to Section 7.7 and to Ginibre and Velo [6,7]
for more general results.
Theorem 7.6.1. Assume n ≥ 3. Let g be as above, and assume that α >4
n. If
ϕ ∈ H1(Rn) and if u ∈ C(R, H1(Rn)) is the maximal solution of (7.1.1), then there exist
u+, u− ∈ H1(Rn) such that ‖T (−t)u(t)− u±‖H1 −→t→±∞
0. In addition,
‖u+‖L2 = ‖u−‖L2 = ‖ϕ‖L2 ,
146
and1
2
∫
Rn
|∇u+|2 =1
2
∫
Rn
|∇u−|2 = E(ϕ).
Proof. Let v(t) = T (−t)u(t). We have
v(t) = ϕ+ i
∫ t
0
T (−s)|g(u(s)) ds.
Therefore, for 0 < t < τ ,
v(t)− v(τ) = i
∫ t
τ
T (−s)g(u(s)) ds.
It follows from Theorem 3.2.5 that
‖v(t)− v(τ)‖H1 = ‖T (t)(v(t)− v(τ))‖H1 ≤ C‖g(u)‖Lq′(t,τ ;W 1,r′),
where (q, r) is the admissible pair such that r = α+ 2. It follows that
‖v(t)− v(τ)‖H1 −→t,τ→∞
0,
(see the end of the proof of Theorem 7.5.3). Therefore, there exists u+ ∈ H1(Rn) such
that v(t) → u+ in H1, as t → ∞. One shows as well that there exists u− ∈ H1(Rn)
such that v(t) → u− in H1, as t→ −∞. The other properties follow from conservation of
charge and energy.
Remark 7.6.2. Theorem 7.6.1 means that every H1 solution is asymptotic, as t→ ±∞,
to a solution of the linear Schrodinger equation.
Remark 7.6.3. The mappings U+ : ϕ 7→ u+ and U− : ϕ 7→ u− defined by Theorem 7.6.1
map H1(Rn) → H1(Rn). In fact, one can show with the same estimates that U+ and U−
are continuous H1(Rn) → H1(Rn).
Remark 7.6.4. It follows from Corollary 3.2.7 that one has the following formula
u± = ϕ+ i
∫ ±∞
0
T (−s)g(u(s)) ds.
147
In particular,
u(t) = T (t)u± − i
∫ ±∞
t
T (t− s)g(u(s)) ds, (7.6.1)
for all t ∈ R.
Theorem 7.6.5. Assume n ≥ 3, and let g be as in Theorem 7.6.1. Then,
(i) for every u+ ∈ H1(Rn), there exists a unique ϕ ∈ H1(Rn) such that the maximal
solution u ∈ C(R, H1(Rn)) of (7.1.1) verifies ‖T (−t)u(t)− u+‖H1 → 0, as t→ +∞;
(ii) for every u− ∈ H1(Rn), there exists a unique ϕ ∈ H1(Rn) such that the maximal
solution u ∈ C(R, H1(Rn)) of (7.1.1) verifies ‖T (−t)u(t)− u−‖H1 → 0, as t→ −∞;
Proof. We prove (i), the proof of (ii) being similar. The idea of the proof is to solve
equation (7.6.1) by a fixed point argument. To that end, we introduce the function ω(t) =
T (t)u+. Let (q, r) be the admissible pair such that r = α+2. It follows from Theorem 3.2.5
and Corollary 3.2.8 that ω ∈ Lq(R,W 1,r(Rn)) and that ‖ω(t)‖Lr → 0, as t→∞. Consider
S > 0 and let
KS = ‖ω‖Lq(S,∞;W 1,r) + supt≥S
‖ω(t)‖Lr . (7.6.2)
Note that
KS −→S→∞
0. (7.6.3)
Let
X = u ∈ Lq(S,∞;W 1,r(Rn)); ‖u‖Lq(S,∞;W 1,r) + supt≥S
‖ω(t)‖Lr ≤ 2KS,
and
d(u, v) = ‖v − u‖Lq(S,∞;Lr), for u, v ∈ X.
It is easily checked that (X, d) is a complete metric space. Given u ∈ X, we have (see the
proof of Theorem 7.6.3)
‖g(u)‖Lq′(S,∞;W 1,r′) ≤ C(2KS)α+1.
It follows from Corollary 3.2.7 that the function
Fu(t) = −i∫ ∞
t
T (t− s)g(u(s)) ds, (7.6.4)
148
is well defined, that
Fu ∈ C([S,∞), H1(Rn)) ∩ Lq(S,∞;W 1,r(Rn)), (7.6.5)
and that
‖Fu‖Lq(S,∞;W 1,r) + ‖Fu‖L∞(S,∞;H1) ≤ C(2KS)α+1. (7.6.6)
Applying (7.6.3), (7.6.6) and Sobolev’s inequality, it follows that
‖Fu‖Lq(S,∞;W 1,r) + ‖Fu‖L∞(S,∞;Lr) ≤ KS, for S large enough. (7.6.7)
Putting together (7.6.2) and (7.6.7), it follows that A defined by
Au(t) = T (t)u+ + Fu(t), for t ≥ S,
maps X to itself if S is large enough. One verifies easily with the same type of estimates
that if S is large enough, one has
d(Au,Av) ≤1
2d(u, v), for all u, v ∈ X. (7.6.8)
It follows from Banach’s fixed point theorem thatA has a fixed point u ∈ X, which satisfies
equation (7.6.1) on [S,∞). One concludes like for Theorem 7.3.5.
Remark 7.6.6. The mappings Ω+ : u+ 7→ ϕ and Ω− : u− 7→ ϕ defined by Theo-
rem 7.6.5 map H1(Rn) → H1(Rn). In fact, one can show with the same estimates that
Ω+ and Ω− are continuous. These operators are called the wave operators. It follows
from Theorems 7.6.1 and 7.6.5 that U±Ω± = Ω±U± = I on Σ, where U± is defined by
Remark 7.6.3. In particular, Ω± : H1(Rn) → H1(Rn) is one to one, with continuous
inverse (Ω±)−1 = U±.
Theorem 7.6.7. Assume n ≥ 3 and let g be as in Theorem 7.6.1. For every u− ∈H1(Rn), there exists a unique u+ ∈ H1(Rn) and a unique ϕ ∈ H1(Rn), such that the
maximal solution u ∈ C(R, H1(Rn)) of (7.1.1) verifies T (−t)u(t) → u± in H1(Rn), as
t→ ±∞. The scattering operator
S :H1(Rn) → H1(Rn)
u− 7→ u+
149
is continuous, one to one, and its inverse is continuous H1(Rn) → H1(Rn). In addition,
‖u+‖L2 = ‖u−‖L2 and ‖∇u+‖L2 = ‖∇u−‖L2 , for every u− ∈ H1(Rn).
Proof. The result follows from Theorems 7.6.1 and 7.6.5, and Remark 7.6.6, by setting
S = U+Ω−. Note that S−1 = U−Ω+.
7.7. Comments. The estimates of Theorem 7.2.1 remain valid for more general non-
linearities. In particular, consider g(u) = f(u(·)), where f is as in the beginning of Sec-
tion 7.1. Assume that F (s) ≤ 0 for all s ≥ 0, and that there exists 0 < δ ≤ 4
nsuch that
−s−(2+4/n)F (s) is a nondecreasing function of s ≥ 0. We have the following result.
Proposition 7.7.1. Let g be as above. If ϕ ∈ H1(Rn) is such that | · |ϕ(·) ∈ L2(Rn),
and if u be the maximal solution of (7.1.1) (see Proposition 6.4.1), then
∫
Rn
F (u(t)) dx ≤ C|t|−nδ2 ,
and
‖u(t)‖Lr ≤ C|t|−n2δ4 ( 1
2− 1r ),
for all t ∈ R.
Proof. It follows from Theorem 7.1.1 that v defined by (7.1.5) verifies
t2E(v) ≤ ‖xϕ‖2L2 −
∫ t
0
s
∫
Rn
(8(n+ 2)F (u)− 4nRe(f(u)u)) dx ds.
By assumption, we have −sf(s) ≤ −(2 + δ)F (s). It follows that
∫
Rn
|∇v(t)|2 dx− 2t2∫
Rn
F (u(t)) dx ≤ (4− nδ)
∫ t
0
∫
Rn
sF (u(s)) dx ds.
One concludes like for Theorem 7.2.1 that∫Rn F (u(t)) dx ≤ C|t|−nδ
2 . It follows that
‖∇v(t)‖L2 ≤ C|t|−nδ4 . The result now follows from Gagliardo-Nirenberg’s inequality (The-
orem 2.3.7) and conservation of charge.
Applying these estimates, one can extend the scattering theory in Σ to more general
nonlinearities (see Ginibre and Velo [1,2]).
150
The result of Theorem 7.5.1 can be generalized in the following way. Consider g(u) =
f(u(·)), where f is as in the beginning of Section 7.1. Assume that there exists δ <4
nsuch
that
F (s) ≤ C(s2 + sδ+2), (7.7.1)
so that all solutions of (7.1.1) are global (see Example 1 of Section 6.2). Assume further
that
|f(s)| ≤ C(sµ+1 + sν+1), for all s ≥ 0, (7.7.2)
for some4
n< µ ≤ ν <
4
n− 2. Assume finally that
2F (s)− sf(s) ≥ cminsµ+2, sν+2, for all s ≥ 0. (7.7.3)
We have the following result.
Theorem 7.7.2. Assume n ≥ 3 and let g be as above. For every ϕ ∈ H1(Rn), the
maximal solution u ∈ C(R, H1(Rn)) of (7.1.1) verifies
‖u(t)‖Lr −→t→±∞
0,
for every 2 < r <2n
n− 2.
Proof. The proof is an adaptation of the proof of Theorem 7.5.1. We only prove the result
for t → +∞, since the case t → −∞ follows from the same argument. Note also that we
need only to establish the result for r = ν+2, the general case following immediately from
the boundedness of the solution in H1(Rn) and Holder’s inequality.
Step 1. We have the following estimate.
∫
|x|≥tLogt|u(t, x)|α+2 dx −→
t→+∞0. (7.7.4)
The proof is the same as that of Step 1 of the proof of Theorem 7.4.1.
Step 2. For every ε > 0, t > 1, τ > 0, there exists t0 > mint, 2τ such that
∫ t0
t0−2τ
∫
|x|≤tLogtmin|u|µ+2, |u|ν+2 dxdt ≤ ε. (7.7.5)
151
This follows Morawetz’ estimate and (7.7.3), by following the proof of Theorem 7.4.1,
Step 2.
Step 3. For every ε > 0, t > 1, τ > 0, and 2 < r <2n
n− 2there exists t0 > mint, 2τ
such that ∫ t0
t0−2τ
∫
|x|≤tLogt|u(t, x)|r dxdt ≤ ε. (7.7.6)
Note that we need only to establish the result for r = µ + 2, the general case following
immediately from the boundedness of the solution in H1(Rn) and Holder’s inequality.
Consider ε > 0, t > 1, τ > 0. Define
v(t, x) =
u(t, x), if |u(t, x)| ≤ 1,
0, if |u(t, x)| > 1;w(t, x) =
u(t, x), if |u(t, x)| > 1,
0, if |u(t, x)| ≤ 1;
It follows from Step 2 that for every ε′ > 0, there exists t0 > mint, 2τ such that
∫ t0
t0−2τ
∫
|x|≤tLogt|v(t, x)|ν+2 dxdt+
∫ t0
t0−2τ
∫
|x|≤tLogt|w(t, x)|µ+2 dxdt ≤ ε′. (7.7.7)
For t > 2τ > 0, we have
∫ t
t−2τ
∫
|x|≤tLogt|u(t, x)|µ+2 dxdt =
∫ t
t−2τ
∫
|x|≤tLogt|v(t, x)|µ+2 dxdt
+
∫ t
t−2τ
∫
|x|≤tLogt|w(t, x)|µ+2 dxdt.
(7.7.8)
Applying Holder’s inequality in space and time, and conservation of charge, we obtain
∫ t
t−2τ
∫
|x|≤tLogt|w(t, x)|µ+2 ≤ Cτ
2(ν−µ)ν(µ+2)
(∫ t
t−2τ
∫
|x|≤tLogt|w(t, x)|ν+2
)µ(ν+2)ν(µ+2)
. (7.7.9)
Choosing ε′ such that
ε′ + Cτ2(ν−µ)ν(µ+2) (ε′)
µ(ν+2)ν(µ+2) ≤ ε, (7.7.10)
the result follows from putting together (7.7.8), (7.7.9), (7.7.7) and (7.7.10).
Step 4. For every ε > 0 and t, τ > 0, there exists t0 > mint, τ such that
sups∈[t0−τ,t0]
‖u(s)‖Lν+2 ≤ ε. (7.7.11)
152
Consider t > τ > 0, and write
u(t) = T (t)ϕ+ i
∫ t−τ
0
T (t− s)g(u(s)) ds+ i
∫ t
t−τ
T (t− s)g(u(s)) ds
= v(t) + w(t, τ) + z(t, τ).
(7.7.12)
It follows from Corollary 3.2.8
‖v(t)‖Lν+2 −→t→∞
0. (7.7.13)
Arguing like in the proof of Theorem 7.5.1, Step 3, one shows easily that there exists K
such that
‖w(t, τ)‖Lν+2 ≤ Kτ−ν(nµ−2 maxµ,1)
2µ(ν+2) , (7.7.14)
for all t > τ > 0. Let now ρ =(µ+ 1)(ν + 2)
ν + 1∈ [µ+ 2, ν + 2]. Arguing like in the proof of
Theorem 7.5.1, Step 3, one shows easily that there exists L <∞ and a, b, c > 0 such that
‖z(t, τ)‖Lα+2 ≤ L(1 + τa)
[(∫ t
t−τ
‖u‖ν+2Lν+2
)b (∫ t
t−τ
‖u‖ρLρ
)c]. (7.7.15)
One concludes like in the proof of Theorem 7.5.1, Step 3.
Step 5. Conclusion. We need to show that for every ε > 0, we have ‖u(t)‖Lν+2 ≤ ε for
t large. Consider ε > 0, and let τε be defined by
Kτ−ν(
nµ−2 maxµ,1)2µ(ν+2)
ε =ε
4. (7.7.16)
It follows from (7.7.13) that there exists t1 > 0 such that
‖v(t)‖Lν+2 ≤ ε
4, for t ≥ t1. (7.7.17)
Applying (7.7.2), (7.7.13), (7.7.14) and (7.7.16), we obtain
‖u(t)‖Lν+2 ≤ ε
2+ ‖z(t, τε)‖Lν+2 , (7.7.18)
for t ≥ t1. Note also that by Proposition 3.2.1, we have for t ≥ τε,
‖z(t, τε)‖Lν+2 ≤ Cτ1− nν
2(µ+2)ε sup
t−τε,t(‖u‖µ+1
Lρ + ‖u‖ν+1Lν+2), (7.7.19)
where ρ =(µ+ 1)(ν + 2)
ν + 1∈ [µ + 2, ν + 2] (compare the proof of Theorem 7.5.1, Step 4).
By Holder’s inequality and conservation of charge, we have
‖u‖µ+1Lρ ≤ C‖u‖
µ(ν+2)−ν
ν
Lν+2 . (7.7.20)
153
Note thatµ(ν + 2)− ν
ν≤ µ(ν + 1)
ν≤ ν + 1; and so, it follows from (7.7.19), (7.7.20) and
the boundedness of u in H1(Rn) that there exists M such that
‖z(t, τε)‖Lν+2 ≤Mτ1− nν
2(µ+2)ε sup
t−τε,t(‖u‖
µ(ν+2)−ν
ν
Lν+2 ). (7.7.21)
By Step 4, there exists t0 ≥ minτε, t1 such that ‖u(t)‖Lν+2 ≤ ε, for t ∈ [t0 − τε, t0].
Therefore, we can define
tε = supt ≥ t0; ‖u(s)‖Lν+2 ≤ ε for all s ∈ [t0 − τε, t]
Assume that tε <∞. Then,
‖u(tε)‖Lν+2 = ε. (7.7.22)
Applying (7.7.21), (7.7.22) and (7.7.18) with t = tε, it follows that
ε ≤ ε
2+Mτ
1− nα2(α+2)
ε εµ(ν+2)−ν
ν ,
which implies τ1− nα
2(α+2)ε ε
µ(ν+2)−2ν
ν ≥ 1
2M. Applying (7.7.16), it follows that
τ−γε ≥ 1
2M(4K)µ(ν+2)−2ν
ν
, (7.7.23)
where
γ =(µ(ν + 2)− 2ν)(nµ− 2− 2 maxµ, 1) + ν(nµ− 4)
2µ(ν + 2).
One can conclude like ine the proof of Theorem 7.5.1, Step 4, provided γ > 0. If µ ≤ 1,
we have
γ =(µ(ν + 2)− ν)(nµ− 4)
2µ(ν + 2).
Note that µ >2µ
µ+ 2≥ 2ν
ν + 2; and so, µ(ν + 2)− ν > ν > 0. Since also nµ > 4, we have
γ > 0. If µ > 1, we have γ =n− 2
2(µ− φ(ν)), where
φ(x) =(n− 2)x+ 4
(n− 2)(x+ 2).
When n ≥ 4, φ is nondecreasing; and so, φ(ν) ≤ φ(4
n− 2) =
4
n. This implies again that
γ > 0. When n = 3, φ(x) is decreasing and φ( 4n−2 ) = 4
n . Since nµ > 4, there exists
154
ν ≤ ν <4
n− 2such that µ − φ(ν) > 0. Observe that f satisfies as well assumptions
(7.7.2) and (7.7.3) with ν replaced by ν. Therefore, in this case also we have γ > 0. This
completes the proof.
Remark 7.7.3. It is not difficult to extend the results of Theorems 7.5.3, 7.6.1, 7.6.5 and
7.6.7 to the case where g is as in Theorem 7.7.2. Therefore, one can construct a scattering
theory in H1(Rn) for such nonlinearities (see Ginibre and Velo [6,7]).
Remark 7.7.4. When g(u) = −λ|u|αu, λ > 0, the scattering operator is well defined
on Σ for α > a(n) (Theorem 7.3.7) and it is well defined on H1(Rn) when α > 4/n
(Theorem 7.6.7). Here are several extensions of these results. First of all, note that
when α ≤ 2/n, no scattering operator can be defined, even on Σ. More precisely, for
any nontrivial solution u in Σ, T (−t)u(t) does not have any strong limit even in L2(Rn)
(see Glassey [2], Strauss [5,6], Barab [1], Tsutsumi and Yajima [1]). For α > 2/N every
solution in Σ has a scattering states in L2(Rn), i.e. there exist u± ∈ L2(Rn) such that
T (−t)u(t) −→t→±∞
u± in L2(Rn). In other words, the operators U± are well defined Σ →L2(Rn). It follows from Theorem 7.3.7 that U± is well defined Σ → Σ if α > a(n), and this
still holds for α = a(n) when n ≥ 3 (see Cazenave and Weissler [6], Theorem 4.10). For
α >4
n− 2, U± is defined from a neghborhood of 0 in Σ to Σ (see Cazenave and Weissler [6],
Theorem 4.2). As for the wave operators Ω±, they are well defined Σ → Σ for α >4
n+ 2(see Cazenave and Weissler [6], Proposition 4.6). For the intermediate values, the existence
of these operators is still an open question. Note that the decay estimates of Sections 7.2
and 7.5 are optimal (see Hayashi and Ozawa [5]). Concerning the decay of solutions in
L∞, see Ginibre and Velo [2], Dong and Li [1] (one dimensional case), Cazenave [1] (two
dimensional case) and Lin and Strauss [1] (three dimensional case).
Remark 7.7.5. When g(u) = −λ|u|4/nu, λ ∈ R, a scattering theory can be constructed
in a subset of L2(Rn), containing for example all functions with small L2 norm and also
all functions in u ∈ L2(Rn) such that xu ∈ L2(Rn) in the case λ > 0 (see Cazenave and
Weissler [5], and also Weinstein [4] for a related result).
Remark 7.7.6. When g(u) = λ|u|αu, with λ > 0 and α ≥ 4
n, all solutions with small
initial data converge weakly to 0, as t → ±∞. Indeed, we know (see Section 6.2) that
155
there exists ε > 0 such that if ‖ϕ‖H1 ≤ ε, then the solution of (7.7.1) is global and
supt∈R
‖u(t)‖H1 ≤ C(ε). It follows that for such a ϕ we have for every S > 0 (see the proof
of Theorem 7.5.3),
‖u‖Lγ(−S,S;W 1,ρ) ≤ C‖ϕ‖H1 + C(ε)‖u‖γ−1Lγ(−S,S;W 1,ρ).
Applying Lemma 7.5.4, it follows that if ‖ϕ‖H1 is small enough, then ‖u‖Lγ(−S,S;W 1,ρ) ≤ K,
for some K independent of S. Therefore, u ∈ Lγ(R,W 1,ρ(Rn)). One concludes easily to
the weak convergence to 0 (see Corollary 7.4.6).
Remark 7.7.7. When g(u) = λ|u|αu, with λ > 0 and α >4
n, one can adapt easily the
proofs of Theorems 7.6.1, 7.6.5 and 7.6.7 (see Remark 7.7.6) to construct the scattering
operator S on the set u ∈ H1(Rn); ‖u‖H1 ≤ ε, for ε small enough. Obviously, the
scattering operator cannot be defined on the whole space H1(Rn), since some solutions
blow up in finite time (see Remark 6.5.1). As well, when α > a(n), one can adapt easily the
proof of Theorem 7.3.7 to construct the scattering operator S on the set u ∈ Σ; ‖u‖Σ ≤ ε,for ε small enough. As a matter of fact, the same result holds for α >
4
n+ 2provided
n ≥ 3 (see Cazenave and Weissler [6], Corollary 4.3). The assumptions α > 4/n (for the
H1 theory) and α >4
n+ 2(for the Σ theory) are optimal (see Cazenave and Weissler [6],
Remark 4.4).
Remark 7.7.8. When g(u) = λ|u|αu, with λ > 0 and 0 < α <4
n, the conclusion of
Remark 7.7.6 becomes false. Indeed, there exists solutions of (7.7.1) of the form uω(t, x) =
eiωtϕω(x), where ‖ϕω‖H1 → 0, as ω ↓ 0 (see Remark 8.1.8). In particular, ‖uω‖H1 can be
chosen arbitrarily small, but obviously uω(t) 6 0, as t→ ±∞.
Remark 7.7.9. The results of Sections 7.2 and 7.3 can be extended to Hartree type
nonlinearities. See Cazenave, Dias and Figueira [1], Chadam and Glassey [1], Dias [1],
Dias and Figueira [1], Ginibre and Velo [3], Hayashi [1], Hayashi and Ozawa [1,2,3,4],
Hayashi and Tsutsumi [1], Lange [1,2], P.-L. Lions [1,2], Pecher and Von Wahl [1]. The
results of Sections 7.4, 7.5 and 7.6 can also be extended, see Ginibre and Velo [9].
156
8. Stability of bound states in the attractive case. In this chapter, we study the
stability of standing waves of the nonlinear Schrodinger equation, for a class of attractive
nonlinearities. Throughout the chapter, we assume that Ω = Rn, and we consider the
problem iut +4u+ g(u) = 0;
u(0) = ϕ;(8.1)
For the sake of simplicity, we consider the model case
g(u) = λ|u|αu, (8.2)
where
λ > 0, (8.3)
and 0 < α <4
n− 2(0 < α < ∞, if n = 1, 2), and we indicate references concerning
more general nonlinearities. We have seen in the preceding chapter that when λ < 0, all
solutions converge weakly to 0, as t → ±∞. When λ > 0, we have a completely different
situation. Indeed, in the case α ≥ 4/n, it follows from Remark 7.7.6 that all solutions
with small initial data converge weakly to 0, as t→ ±∞; and on the other hand, it follows
from Remark 6.5.1 that solutions with “large” initial data blow up in finite time. In fact,
in both the case α ≥ 4/n and the case α < 4/n, we show in Section 8.1 the existence of a
third type of solutions, that are global but do not converge weakly to 0. More precisely,
we construct solutions of (8.1) of the form
u(t, x) = eiωtϕ(x),
where ω ∈ R and ϕ ∈ H1(Rn), ϕ 6= 0. Such solutions are called standing waves, or
stationary states, or localized solutions. In Section 8.2, we show that when α ≥ 4/n a
class of standing waves is instable, and in Section 8.3, we show that when α < 4/n a
class of standing waves is stable. We apply purely variational methods, and we refer to
Section 8.4 for other methods.
8.1. Nonlinear bound states. Throughout this section, we consider g of the form
(8.2)-(8.3). We look for solutions of (8.1) of the form
u(t, x) = eiωtϕ(x), (8.1.1)
157
where ω ∈ R is a given parameter and ϕ ∈ H1(Rn), ϕ 6= 0. It is clear that ϕ must solve
the following problem. ϕ ∈ H1(Rn), ϕ 6= 0;
−4ϕ+ ωϕ = |ϕ|αϕ.(8.1.2)
We refer to Strauss [4], Berestycki and Lions [1], Berestycki, Gallouet and Kavian [1],
Berestycki, Lions and Peletier [1] and Jones and Kupper [1] for a complete study of (8.1.2)
as well as for similar problems with more general nonlinearities. In particular, it is known
that if ω ≤ 0, then (8.1.2) does not have any solution. Therefore, from now on we assume
that
ω > 0. (8.1.3)
We begin with a regularity result.
Theorem 8.1.1. Let 0 < α <4
n− 2(0 < α < ∞, if n = 1, 2) and ω > 0. If ϕ
satisfies (8.1.2), then the following properties hold.
(i) ϕ ∈W 2,p(Rn), for every 2 ≤ p <∞;
(ii) ϕ(x) −→|x|→∞
0;
(iii) ϕ ∈ C2(Rn);
(iv) there exists ε > 0 such that eε|x|ϕ(x) ∈ L∞(Rn).
Proof. It is sufficient to establish these properties for u defined by ϕ(x) = ω1/αu(x√ω),
which satisfies equation
−4u+ u = |u|αu. (8.1.4)
Note that (8.1.4) can be written in the form
F−1((1 + 4π2|ξ|2)Fu) = |u|αu, (8.1.5)
where F is the Fourier transform and (8.1.5) makes sense in the space of tempered distri-
butions S ′(Rn).
(i) Note that if u ∈ Lp(Rn) for some α+ 1 < p <∞, we have |u|αu ∈ L pα+1 (Rn). It
follows from Bergh and Lofstrom [1], Theorem 6.2.3 p.141 that u ∈W 2, pα+1 (Rn). Applying
Sobolev’s embedding theorem, this implies that
u ∈ Lq(Rn), for all q ≥ p
α+ 1such that
1
q≥ α+ 1
p− 2
n. (8.1.6)
158
Consider the sequence qj defined by
1
qj= (α+ 1)j
(1
α+ 1− 2
nα+
2
nα(1 + α)j
).
Since (n− 2)α < 4, it follows that1
α+ 2− 2
nα= −δ, where δ > 0. We have
1
qj+1− 1
qj= −α(α+ 1)jδ ≤ −δ;
and so,1
qjis decreasing and
1
qj−→j→∞
−∞. Since q0 = α + 2, it follows that there exists
j ≥ 0 such that1
q`> 0, for 0 ≤ ` ≤ j;
1
qj+1≤ 0.
We claim that u ∈ Lqj (Rn). Indeed, arguing by induction, we have u ∈ H1(Rn), therefore
u ∈ Lq0(Rn); and if u ∈ Lq`(Rn), for some ` ≤ j − 1, we have by (8.1.6)
u ∈ Lq(Rn), for all q ≥ q`α+ 1
such that1
q≥ α+ 1
q`− 2
n=
1
q`+1.
In particular, u ∈ Lq`+1(Rn). Hence the result. Applying once again (8.1.6), it follows
that u ∈ Lq(Rn), for all q ≥ qjα+ 1
such that1
q≥ 1
qj+1. In particular, we can take q = ∞.
Therefore, |u|αu ∈ L2(Rn) ∩ L∞(Rn). The result follows by applying (8.1.5) and Bergh
and Lofstrom [1], Theorem 6.2.3 p.141.
(ii) By (i) and Sobolev’s embedding, u ∈W 1,∞(Rn). In particular, u is (uniformly)
Lipschitz continuous. Since u ∈ L2(Rn), this implies easily (ii).
(iii) By (i) and Sobolev’s embedding, u ∈ W 1,n+1(Rn) → L∞(Rn). Applying
Remark 3.2.1 (viii), it follows that |u|αu ∈ W 1,n+1(Rn). In particular, it follows from
(8.1.4) that for every j ∈ 1, · · · , n, we have (−4 + I)∂ju ∈ Ln+1(Rn), or as well
F−1((1 + 4π2|ξ|2)F∂ju) ∈ Ln+1(Rn). Applying Bergh and Lofstrom [1], Theorem 6.2.3
p.141, we obtain that ∂ju ∈W 2,n+1(Rn). Therefore, u ∈W 3,n+1(Rn) → C2(Rn).
(iv) Let ε > 0 and θε(x) = e|x|
1+ε|x| . θε is bounded, Lipschitz continuous, and∇θε ≤ θε
almost everywhere. Taking the scalar product of the equation with θεu ∈ H1(Rn), it
follows that
Re
∫
Rn
∇u · ∇(θεu) +
∫
Rn
θε|u|2 ≤∫
Rn
θε|u|α+2. (8.1.7)
159
Note that ∇(θεu) = u∇θε + θε∇u. Therefore,
Re (∇u · ∇(θεu)) ≥ θε|∇u|2 − θε|u| |∇u|.
Applying (8.1.7) and Cauchy-Schwarz inequality, we obtain easily
∫
Rn
θε|u|2 ≤ 2
∫
Rn
θε|u|α+2. (8.1.8)
By (ii), there exists R > 0, such that |u(x)|α ≤ 1/4 for |x| ≥ R. Therefore,
2
∫
Rn
θε|u|α+2 ≤ 2
∫
|x|≤Re|x||u|α+2 +
1
2
∫
Rn
θε|u|2. (8.1.9)
Putting together (8.1.8) and (8.1.9), it follows that
∫
Rn
θε|u|2 ≤ 4
∫
|x|≤Re|x||u|α+2.
Letting ε ↓ 0, it follows that ∫
Rn
e|x||u|2 <∞. (8.1.10)
By (ii), there exists R > 0, such that |u(x)| ≤ 1 for |x| ≥ R. We have
e|x|
n+2 |u(x)| ≤ eR
n+2 ‖u‖L∞ , for |x| ≤ R. (8.1.11)
Consider now x ∈ Rn such that |x| ≥ R. We know (see the proof of (i)) that u is Lipschitz
continuous. Therefore, there exists L such that
|u(y)| ≥ |u(x)| − L|x− y|, for all y ∈ Rn.
It follows that
|u(x)|2 ≤ 2(|u(y)|2 + L2|x− y|2).
If |x− y| ≤ 1
2L|u(x)|, we obtain
|u(x)|2 ≤ 2|u(y)|2 +1
2|u(x)|2;
and so
|u(x)|2 ≤ 4|u(y)|2, for |x− y| ≤ 1
2L|u(x)|.
160
Integrating the above inequality on the set B(x)y; 2|x− y| ≤ |u(x)|, we obtain
|B(x)| |u(x)|2 ≤ 4
∫
B(x)
|u(y)|2 dx.
Note that |B(x)| = c|u(x)|n and that on B(x), we have 1 ≤ e1/2Le−|x|e|y|. Therefore, we
obtain
e|x||u(x)|n+2 ≤ C
∫
B(x)
e|y||u(y)|2 ≤ C
∫
Rn
e|y||u(y)|2,
where C is independent of x. Applying (8.1.10), it follows that
e|x|
n+2 |u(x)| ≤ C, for |x| ≥ R.
(iv) now follows from the above inequality and (8.1.11).
Remark 8.1.2. The conclusions of Theorem 8.1.1 hold for all solutions u ∈ H1(Rn) of
−4u+ au = b|u|αu, where a > 0 and b ∈ R. The proof is the same.
Lemma 8.1.3. Let u ∈ H1(Rn) satisfy −4u + au = b|u|αu, where a > 0 and b ∈ R.
Then,
(i)
∫
Rn
|∇u|2 + a
∫
Rn
|u|2 =
∫
Rn
|u|α+2;
(ii) (Pohozaev’s identity) (n− 2)
∫
Rn
|∇u|2 + na
∫
Rn
|u|2 =2n
α+ 2
∫
Rn
|u|α+2.
Proof. (i) follows immediately by taking the L2 scalar product of the equation with u.
(ii) is the so-called Pohozaev’s identity, which is obtained formally by taking the L2 scalar
product of the equation with x · ∇u (see Berestycki and Lions [1], Proposition 1 p.320).
Before stating the main result of this section, we need to introduce some notation.
Given 0 < α <4
n− 2(0 < α < ∞, if n = 1, 2) and ω > 0, we introduce the following
functionals on H1(Rn).
T (u) =
∫
Rn
|∇u|2 dx; (8.1.12)
V (u) =1
α+ 2
∫
Rn
|u|α+2 dx− ω
2
∫
Rn
|u|2 dx; (8.1.13)
S(u) =1
2T (u)− V (u); (8.1.14)
161
E(u) =1
2
∫
Rn
|∇u|2 dx− 1
α+ 2
∫
Rn
|u|α+2 dx = S(u)− ω
2
∫
Rn
|u|2 dx. (8.1.15)
It is easily verifies that these functionals are in C1(H1(Rn),R), and that T ′(u) = −24u,V ′(u) = |u|αu− ωu. We introduce the sets A and G defined by
A = u ∈ H1(Rn); u 6= 0 and −4u+ ωu = |u|αu; (8.1.16)
G = u ∈ A; S(u) ≤ S(v) for all v ∈ A. (8.1.17)
We have the following result.
Corollary 8.1.4. Let 0 < α <2n
n− 2(0 < α <∞, if n− 1, 2) and ω > 0. If u ∈ H1(Rn)
satisfies (8.1.2), then
S(u) =1
nT (u); (8.1.18)
(n− 2)T (u) = 2nV (u); (8.1.19)
E(u) =nα− 4
2nαT (u); (8.1.20)
ω
2
∫
Rn
|u|2 =4− (n− 2)α
2nαT (u). (8.1.21)
Proof. These identities follow immediately from Lemma 8.1.3.
Our goal is to show that A and G are nonempty, and to characterize G. For technical
reasons, we consider separately the cases n ≥ 3, n = 2 and n = 1. Our results in these
three cases are the following.
Theorem 8.1.5. Assume n ≥ 3. If α, ω are as above, then
(i) A and G are nonempty;
(ii) u ∈ G if, and only if u solves the following minimization problem
V (u) = Λn/2;
S(u) = minS(w); V (w) = Λn/2;(8.1.22)
where Λ =n− 2
2ninfT (v); V (v) = 1. In addition, minS(w); V (w) = Λn/2 =
2
n− 2Λn/2;
162
(iii) there exists a real valued, positive, spherically symmetric and decreasing function
ϕ ∈ G such that G = ∪eiθϕ(· − y); θ ∈ R, y ∈ Rn.
Theorem 8.1.6. Assume n = 2. If α, ω are as above, then
(i) A and G are nonempty;
(ii) u ∈ G if, and only if u solves the following minimization problem
u ∈ N and
∫
Rn
|u|2 = γ;
S(u) = minS(w); w ∈ N;(8.1.23)
where N = u ∈ H1(Rn); V (u) = 0 and u 6= 0 and γ =4
ωαminS(w); w ∈ N;
(iii) there exists a real valued, positive, spherically symmetric and decreasing function
ϕ ∈ G such that G = ∪eiθϕ(· − y); θ ∈ R, y ∈ Rn.
Theorem 8.1.7. Assume n = 1. If α, ω are as above, then
(i) A and G are nonempty;
(ii) A = G;
(iii) there exists a real valued, positive, spherically symmetric and decreasing function
ϕ ∈ G such that G = ∪eiθϕ(· − y); θ ∈ R, y ∈ R.
Let us first consider the case n = 1, which is especially simple.
Proof of Theorem 8.1.7. Note that (8.1.2) is the ordinary differential equation
−u′′ + ωu = |u|αu. (8.1.24)
Define c =
(ω(α+ 2)
2
)1/α
, and let ϕ be the maximal, real valued solution of (8.1.24) such
that ϕ(0) = c and ϕ′(0) = 0. It is clear that ϕ is an even function of x. Furthermore, on
multiplying the equation by ϕ′, we obtain
d
dx
(1
2ϕ′2 − ω
2ϕ2 +
1
α+ 2|ϕ|α+2
)= 0;
and so,1
2ϕ′2 − ω
2ϕ2 +
1
α+ 2|ϕ|α+2 = 0, (8.1.25)
163
throughout the existence interval. It follows easily that ϕ is bounded and therefore exists
for all x ∈ R. Furthermore, we have ϕ′′(0) = −ωαc/2 < 0. Therefore, there exists a > 0
such that ϕ′ < 0 on (0, a). We claim that ϕ′ < 0 on (0,∞). Otherwise, there would exist
b > 0 such that ϕ′ < 0 on (0, b) and ϕ′(b) = 0. Applying (8.1.25), this would imply that
ϕ(b) = −c. Therefore, there would exist d ∈ (0, b) such that ϕ(b) = 0. Applying again
(8.1.25), we would obtain ϕ′(d) = 0, which would imply that ϕ ≡ 0. Therefore, ϕ decreases
to a limit ` ∈ [0, c). In particular, there exists xm →∞ such that ϕ′(xm) → 0. Passing to
the limit in (8.1.25), it follows that `2(
`α
α+ 2− 1
2
)= 0, which implies ` = 0. Therefore ϕ
decreases to 0, as x→ +∞, and it follows easily that the decay is exponential. Therefore
ϕ′′, hence ϕ′ also decay exponentially to 0. Therefore, ϕ ∈ A, which proves (i). Let now
v ∈ A. On multiplying the equation by v′, it follows easily that
1
2|v′|2 − ω
2|v|2 +
1
α+ 2|v|α+2 = K. (8.1.26)
Since v ∈ H1(R), it follows that v(x) → 0, as |x| → ∞. Therefore, by the equation,
v′′(x) → 0, as |x| → ∞; and so, v′(x) → 0, as |x| → ∞. Letting |x| → 0 in (8.1.26), it
follows that K = 0; and so,
1
2|v′|2 − ω
2|v|2 +
1
α+ 2|v|α+2 = 0. (8.1.27)
In particular, we have |v| > 0, for if v would vanish, then by (8.1.27) v′ would vanish at
the same time and we would have v ≡ 0. Therefore, we can write v = ρeiθ, where ρ > 0
and ρ, θ ∈ C2(R). Writing down the system of equations satisfied by ρ, θ, it follows in
particular that ρθ′′+2ρ′θ′ ≡ 0, which implies that there exists K ∈ R such that ρ2θ′ ≡ K;
and so, θ′ ≡ K/ρ2. On the other hand, since |v′| is bounded, it follows that ρ2θ′2 is
bounded. This means that Kθ′ is bounded, or equivalently that K2/ρ2 is bounded. Since
ρ(x) −→|x|→∞
0, we must have K = 0. Therefore (remember that ρ > 0) θ ≡ θ0, for some
θ0 ∈ R. It follows that v = eiθ0ρ. Since ρ ∈ H1(Rn), there must exist x0 ∈ R such that
ρ′(x0) = 0; and by (8.1.27), it follows that ρ(x0) = c. Let now w(x) = ρ(x− x0). Then, w
satisfies (8.1.24), w(0) = c and w′(0) = 0. By uniqueness of the initial value problem for
(8.1.24), it follows that w ≡ ϕ; and so, v(x) = eiθ0ϕ(x+ x0), which completes the proof.
We next consider the case n ≥ 3, and we begin with the following lemma.
164
Lemma 8.1.8. Assume that n ≥ 3, and let λ, ω be as above. Then, the minimization
problem V (u) = 1;
T (u) = minT (w); V (w) = 1;(8.1.28)
has a solution. Every solution u of (8.1.28) satisfies the equation
−4u+ Λωu = Λ|u|αu,
where
Λ =n− 2
2ninfT (v); V (v) = 1. (8.1.29)
Proof. We repeat the proof of Berestycki and Lions [1]. We recall the definition of the
Schwarz symmetrization. If u ∈ L2(Rn) is a nonnegative function, we denote by u∗ the
unique spherically symmetric, nonnegative, nonincreasing function such that
|x ∈ Rn; u∗(x) > λ| = |x ∈ Rn; u(x) > λ|,
for all λ > 0. We refer to Berestycki and Lions [1], Appendix A.III for the main properties
of ∗. In particular, ∫
Rn
|u∗|p =
∫
Rn
|u|p, (8.1.30)
for all 1 ≤ p <∞ such that u ∈ Lp(Rn), and
∫
Rn
|∇u∗|2 ≤∫
Rn
|∇u|2, (8.1.31)
if u ∈ H1(Rn). The proof proceeds in four steps.
Step 1. Selection of a minimizing sequence. Let u ∈ H1(Rn). Then, one can easily find
λ > 0 such that V (λu) = 1. Therefore, the set u ∈ H1(Rn); V (u) = 1 is nonempty. Let
(vm)m∈N be a minimizing sequence of (8.1.28). Let um = (vm)∗. Then, it follows from
(8.1.30) and (8.1.31) that (um)m∈N is also a minimizing sequence of (8.1.28).
Step 2. Estimates of (um)m∈N. By definition, ‖∇um‖L2 is bounded, and by Sobolev’s
inequality, (um)m∈N is bounded in L2n
n−2 (Rn). On the other hand, V (um) = 1 implies
thatω
2
∫
Rn
|um|2 ≤1
α+ 2
∫
Rn
|um|α+2.
165
By Holder’s inequality, this implies that
ω
2‖um‖2
L2 ≤ 1
α+ 2‖um‖nα/2
L2n
n−2‖um‖α+2−nα/2
L2 .
Since α+2−nα/2 < 2, it follows that (um)m∈N is bounded in L2(Rn), hence in H1(Rn).
Step 3. Passage to the limit. By Step 2, there exists a subsequence, which we still denote
by (um)m∈N, and there exists u ∈ H1(Rn) such that um u in H1(Rn), as m → ∞.
Consider now any nonnegative, spherically symmetric, nonincreasing function w ∈ L2(Rn).
For every r > 0, we have
‖w‖2L2 ≥
∫
|x|≥r|w(x)|2 dx ≥ |w(r)|2||x| ≥ r| = crn|w(r)|2;
and so,
w(r) ≤ C1
rn/2‖w‖L2 . (8.1.32)
Applying (8.1.32) to um and u, it follows that there exists K independent of m such that
|um(r)|+ |u(r)| ≤ Kr−n/2, for all r > 0. (8.1.33)
Let us show that um → u in Lα+2(Rn). Indeed, given r > 0,
∫
Rn
|um − u|α+2 ≤∫
|x≤r||um − u|α+2 +
∫
|x≥r||um − u|α+2
≤∫
|x≤r||um − u|α+2 +
C
rnα/2
∫
|x≥r||um − u|2,
by (8.1.33). Given ε > 0, there exists r > 0 such that
C
rnα/2
∫
|x≥r||um − u|2 ≤ ε/2.
Furthermore, if Br is the ball of Rn of radius r, the embedding H1(Rn) → Lα+2(Br) is
compact; and so, for m large enough,
∫
|x≤r||um − u|α+2 ≤ ε/2.
Therefore, for m large enough,
∫
Rn
|um − u|α+2 ≤ ε.
166
This implies that um → u in Lα+2(Rn). By the weak lower semicontinuity of the L2 norm,
it follows that
V (u) ≥ 1, and T (u) ≤ lim infm→∞
T (um) =2n
n− 2Λ, (8.1.34)
where Λ is defined by (8.1.29). Since V (u) ≥ 1, it follows that u 6= 0. We claim that
in fact V (u) = 1. Indeed, if V (u) > 1, then there exists λ > 1 such that v(x) = u(λx)
verifies V (v) = 1. Then T (v) = λ2−nT (u) < T (u) ≤ 2nn−2
Λ, which is a contradiction with
the definition of Λ. Thus, V (u) = 1, which implies by definition of Λ that T (u) ≥ 2nn−2
Λ.
Comparing with (8.1.34), it follows that T (u) = 2nn−2Λ. Therefore, u satisfies (8.1.28).
Step 4. Conclusion. Let u be any solution of (8.1.28). There exists a Lagrange multiplier
λ such that
−4u = λ(|u|αu− ωu). (8.1.35)
Taking the L2 scalar product of (8.1.35) with u, we obtain
T (u) = λ
((α+ 2)V (u) +
αω
2
∫
Rn
|u|2)
= λµ,
with µ > 0. Therefore, λ > 0. Applying Lemma 7.1.3 (ii), it follows that
T (u) =2n
n− 2λV (u) =
2n
n− 2λ.
Since T (u) =2n
n− 2Λ, it follows that λ = Λ. This completes the proof.
Corollary 8.1.9. Assume that n ≥ 3, and let α, ω be as above. If Λ is defined by
(8.1.29), then the minimization problem
V (u) = Λn/2;
T (u) = minT (w); V (w) = Λn/2;(8.1.36)
has a solution. Every solution u of (8.1.36) satisfies equation (8.1.2). In addition,
minT (w); V (w) = Λn/2 =2n
n− 2Λn/2. (8.1.37)
Proof. Given u ∈ H1(Rn), let Au ∈ H1(Rn) be defined by u(x) = Au(Λ1/2x). One
verifies quite easily that u satisfies (8.1.28) if, and only if Au satisfies (8.1.36). Therefore,
167
it follows from Lemma 8.1.8 that (8.1.36) has a solution. Finally, given a solution u of
(8.1.36), let v be defined by Av = u. Then, v satisfies (8.1.28); and so T (v) = 2nn−2Λ, by
(8.1.29). This implies that T (u) = Λn/2−1T (v) = 2nn−2
Λn/2. Hence (8.1.37). Furthermore,
since v satisfies
−4u+ Λωu = Λ|u|αu,
it follows that u satisfies (8.1.2). This completes the proof.
Corollary 8.1.10. Assume that n ≥ 3, and let α, ω be as above. If Λ is defined by
(8.1.29), then the minimization problem
V (u) = Λn/2;
S(u) = minS(w); V (w) = Λn/2;(8.1.38)
has a solution. Every solution u of (8.1.38) satisfies equation (8.1.2). In addition,
minS(w); V (w) = Λn/2 =2
n− 2Λn/2. (8.1.39)
Finally, u satisfies (8.1.38) if, and only if u satisfies (8.1.36)
Proof. Let u ∈ H1(Rn) be such that V (u) = Λn/2. Then,
S(u) =1
2T (u)− Λn/2.
It follows immediately that u satisfies (8.1.36) if, and only if u satisfies (8.1.38); and so,
(8.1.38) has a solution, by Corollary 8.1.9. Finally, let u satisfy (8.1.38). Then, u satisfies
(8.1.36), and by Corollary 8.1.9, u satisfies (8.1.2). Furthermore, (8.1.39) follows from
(8.1.37) and (8.1.18).
Corollary 8.1.11. Assume that n ≥ 3. If α, ω are as above, then G is nonempty.
Furthermore, u ∈ G if and only if u satisfies (8.1.38).
Proof. Consider a solution u of (8.1.38). It follows from Corollary 8.1.10 that u satisfies
(8.1.36) and (8.1.2). In particular, it follows from (8.1.37) and (8.1.39) that
V (u) = Λn/2; T (u) =2n
n− 2Λn/2; S(u) =
2
n− 2Λn/2. (8.1.40)
168
It follows in particular from Corollary 8.1.10 that A is nonempty. Consider any v ∈ A. It
follows from Corollary 8.1.4 that if we set
V (v) = γn/2, (8.1.41)
we have
T (v) =2n
n− 2γn/2 and S(v) =
2
n− 2γn/2. (8.1.42)
Let σ = Λ/γ, and let v(x) = w(σ1/2x). We have V (w) = Λn/2; and so, by (8.1.37),
T (w) ≥ 2n
n− 2Λn/2. (8.1.43)
By (8.1.42), we have
T (w) = σn/2−1T (v) =2n
n− 2Λn/2 γ
Λ.
Applying (8.1.43), it follows that γ ≥ Λ. By (8.1.40) and (8.1.42), this implies that
S(v) ≥ S(u); (8.1.44)
and so, u ∈ G. In particular, G is nonempty. If we assume further that v ∈ G, then we
must have S(v) ≤ S(u), since u satisfies (8.1.2). In view of (8.1.44), this means that
S(v) = S(v).
Applying (8.1.40), (8.1.41) and (8.1.42), it follows that
V (v) = Λn/2 and S(u) =2
n− 2Λn/2.
It follows from Corollary 8.1.10 that v satisfies (8.1.39). This completes the proof.
Finally, before completing the proof of Theorem 8.1.5, we need the following lemma.
Lemma 8.1.12. Let a : Rn → R be continuous, and assume that a(x) −→|x|→∞
0. If there
exists v ∈ H1(Rn) such that
∫
Rn
(|∇v|2 − a|v|2
)dx < 0, (8.1.45)
169
then there exists λ > 0 and a positive solution u ∈ H1(Rn) ∩ C(Rn) of equation
−4u+ λu = au. (8.1.46)
In addition, if w ∈ H1(Rn) is nonnegative, w 6= 0, and if there exists ν ∈ R such that
−4w + νw = aw, then there exists c > 0 such that w = cu. In particular, µ = λ.
Proof. Let us first show that the following minimization problem has a nonnegative so-
lution. ‖u‖L2 = 1,
J(u) = minJ(v); ‖v‖L2 = 1;(8.1.47)
where
J(u) =
∫
Rn
(|∇u|2 − a|u|2
)dx.
Indeed, let (vm)m∈N be a minimizing sequence of (8.1.46), and let um = |vm|. Since
|um| = |vm| and |∇um| ≤ |∇vm|, it follows that (um)m∈N is also a minimizing sequence.
Since a ∈ L∞(Rn) by assumption, it follows easily that (um)m∈N is bounded in H1(Rn).
Therefore, there exists a subsequence, which we still denote by (um)m∈N, and there exists
u ∈ H1(Rn) such that um u in H1(Rn). Note that u ≥ 0. Let us show that u
satisfies (8.1.47). For every r > 0, we have
∫
Rn
|a| |u2m − u2| ≤
∫
|x|≤r|a| (um + u)|um − u|+ sup|a(x)|; |x| ≥ r
∫
|x|≥r(u2
m + u2).
It follows that
∫
Rn
|a| |u2m − u2| ≤ 2‖a‖L∞
(∫
|x|≤r|um − u|2
)1/2
+ 2 sup|a(x)|; |x| ≥ r.
Consider ε > 0. There exists r > 0 such that
2 sup|a(x)|; |x| ≥ r ≤ ε/2.
Since the embedding H1(Rn) → L2(Br) is compact, it follows that for m large enough,
we have
2‖a‖L∞
(∫
|x|≤r|um − u|2
)1/2
≤ ε/2.
170
Therefore, ∫
Rn
|a| |u2m − u2| ≤ ε,
for m large enough. It follows that
∫
Rn
|a|u2m −→
m→∞
∫
Rn
|a|u2.
Using the weak lower semicontinuity of the L2 norm, it follows that
J(u) ≤ −µ, and ‖u‖L2 ≤ 1;
where −µ = infJ(v); ‖v‖L2 = 1. Note that by (8.1.45), µ > 0; and so, u 6= 0. We
claim that ‖u‖L2 = 1. Otherwise, there would exist k > 1 such that w = ku verifies
‖w‖L2 = 1. We would obtain J(w) = k2J(u) < −µ, which is a contradiction by definition
of µ. Therefore, ‖u‖L2 = 1, and again by definition of µ, we must have J(u) = −µ. It
follows that u satisfies (8.1.47). Therefore, there exists a lagrange multiplier λ such that
−4u+ λu = au. (8.1.48)
On taking the L2 scalar product of the equation with u, we obtain
λ = µ > 0. (8.1.49)
It follows easily from (8.1.48) that u ∈ H2(Rn)∩C(Rn) (see the proof of Theorem 8.1.1);
and since u ≥ 0, it follows from the strong maximum principle (Gilbarg and Trudinger [1],
Corollary 8.21, p.199) that
u > 0, on Rn. (8.1.50)
So far, we have proved the first part of the statement of the lemma. Let now ν ∈ R be
such that there exists a solution w ∈ H1(Rn), w ≥ 0 of equation
−4w + νw = aw. (8.1.51)
We may assume that w 6= 0. On multiplying (8.1.48) by w, (8.1.51) by u and computing
the difference, it follows that
(λ− ν)
∫
Rn
wu = 0.
171
Since wu ≥ 0 and wu 6≡ 0, this implies that ν = λ. We claim now that there exists c > 0
such that w = cu; for if this was not the case, there would exist c > 0 such that z = w− cutakes both positive and negative values. Note that
−4z + λz = az.
On multiplying the equation by z, it follows that
J(z) = −λ‖z‖2L2 .
Therefore, y defined by
y =z
‖z‖L2
,
satisfies (8.1.47). It follows that |y| also satisfies (8.1.47). Repeating the argument that
we made for u, it follows that |y| satisfies (8.1.48), and that |y| > 0. Therefore, z has a
constant sign, which is a contradiction. this completes the proof.
Proof of Theorem 8.1.5. Parts (i) and (ii) follow immediately from Corollary 8.1.11. It
remains to show (iii). Consider u ∈ G. It follows that u satisfies (8.1.38). Let f = |Re(u)|,g = |Im(u)| and v = f + ig. We have |v| = |u| and |∇v| = |∇u|. It follows that v also
satisfies (8.1.38). Applying Corollary 8.1.11, this implies that
−4v + ωv = |v|αv;
and so, −4f + ωf = af ;
−4g + ωg = ag;
where a = |v|α. Applying Theorem 8.1.1, it follows that a verifies the assumption of
Lemma 8.1.12. Furthermore,
J(v) = −ω‖v‖2L2 < 0.
It follows from Lemma 8.1.12 that there exists a positive function z and two nonnegative
constants µ, ν such that f = µz and g = νz. In particular, Re(u) and Im(u) do not change
sign; and so, there exists c, d ∈ R such that u = cz + idz. This implies that there exists a
positive function ψ and θ ∈ R such that u = eiθψ. It follows that ψ also satisfies (8.1.38),
hence (8.1.2) by Corollary 8.1.11. By Theorem 8.1.1, ψ ∈ C2(Rn) and |ψ(x)| −→|x|→∞
0.
172
Applying Gidas, Ni and Nirenberg [1], Theorem 2 p.370, it follows that there exists a
positive, spherically symmetric solution ϕ of (8.1.2) and y ∈ Rn such that ψ(·) = ϕ(· − y).Therefore, u(·) = eiθϕ(·− y). Note that ϕ, being radially symmetric, satisfies the ordinary
differential equation
ϕ′′ +n− 1
rϕ′ + ϕα+1 − ωϕ = 0.
It follows from Kwong [1] that such a solution ϕ is unique. This completes the proof.
Remark 8.1.13. Note that we gave a self-contained proof of statements (i) and (ii).
On the contrary, the proof of (iii) relies on the two difficult results of Gidas, Ni and
Nirenberg [1] and of Kwong [1]. We use property (iii) to prove a strong version of the
stability property (cf Section 8.3).
We finally consider the case n = 2. Note that the method for n ≥ 3 does not apply
to this case, since by Corollary 8.1.4, we have V (u) = 0, for every u ∈ A.
Proof of Theorem 8.1.6. We proceed in four steps. We define
N = u ∈ H1(Rn); V (u) = 0 and u 6= 0, (8.1.52)
c = infS(w); w ∈ N, (8.1.53)
and
γ =4
ωαinfS(w); w ∈ N. (8.1.54)
Let us first observe that γ > 0. Indeed, consider u ∈ N . It follows that
∫
Rn
|u|2 ≤ 2
ω(α+ 2)
∫
Rn
|u|α+2.
On the other hand, it follows from Gagliardo-Nirenberg’s inequality that there exists C
independent of u such that
∫
Rn
|u|α+2 ≤ C(T (u))α/2
∫
Rn
|u|2.
This implies that there exists σ > 0 such that T (u) ≥ σ; and so,
S(u) ≥ σ
2, for all u ∈ N.
173
Therefore, γ > 0.
Step 1. The minimization problem (8.1.23) has a solution. We repeat the proof of
Berestycki, Gallouet and Kavian [1]. It is clear that N 6= ∅. Let (vm)m∈N be a mini-
mizing sequence. In other words, vm 6= 0, V (vm) = 0 and S(vm) → c. Let wm = |vm|∗
(see the beginning of the proof of Lemma 8.1.8). It follows that (wm)m∈N verifies the same
properties as (vm)m∈N. Define now (um)m∈N by um(x) = wm(λ1/2m x), where
λm =‖wm‖2
L2
γ.
We have ∫
Rn
u2m = γ, (8.1.55)
V (um) = 0, (8.1.56)
and
S(um) = S(wm) −→m→∞
c. (8.1.57)
In particular, (um)m∈N is also a minimizing sequence. It follows from (8.1.55), (8.1.56)
and (8.1.57) that (um)m∈N is bounded in H1(Rn). Therefore, there exists a subsequence,
which we still denote by (um)m∈N, and there exists u ∈ H1(Rn) such that um u in
H1(Rn), as m→∞. It follows in particular (see the proof of Lemma 8.1.8) that
∫
Rn
uα+2m −→
m→∞
∫
Rn
uα+2,
∫
Rn
u2 ≤ lim infm→∞
∫
Rn
u2m = γ,
and
T (u) ≤ lim infm→∞
T (um).
Therefore,
V (u) ≥ 0, and S(u) ≤ c.
We claim that V (u) = 0. To see this, we argue by contradiction. If V (u) > 0, then in
particular u 6= 0. It follows that there exists λ ∈ (0, 1) such that v = λu verifies V (v) = 0;
and so v ∈ N . Furthermore, T (v) = λ2T (u) < T (u). It follows that S(v) < S(u),
174
which implies that S(v) < c. This is in contradiction with the definition of c. Therefore,
V (u) = 0. It follows that V (um) → V (u), which implies that
∫
Rn
u2 = limm→∞
∫
Rn
u2m = γ;
and so, u satisfies (8.1.23).
Step 2. Every solution of (8.1.23) belongs to A. Indeed, consider a solution u of (8.1.23)
(which exists by Step 1). There exists a Lagrange multiplier λ such that
−4u = λ(|u|αu− ωu).
On taking the L2 scalar product of the equation with u, we obtain
T (u) = λ
(∫
Rn
|u|α+2 − ω
∫
Rn
|u|2).
Since u satisfies (8.1.23), this implies that
2c =λωαγ
2;
and so, λ = 1. Therefore, u satisfies (8.1.2). Hence the result.
Step 3. u satisfies (8.1.23) if, and only if u ∈ G. Consider any solution u of (8.1.23) and
any v ∈ A (A 6= ∅, by Step 2). It follows from Corollary 8.1.4 that v ∈ N and
∫
Rn
|v|2 =4
ωαS(v) = γ
S(v)
S(u). (8.1.58)
Since v ∈ N , it follows that S(v) ≥ S(u); and so u ∈ G 6= ∅.Assume further that v ∈ G. Since also u ∈ G, we have S(u) = S(v). It follows from
(8.1.58) that ∫
Rn
|v|2 = γ,
which means that v satisfies (8.1.23). Hence the result.
Step 4. Conclusion. properties (i) and (ii) follow from Step 3; and so, it remains to
establish (iii). Consider u ∈ G. In particular, u satisfies (8.1.23). Let f = |Re(u)|,g = |Im(u)| and v = f + ig. We have |v| = |u| and |∇v| = |∇u|. It follows that v also
satisfies (8.1.23). Therefore, by Step 2
−4v + ωv = |v|αv;
175
and so, −4f + ωf = af ;
−4g + ωg = ag;
where a = |v|α. Applying Theorem 8.1.1, it follows that a verifies the assumption of
Lemma 8.1.12. Furthermore,
J(v) = −ω‖v‖2L2 < 0.
It follows from Lemma 8.1.12 that there exists a positive function z and two nonnegative
constants µ, ν such that f = µz and g = νz. In particular, Re(u) and Im(u) do not change
sign; and so, there exists c, d ∈ R such that u = cz + idz. This implies that there exists a
positive function ψ and θ ∈ R such that u = eiθψ. It follows that ψ also satisfies (8.1.23),
hence (8.1.2) by Step 2. By Theorem 8.1.1, ψ ∈ C2(Rn) and |ψ(x)| −→|x|→∞
0. Applying
Gidas, Ni and Nirenberg [1], Theorem 2 p.370, it follows that there exists a positive,
spherically symmetric solution ϕ of (8.1.2) and y ∈ Rn such that ψ(·) = ϕ(·−y). Therefore,
u(·) = eiθϕ(·−y). Note that ϕ, being radially symmetric, satisfies the ordinary differential
equation
ϕ′′ +n− 1
rϕ′ + ϕα+1 − ωϕ = 0.
It follows from Kwong [1] that such a solution ϕ is unique. This completes the proof.
Definition 8.1.14. A function u ∈ A is called a bound state of (8.1.2). A function u ∈ Gis called a ground state of (8.1.2). By definition, this is a bound state that minimizes the
action S among all other bound states.
Remark 8.1.15. Note that the ground state is unique, modulo space translations and
multiplication by eiθ, as follows from Theorems 8.1.5 to 8.1.7.
Remark 8.1.16. In the litterature, one calls sometimes ground state any positive so-
lution of (8.1.2). It follows from Theorems 8.1.5 to 8.1.7 that these two definitions are
equivalent, modulo multiplication by eiθ.
Remark 8.1.17. In the case n = 1, every u ∈ A is a ground state, since A = G. This is
not true anymore when n ≥ 2. Indeed, in this case, it follows from Berestycki and Lions [1],
176
and Berestycki, Gallouet and Kavian [1] that there exists a sequence (um)m∈N ⊂ A such
that S(um) −→m→∞
∞. This implies that for m large, um 6∈ G.
Remark 8.1.18. Let u be the (unique) positive, spherically symmetric ground state of
(8.1.2) with ω = 1. For ω > 0, let uω(x) = ω1/αu(ω1/2x). It follows that uω satisfies (8.1.2);
and so, uω is the unique positive, spherically symmertic ground state of (8.1.2). We have
‖uω‖2H1 = ω
2α−n
2
∫
Rn
u2 + ω2α−n−2
2
∫
Rn
|∇u|2.
Therefore, if α ≥ 4/n, there exists σ > 0 such that ‖uω‖H1 ≥ σ, for all ω > 0. On the
other hand, when α < 4/n, we have ‖uω‖H1 → 0, as ω → 0. In particular, there exist
ground states of (8.1.2) of arbitrarily small H1 norm (when ω varies).
8.2. An instability result. We begin with the following result of Weinstein [1].
Theorem 8.2.1. Let α = 4/n, ω > 0. If ϕ ∈ A (cf Theorems 8.1.5, 8.1.6 and 8.1.7),
then u(t, x) = eiωtϕ(x) is an instable solution of (8.1) in the following sense. There exists
(ϕm)m∈N ⊂ H1(Rn) such that
ϕm −→m→∞
ϕ, in H1(Rn),
and such that the corresponding maximal solution um of (8.1) blows up in finite time for
both t > 0 and t < 0.
Proof. It follows from Corollary 8.1.4 that E(ϕ) = 0. Therefore, E(λϕ) < 0, for every
λ > 1. On the other hand, it follows from Theorem 8.1.1 that xϕ(x) ∈ L2(Rn). Applying
Theorem 6.4.7, it follows that the maximal solution of (8.1) with initial datum λϕ blows
up in finite time for both t > 0 and t < 0. The result follows by taking for example
ϕm = (1 + 1m )ϕ.
In the case α > 4/n, we have the following result of Berestycki and Cazenave [1] (see
also Cazenave [3,4]).
Theorem 8.2.2. Let4
n< α <
4
n− 2(4/n < α < ∞, if n = 1, 2) and ω > 0. If ϕ ∈ G
(cf Theorems 8.1.5, 8.1.6 and 8.1.7), then u(t, x) = eiωtϕ(x) is an instable solution of (8.1)
in the following sense. There exists (ϕm)m∈N ⊂ H1(Rn) such that
ϕm −→m→∞
ϕ, in H1(Rn),
177
and such that the corresponding maximal solution um of (8.1) blows up in finite time for
both t > 0 and t < 0.
Remark 8.2.3. As we will see, the proof of Theorem 8.2.2 is much more complicated
than the proof of Theorem 8.2.1. On the other hand, the result is much weaker (except
when n = 1), since it concerns only the ground states (see Remark 8.1.17). Up to now, it
is unknown whether or not the other stationary states are instable.
Let us define the functional Q ∈ C1(H1(Rn),R) by
Q(u) =
∫
Rn
|∇u|2 − nα
2(α+ 2)
∫
Rn
|u|α+2, for u ∈ H1(Rn), (8.2.1)
and let
M = u ∈ H1(Rn); u 6= 0 and Q(u) = 0. (8.2.2)
The proof of Theorem 8.2.2 relies on the following result.
Proposition 8.2.4. Let α, ω be as in Theorem 8.2.2. If u ∈ H1(Rn), then u ∈ G if and
only if u solves the following minimization problem.
u ∈M ;
S(u) = minS(v); v ∈M.(8.2.3)
Before proceeding to the proof of Proposition 8.2.4, we need the following lemma.
Lemma 8.2.5. For u ∈ H1(Rn), u 6= 0 and λ > 0, define P(λ, u)(x) = λn/2u(λx). Then,
the following properties hold.
(i) There exists a unique λ∗(u) > 0 such that P(λ∗(u), u) ∈M ;
(ii) the function λ 7→ S(P(λ, u)) is concave on (λ∗(u),∞);
(iii) λ∗(u) < 1 if, and only if Q(u) < 0;
(iv) λ∗(u) = 1 if, and only if u ∈M ;
(v) S(P(λ, u)) < S(P(λ∗(u), u)) for every λ > 0, λ 6= λ∗(u);
(vi)d
dλS(P(λ, u)) =
1
λQ(P(λ, u)) for every λ > 0;
178
(vii) |P(λ, u)|∗ = P(λ, |u|∗) for every λ > 0, where ∗ is the Schwarz symmetrization.
In addition, for every λ > 0, we have the following property.
(viii) If um → u in H1(Rn) weakly and in Lα+2(Rn) strongly, then P(λ, um) → P(λ, u) in
H1(Rn) weakly and in Lα+2(Rn) strongly.
Proof. Let u ∈ H1(Rn), u 6= 0 and let uλ = P(λ, u). We have
S(uλ) =λ2
2
∫
Rn
|∇u|2 +ω
2
∫
Rn
|u|2 − λnα/2
α+ 2
∫
Rn
|u|α+2. (8.2.4)
(vi) follows easily. Let λ∗(u) be defined by
λ∗(u)nα−4
2 =α+ 2
2
(∫
Rn
|∇u|2)(∫
Rn
|u|α+2
)−1
.
Elementary calculations show that with λ∗(u) defined as above, properties (i), (ii), (iii),
(iv) and (v) are verified. (vii) follows easily from the definition of Schwarz’ symmetrization
(see the beginning of the proof of Lemma 8.1.8). Finally, given λ > 0, the operator
u 7→ P(λ, u) is linear and strongly continuous H1(Rn) → H1(Rn). Therefore, it is also
weakly continuous. The Lα+2 continuity is immediate. Hence (viii).
Corollary 8.2.6. M is nonempty. If we set
m = infS(u); u ∈M, (8.2.5)
then for every u ∈ H1(Rn) such that Q(u) < 0, we have Q(u) ≤ S(u)−m.
Proof. It follows from Lemma 8.2.5 that M is nonempty. Let u ∈ H1(Rn) be such that
Q(u) < 0, and let f(λ) = S(P(λ, u)). By Lemma 8.2.4 (iii), we have λ∗(u) < 1, and by
(ii) f is concave on (λ∗(u), 1). Therefore,
f(1) ≥ f(λ∗(u)) + (1− λ∗(u))f ′(1).
Applying (vi), we obtain
S(u) ≥ f(λ∗(u)) + (1− λ∗(u))Q(u) ≥ f(λ∗(u)) +Q(u).
179
Since by (i) P(λ∗(u), u) ∈M , it follows that f(λ∗(u)) ≥ m; and so,
S(u) ≥ m+Q(u),
which completes the proof.
Proof of Proposition 8.2.4. We proceed in three steps.
Step 1. The minimization problem (8.2.3) has a solution. By Corollary 8.2.6, M 6= ∅.Therefore, (8.2.3) has a minimizing sequence (vm)m∈N. In particular, Q(vm) = 0 and
S(vm) → m, where m is defined by (8.2.5). Let wm = |vm|∗, and um = P(λ∗(wm), wm).
It follows from Lemma 8.2.4 (i) that um ∈ M . Furthermore, it follows from (vii) that
um = |P(λ∗(wm), vm)|∗. Therefore,
S(um) ≤ S(P(λ∗(wm), vm)) ≤ S(P(λ∗(vm), vm)) ≤ S(vm),
where the last two inequalities follow from (v) and (i). In particular, (um)m∈N is a nonneg-
ative, spherically symmetric, nonincreasing minimizing sequence of (8.2.3). Furthermore,
note that
S(um) =2
nαQ(um) +
nα− 4
2nα
∫
Rn
|∇um|2 +ω
2
∫
Rn
u2m
=nα− 4
2nα
∫
Rn
|∇um|2 +ω
2
∫
Rn
u2m.
It follows that (um)m∈N is bounded in H1(Rn). Since Q(um) = 0, it follows from
Gagliardo-Nirenberg’s inequality and the boundedness of (um)m∈N in L2(Rn) that there
exists C such that
‖∇um‖L2 ≤ C‖∇um‖nα4
L2 .
Since nα > 4, it follows that ‖∇um‖L2 is bounded from below; and since Q(um) = 0, there
exists σ > 0 such that
‖um‖Lα+2 ≥ σ, for all m ≥ 0. (8.2.6)
Arguing like in the proof of Lemma 8.1.8, it follows that there exists a subsequence, which
we still denote by (um)m∈N, and v ∈ H1(Rn) such that um → v as m → ∞, in H1(Rn)
weakly and in Lα+2(Rn) strongly; and so, by (8.2.6), v 6= 0. Therefore, we can define
180
u = P(λ∗(v), v). It follows from Lemma 8.2.4 (i) that u ∈M . Furthermore, it follows from
(vii) that P(λ∗(v), um) → u in H1(Rn) weakly and in Lα+2(Rn) strongly. Therefore,
S(u) ≤ lim infm→∞
S(P(λ∗(v), um)) ≤ lim infm→∞
S(P(λ∗(um), um)) = lim infm→∞
S(um) = m,
where the last three inequalities follow from (v), (iv) and (8.2.5); and so, u satisfies (8.2.3).
Step 2. Every solution of (8.2.3) satisfies (8.1.2). Consider any solution u of (8.2.3). For
σ > 0, let u(x) = σ2/αuσ(σx). One verifies easily that
Q(uσ) = σn−2− 2αQ(u) = 0;
and so, uσ ∈ M . Since u = u1 satisfies (8.2.3), it follows that f(σ) = S(uσ) verifies
f ′(1) = 0. One computes easily by using the property uσ ∈M , that
f ′(1) = 〈S′(u), u〉H−1,H1 ,
where S′ is the gradient of the C1 functional S (i.e. S′(u) = −4u+ωu−|u|αu). It follows
that
〈S′(u), u〉H−1,H1 = 0. (8.2.7)
On the other hand, Q′(u) = −24u− nα2|u|αu; and so, since u ∈M we obtain
〈Q′(u), u〉H−1,H1 = −αT (u) < 0. (8.2.8)
Finally, since u satisfies (8.2.3), there exists a Lagrange multiplier λ such that S ′(u) =
λQ′(u). Applying (8.2.7) and (8.2.8), it follows that λ = 0; and so, S ′(u) = 0, which
means that u satisfies (8.1.2).
Step 3. Conclusion. Consider
` = minS(u); u ∈ A. (8.2.9)
Let u ∈ G. It follows that S(u) = `. Applying Corollary 8.1.4, one obtains easily that
u ∈M . Therefore, S(u) ≥ m, where m is defined by (8.2.5). In particular,
` ≥ m. (8.2.10)
181
Consider now a solution u of (8.2.3). By Step 2, u ∈ A. Since S(u) = m, it follows from
(8.2.9) that m ≥ `. Comparing with (8.2.10), we obtain m = `. The equivalence of the
two problems follows easily.
Proof of Theorem 8.2.2. Let ϕ ∈ G, and let ϕλ = P(λ, ϕ), for λ > 0. It follows from
Lemma 8.2.5 (iv) and (v) that
Q(ϕλ) < 0, (8.2.11)
and
S(ϕλ) < m = S(ϕ), (8.2.12)
for all λ > 1. Let uλ be the maximal solution of (8.1) with initila datum ϕλ. By conser-
vation of charge and energy, we have
S(uλ(t)) = S(ϕλ), for all t ∈ (−T∗(ϕλ), T ∗(ϕλ)). (8.2.13)
By continuity, it follows from (8.2.11) that Q(uλ(t)) < 0 for |t| small. On the other hand,
if t is such that Q(uλ(t)) < 0, it follows from Corollary 8.2.6, (8.2.13) and (8.2.12) that
Q(uλ(t)) ≤ S(ϕλ)−m = −δ < 0. (8.2.14)
By continuity, it follows easily that (8.1.14) holds for all t ∈ (−T∗(ϕλ), T ∗(ϕλ)). Applying
Proposition 6.4.2, it follows that f defined by (6.4.2) verifies
f ′′(t) = 8Q(uλ(t)) ≤ −8δ, for all t ∈ (−T∗(ϕλ), T ∗(ϕλ)).
It follows easily that both T∗(ϕλ) and T ∗(ϕλ) are finite (see the proof of Theorem 6.4.7).
The result follows, since ϕλ → ϕ in H1(Rn), as λ ↓ 1 (apply Theorem 8.1.1).
Remark 8.2.7. Theorems 8.2.1 and 8.2.2 show the instability of ground states when
α ≥ 4/n. When α < 4/n, it follows from the results of Section 8.3 that the ground states
are on the contrary stable.
Remark 8.2.8. The result of Theorem 8.2.2 still holds for more general nonlinearities,
see Berestycki and Cazenave [1].
182
8.3. A stability result. Our goal in this section is to establish the following result of
Cazenave and Lions [1] (see also P.-L. Lions [3,4] and Cazenave [2])
Theorem 8.3.1. Let 0 < α < 4/n, ω > 0. If ϕ ∈ G (cf Theorems 8.1.5, 8.1.6 and
8.1.7), then u(t, x) = eiωtϕ(x) is a stable solution of (8.1) in the following sense. For every
ε > 0, there exists δ(ε) > 0 such that if ψ ∈ H1(Rn) verifies ‖ϕ − ψ‖H1 ≤ δ(ε), then the
corresponding maximal solution v of (8.1) verifies
supt∈R
infθ∈R
infy∈Rn
‖v(t, ·)− eiθϕ(· − y)‖H1 ≤ ε. (8.3.1)
In other words, there exist functions θ(t) ∈ R and y(t) ∈ Rn such that
supt∈R
‖u(t)− eiθ(t)ϕ(· − y(t))‖H1 ≤ ε. (8.3.2)
Remark 8.3.2. Theorem 8.3.1 means that if ψ is close to ϕ in H1(Rn), then the solution
of (8.1) with initial datum ψ remains close to the orbit of ϕ, modulo space translations.
Note that α < 4/n, which implies that all solutions of (8.1) are global (see Remark 6.5.1).
Remark 8.3.3. The space translations appearing in (8.3.1) and (8.3.2) are necessary.
Indeed, let ϕ ∈ G. Given ε > 0 and y ∈ Rn such that |y| = 1, let
ϕε(x) = eiεx·yϕ(x),
and
uε(t, x) = eiε(x·y−εt)eiωtϕ(x− 2εty).
One verifies easily that uε is the solution of (8.1) with initial datum ϕε. Furthermore,
ϕε → ϕ in H1(Rn), as ε ↓ 0, but one verifies easily that for every ε > 0,
supt∈R
infθ∈R
‖uε(t)− eiθϕ‖H1 = 2‖ϕ‖H1 .
Remark 8.3.3. On the other hand, it is clear that if ϕ ∈ G is spherically symmetric and
if ψ is also spherically symmetric, one can remove the space translations in (8.3.1) and
(8.3.2). In other words,
supt∈R
infθ∈R
‖v(t)− eiθϕ‖H1 ≤ ε.
183
This follows from a trivial adaptation of the proof of the stability theorem in the subspace of
H1(Rn) of spherically symmetric function. Alternatively, this follows from the observation
that if f, g ∈ H1(Rn) are spherically symmetric, then
infy∈Rn
‖f(·)− g(· − y)‖H1 = ‖f − g‖H1 .
Remark 8.3.4. The rotations eiθ appearing in (8.3.1) and (8.3.2) are necessary. Indeed,
let ϕ ∈ G and let u(t, x) = eiωtϕ(x). Given ε > 0, let
ϕε(x) = (1 + ε)1/αϕ((1 + ε)1/2x),
and
uε(t, x) = eiω(1+ε)t(1 + ε)1/αϕ((1 + ε)1/2x).
One verifies easily that uε is the solution of (8.1) with initial datum ϕε. Furthermore,
ϕε → ϕ in H1(Rn), as ε ↓ 0, but one verifies easily that for every ε > 0,
supt∈R
infy∈Rn
‖uε(t, ·)− ϕ(· − y)‖H1 = supt∈R
infy∈Rn
‖uε(t, ·)− u(t, · − y)‖H1 ≥ ‖ϕ‖H1 .
Remark 8.3.5. Theorem 8.3.1 asserts only the stability of ground states. Except when
n = 1, where A = G, one does not know whether or not the other standing waves are
stable.
The proof of Theorem 8.3.1 relies on the following result.
Proposition 8.3.6. Let α, ω be as in Theorem 8.3.1, let µ > 0, and let E be defined by
(8.1.15). If
Γ = u ∈ H1(Rn);
∫
Rn
|u|2 = µ, (8.3.3)
then
(i) the following minimization problem has a solution.
u ∈ Γ;
E(u) = minE(v); v ∈ Γ.(8.3.4)
184
In addition, if we set
−ν = infE(v); v ∈ Γ, (8.3.5)
then the following property holds.
(ii) If (um)m∈N verifies ‖um‖L2 → √µ and E(um) → −ν, then there exists a subsequence
(umk)k∈N and a family (yk)k∈N ⊂ Rn such that (umk
(· − yk))k∈N has a strong limit
u in H1(Rn). In particular, u satisfies (8.3.4).
The proof of Proposition 8.3.6 is delicate and relies on the concentration-compactness
principle introduced by P.-L. Lions [3,4]. We need the following two lemmas, the first of
which is a type of Sobolev’s inequality, and the second a formulation of the concentration-
compactness principle (cf. P.-L. Lions [3], Lemma III.1, p.135).
Lemma 8.3.7. Let 0 < α < 4/n. There exists a constant C such that
∫
Rn
|u|α+2 ≤ C
(sup
y∈Rn
∫
|x−y|≤1|u(x)|2 dx
)α/2
‖u‖2H1 ,
for all u ∈ H1(Rn).
Proof. Cover Rn by a sequence of unit cubes (Qj)j∈N such that Qj ∩ Qk = ∅, if j 6= k.
Then, ∫
Rn
|u|α+2 =
∞∑
j=0
∫
Qj
|u|α+2,
and
‖u‖2H1 =
∞∑
j=0
∫
Qj
(|∇u|2 + |u|2).
Furthermore, it follows from Holder’s inequality and Sobolev’s embedding theorem that
there exists C such that
∫
Qj
|u|α+2 ≤ C
(∫
Qj
|u|2)α/2 ∫
Qj
(|∇u|2 + |u|2).
Note that all Qj ’s are isometric; and so C can be chosen to be independent of j. Summing
on j the above inequality, we obtain
∫
Rn
|u|α+2 ≤ C
(supj∈N
∫
Qj
|u|2)α/2
‖u‖2H1 ,
185
from which the result follows.
Lemma 8.3.8. If µ > 0 and if (um)m∈N is a bounded sequence of H1(Rn) such that
∫
Rn
|um|2 = µ,
then there exists a subsequence, which we still denote by (um)m∈N, for which one of the
following properties holds.
(i) There exists a sequence (ym)m∈N ⊂ Rn such that for every ε > 0, there exists R <∞such that
∫
|x−ym|≤R|um(x)|2 dx ≥ µ− ε.
(ii) limm→∞
supy∈Rn
∫
|x−y|≤R|um(x)|2 dx = 0.
(iii) There exists γ ∈ (0, µ) such that for every ε > 0, there exists m0 ≥ 0 and two
sequences (vm)m∈N, (wm)m∈N ⊂ H1(Rn), with compact, disjoint supports, such that
for m ≥ m0
‖vm‖H1 + ‖wm‖H1 ≤ 4 supm∈N
‖um‖H1 ; (8.3.6)
‖um − vm − wm‖L2 ≤ ε; (8.3.7)
∣∣∣∣∫
Rn
|vm|2 − γ
∣∣∣∣ ≤ ε; (8.3.8)
∣∣∣∣∫
Rn
|wm|2 + γ − µ
∣∣∣∣ ≤ ε; (8.3.9)
∫
Rn
|∇um|2 − |∇vm|2 − |∇wm|2 ≥ −ε. (8.3.10)
Proof. We repeat the proof of P.-L. Lions [3]. Consider the functions
Qm(t) = supy∈Rn
∫
|x−y|≤t|um(x)|2 dx.
Qm is a sequence of nondecreasing functions such that 0 ≤ Qm(t) ≤ µ, for all t ≥ 0,m ∈ N.
Define
Fm(t) =
∫ t
0
Qm(s) ds.
186
Now, Fm is a sequence of C1, convex, nondecreasing, nonnegative, uniformly Lipschitz
continuous functions. It follows easily from Ascoli’s theorem that there exists a subse-
quence, which we still denote by Fm, and a function F such that Fm → F as m → ∞,
uniformly on bounded intervals of R+. It follows that F is convex, nondecreasing, non-
negative, and Lipschitz continuous. In particular, there exists a countable set E ⊂ R+
such that F (t) is differentiable for every t 6∈ E. It follows easily from the convexity that
F ′m(t) → F ′(t) as m → ∞ for all t 6∈ E. Let Q(t) = F ′(t). Q is nonnegative, nondecreas-
ing, and 0 ≤ Q(t) ≤ µ for all t ∈ R + \E. Let γ = limt→∞Q(t). We consider separately
three cases.
The case γ = µ. We claim that in this case, (i) occurs. To see this, consider first
0 < λ < µ. There exists R′ > 0 such that Q(R′) > λ. It follows that Qm(R′) > λ
for m ≥ m0 ≥ 0. For every m ≤ m0, there exists Rm such that Qm(Rm) > λ. Taking
R(λ) = maxR′, R0, · · · , Rm0, it follows that Qm(R(λ)) > λ, for all m ∈ N. Therefore,
there exists xm(λ) such that
∫
|x−xm(λ)|≤R(λ)|um(x)|2 dx > λ.
Let ym = xm(µ/2). Given λ > µ/2, let R = R(µ/2) + 2R(λ). We claim that
∫
|x−ym|≤R|um(x)|2 dx > λ,
for all m ∈ N. Indeed, |xm(λ) − ym| ≤ R(µ/2) + R(λ), for otherwise the sets x; |x −xm(λ)| ≤ R(λ) and x; |x− ym| ≤ R(µ/2) would be disjoint, which would imply that
∫
Rn
|um|2 ≥∫
|x−ym|≤R(µ/2)|um(x)|2 dx+
∫
|x−xm(λ)|≤R(λ)|um(x)|2 dx ≥ µ
2+ λ > µ,
which is absurd. It follows that the set x; |x − xm(λ)| ≤ R(λ) is contained in the set
x; |x− ym| ≤ R; and so,
∫
|x−ym|≤R|um(x)|2 dx ≥
∫
|x−xm(λ)|≤R(λ)|um(x)|2 dx ≥ λ,
which proves (i).
The case γ = 0. γ = 0 means exactly that (ii) occurs.
187
The case γ ∈ (0, µ). We claim that in this case, (iii) occurs. To see this, consider ε > 0.
There exists R0 such that γ− ε < Q(R) < γ+ ε for all R ≥ R0. In particular, there exists
R > 0, m0 ≥ 0 and R ≥ min2R, 1/ε such that
γ − ε < Qm(R) ≤ Qm(R) < γ + ε, for all m ≥ m0;
and so, there exists ym such that
γ − ε <
∫
|x−ym|≤R|um(x)|2 dx < γ + ε.
There exists ρ ∈ D(Rn) be such that ρ ≡ 1 on |x| ≤ R, ρ ≡ 0 on |x| ≥ R/2 and
|∇ρ| ≤ 2ε. As well, there exists θ ∈ D(Rn) be such that θ ≡ 0 on |x| ≤ R/2, θ ≡ 1 on
|x| ≥ R and |∇θ| ≤ 2ε. Let ρm(x) = ρ(x − ym) and θm(x) = θ(x − ym). Finally, let
vm = ρmum and wm = θmum. It follows immediately that (8.3.6) and (8.3.8) hold, and
that the supports or vm and wm are disjoint. Furthermore, it follows from Cauchy-Schwarz
inequality that
∫
Rn
|u− vm − wm|2 ≤∫
R≤|x−ym|≤R|um|2
=
∫
|x−ym|≤R|um|2 −
∫
|x−ym|≤R|um|2
≤ Qm(R)−∫
|x−ym|≤R|um|2
≤ (γ + ε)− (γ − ε) = 2ε.
This proves (8.3.7), and (8.3.9) follows easily. Finally, observe that ∇vm = ρm∇um +
∇ρmum; and so, |∇vm|2 ≤ ρ2m|∇um|2 + 2ε|um| |∇um|+ 4ε2|um|2. It follows that
∫
Rn
|∇vm|2 ≤∫
Rn
ρ2m|∇um|2 + Cε.
as well, ∫
Rn
|∇wm|2 ≤∫
Rn
θ2m|∇um|2 + Cε.
It follows that
∫
Rn
|∇um|2 − |∇vm|2 − |∇wm|2 ≥∫
Rn
(1− ρ2m − θ2
m)|∇um|2 − Cε ≥ −Cε.
This proves (8.3.10) and the proof is now complete.
188
Proof of Proposition 8.3.6. We proceed in four steps.
Step 1. 0 < ν < ∞. It is clear that Γ 6= ∅. Let u ∈ Γ and for λ > 0, let uλ(x) =
λn/2u(λx). It follows easily that uλ ∈ Γ and that
E(uλ) =λ2
2
∫
Rn
|∇u|2 − λnα/2
α+ 2
∫
Rn
|u|α+2.
Since nα < 2, it follows that for λ small we have E(uλ) < 0; and so,
ν > 0. (8.3.11)
Next, we claim that there exists δ > 0 and K <∞ such that
E(u) ≥ δ‖u‖2H1 −K, for all u ∈ Γ. (8.3.12)
This follows immediately from Gagliardo-Nirenberg’s inequality
∫
Rn
|u|α+2 ≤ C
(∫
Rn
|∇u|2)nα
4(∫
Rn
|u|2) 4−(n−2)α
4
,
and property nα < 4. Therefore, ν ≤ K > −∞.
Step 2. Every minimizing sequence of (8.3.4) is bounded in H1(Rn) and bounded from
below in Lα+2(Rn). Let (um)m∈N be a minimizing sequence. Since um ∈ Γ, (um)m∈N
is bounded in L2(Rn), then by (8.3.12) (um)m∈N is bounded in H1(Rn). This proves the
first part of the statement. Furthermore since ν > 0, we have E(um) ≤ −ν/2 for m large
enough. It follows that ∫
Rn
|um|α+2 ≥ α+ 2
2ν.
Hence the result.
Step 3. Let (um)m∈N be a minimizing sequence of (8.3.4). Then, there exists a subse-
quence (umk)k∈N and a family (yk)k∈N ⊂ Rn such that (umk
(· − yk))k∈N has a strong
limit u in H1(Rn).
To see this, let us apply Lemma 8.3.8. By Step 2, um is bounded in H1(Rn) and
bounded from below in Lα+2(Rn). Applying Lemma 8.3.7, it follows that
supy∈Rn
∫
|x−y|≤1|u(x)|2 dx
189
is bounded from below; and so, (ii) cannot occur. Let us prove that (iii) cannot occur
either. To see this, we argue by contradiction, and we assume that (iii) holds. Since vm
and wm have disjoint supports, it follows that E(vm + wm) = E(vm) + E(wm).It follows
easily from (8.3.10), (8.3.6), (8.3.7) and Holder’s inequality that
E(um)−E(vm)− E(wm) ≥ −δ(ε), (8.3.13)
where δ(ε) → 0, as ε ↓ 0. Let u ∈ H1(Rn) and a > 0. We have
E(u) =1
a2E(au) +
aα − 1
α+ 2
∫
Rn
|u|α+2.
Applying that inequality with and am =√µ/‖vm‖L2 , and since amvm ∈ Γ, it follows that
E(vm) ≥ −νa2
m
+aα
m − 1
α+ 2
∫
Rn
|vm|α+2.
It follows from (8.3.8) that there exists c > 0 independent of ε and m such that
aαm − 1
α+ 2≥ c.
Therefore,
E(vm) ≥ −νa2
m
+ c
∫
Rn
|vm|α+2.
As well, after possibly modifying c,
E(wm) ≥ −νb2m
+ c
∫
Rn
|wm|α+2,
with bm =√µ/‖wm‖L2 . Applying (8.3.13), we obtain
E(um) ≥ −νµ
∫
Rn
|vm + wm|2 + c
∫
Rn
|vm + wm|α+2 − δ(ε).
Applying (8.3.7), (8.3.6) and Holder’s inequality, it follows that
E(um) ≥ −ν + δ(ε)
µ+ c(1− δ(ε))
∫
Rn
|um|α+2 − δ(ε),
after possibly modifying δ(ε). Letting ε ↓ 0, it follows that
E(um) ≥ −ν + c
∫
Rn
|um|α+2.
190
Since um is bounded from below in Lα+2(Rn), we obtain a contradiction with the definition
of ν. Therefore, (i) occurs. Let wm = um(· − ym). (wm)m∈N being bounded in H1(Rn),
there exists a subsequence, which we still denote by (wm)m∈N and u ∈ H1(Rn) such that
wm u in H1(Rn), as m→∞. (8.3.14)
Given R > 0, the embedding H1(Rn) → L2(|x| ≤ R) is compact; and so,
∫
|x|≤R|u|2 = lim
m→∞
∫
|x−ym|≤R|um|2.
Together with (i), this implies that
∫
Rn
|u|2 ≥ µ− ε,
for every ε > 0; and so ∫
Rn
|u|2 = µ.
It follows that wm → u in L2(Rn), and in particular u ∈ Γ. Applying Holder’s inequality,
it follows that wm → u in Lα+2(Rn). Together with the weak lower semicontinuity of the
H1 norm, this implies
E(u) ≤ limm→∞
E(wm) = −ν.
By definition of ν, it follows that in fact E(u) = −ν. In particular, E(wm) → E(u), and it
follows that ‖∇wm‖L2 → ‖∇u‖L2 , which implies that wm → u strongly in H1(Rn). Hence
the result.
Step 4. Conclusion. Let (um)m∈N be a minimizing sequence of (8.3.4). By Step 3, there
exists a subsequence (umk)k∈N and a family (yk)k∈N ⊂ Rn such that (umk
(· − yk))k∈N
has a strong limit u in H1(Rn). It follows that u ∈ Γ and that E(u) = −ν; and so, u
satisfies (8.3.4). Finally, let (um)m∈N verify ‖um‖L2 →√µ and E(um) → −ν. Setting
vm =õ
(∫
Rn
|um|2)−1/2
um,
it follows that (vm)m∈N is a minimizing sequence of (8.3.4). By Step 3, there exists a
subsequence (vmk)k∈N and a family (yk)k∈N ⊂ Rn such that (vmk
(· − yk))k∈N has a
strong limit u in H1(Rn). On the other hand, it is immediate that ‖um − vm‖H1 → 0, as
m→∞; and so, umk(· − yk) → u in H1(Rn), as m→∞. This proves (ii).
191
Lemma 8.3.9. Let 0 < α < 4/n and ω > 0. There exists µ > 0 such that
∫
Rn
|u|2 = µ, (8.3.15)
for every ground state u of (8.1.2).
Proof. The result follows from uniqueness of the ground state up to translations and
rotations (cf. Theorems 8.1.5, 8.1.6 and 8.1.7). Alternatively, when n ≥ 2 the result
follows from (8.1.18), (8.1.21) and property (ii) of Theorems 8.1.5 and 8.1.6.
Corollary 8.3.10. Let µ be defined by (8.3.15). If u ∈ H1(Rn), then u is a ground state
of (8.1.2) if and only if u solves the following minimization problem.
u ∈ Γ;
S(u) = minS(v); v ∈ Γ;, (8.3.16)
where Γ is defined by (8.3.3). In addition, problems (8.3.16) and (8.3.4) are equivalent.
Proof. We proceed in four steps.
Step 1. Problem (8.3.16) is equivalent to problem (8.3.4), which has a solution by Propo-
sition 8.3.6. Indeed, if u ∈ Γ, then S(u) = E(u) + ωµ/2; and so, problem (8.3.16) is
equivalent to problem (8.3.4).
Step 2. We have k ≤ `, where ` is defined by (8.2.9) and k is defined by
k = infS(v); v ∈ Γ. (8.3.17)
Indeed, consider u ∈ G. We have S(u) = `, and by Lemma 8.3.9, u ∈ Γ. By definition of
k, this implies k ≤ `.
Step 3. Every solution of (8.3.16) belongs to A. Consider a solution u of (8.3.16), and
let
uλ(x) = λn/2u(λx),
for λ > 0. We have uλ ∈ Γ and u1 = u. It follows from (8.3.16) that f ′(1) = 0, where
f(λ) = S(uλ). This means that
T (u) =nα
2(α+ 2)
∫
Rn
|u|α+2. (8.3.18)
192
Now, since u satisfies (8.3.16), there exists a Lagrange multiplier λ such that S ′(u) = λu;
and so, there exists δ such that
−4u+ δωu = |u|αu. (8.3.19)
On taking the L2 scalar product of (8.3.19) with u and applying (8.3.18), we obtain
δωµ =4− (n− 2)α
nαT (u),
from which it follows that δ > 0. Define now v by
u(x) = δ1/αv(δ1/2x).
It follows from (8.3.19) that v ∈ A, which implies
S(v) ≥ `. (8.3.20)
One computes easily that
S(u) = δ4−(n−2)α
2α S(v) +ωµ
2(1− δ).
Applying (8.3.20) and Step 2, it follows that
` ≥ δ4−(n−2)α
2α `+ωµ
2(1− δ).
On the other hand, it follows from Corollary 8.1.4 that ` > 0, and by (8.3.15) and Corol-
lary 8.1.4,ωµ
2=
4− (n− 2)α
2α`;
and so,
1 ≥ δ4−(n−2)α
2α +4− (n− 2)α
2α(1− δ).
This means that f(δ) > 0, where
f(s) = s4−(n−2)α
2α − 4− (n− 2)α
2αs+
4− nα
2α.
One checks easily that f(s) > 0, if s 6= 1. Therefore, δ = 1, which implies in view of
(8.3.19) that u ∈ A.
193
Step 4. Conclusion. It follows in particular from Steps 2 and 3 that ` = k. Therefore,
if u ∈ G, we have u ∈ Γ and S(u) = k, which implies that u satisfies (8.3.16). Conversely,
let u be a solution of (8.3.16). It follows from Step 3 that u ∈ A, and since S(u) = k = `,
it follows that u ∈ G.
Proof of Theorem 8.3.1. We argue by contradiction. If the conclusion of the theorem
does not hold, there exists a sequence (ψm)m∈N ⊂ H1(Rn), a sequence (tm)m∈N ⊂ R and
ε > 0 such that
‖ψm − ϕ‖H1 −→m→∞
0, (8.3.21)
and such that the maximal solution um of (8.1) with initial datum ψm (which is global,
cf. Remark 8.3.2) verifies
infθ∈R
infy∈Rn
‖um(tm, ·)− eiθϕ(· − y)‖H1 ≥ ε. (8.3.22)
Let us set
vm = um(tm). (8.3.23)
It follows from Corollary 8.3.10, Theorems 8.1.5, 8.1.6 and 8.1.7, and (8.3.23) that (8.3.22)
is equivalent to
infu∈G
‖vm − u‖H1 ≥ ε. (8.3.24)
Applying Corollary 8.3.10, it follows from (8.3.21) that
∫
Rn
|ψm|2 −→m→∞
µ, and S(ψm) −→m→∞
k,
where k is defined by (8.3.17). It follows from conservation of charge and energy that we
have as well ∫
Rn
|vm|2 −→m→∞
µ, and S(vm) −→m→∞
k.
Therefore, (vm)m∈N is a minimizing sequence for problem (8.3.16), hence of problem (8.3.4)
(see Corollary 8.3.10). It follows from Proposition 8.3.6 (ii) that there exists (ym)m∈N ⊂Rn and a solution u of problem (8.3.4) such that ‖vm − u(· − ym)‖H1 → 0. But it follows
from Corollary 8.3.10 that u ∈ G; and so, u(· − ym) ∈ G, which contradicts (8.3.24).
Remark 8.3.11. Note that the proof of Theorem 8.3.1 makes only use of the following
two properties. The conservation laws of (8.1) (charge and energy), and the compactness
194
of any minimizing sequence. Therefore, the method is quite general and can be applied to
many situations (cf. Cazenave [2], Cazenave and Lions [1], P.-L. Lions [3,4]).
Remark 8.3.12. One does not know in general how are the functions θ(t) and y(t)
of (8.3.2). If both ϕ and ψ are spherically symmetric, one can take y(t) ≡ 0 (see Re-
mark 8.3.4). Remarks 8.3.3 and 8.3.5 display examples for which one can take θ and y
to be linear in t. One does not know whether or not this is true in general. Concerning
this question, see the remarkable papers of Soffer and Weinstein [1,2]. They consider in
particular a one dimensional equation with a potential. In this case, y ≡ 0, but they also
show that one can take θ to be linear in t.
8.4. Comments. There are other methods to study the stability of standing waves,
based on the study of a linearized operator. See Shatah and Strauss [1], Grillakis, Shatah
and Strauss [1], Blanchard, Stubbe and Vazquez [1], Weinstein [2,3], Rose and Wein-
stein [1]. The stability of excited states has also been studied, in particular by Jones [1]
and Grillakis [1].
195
9. Further results. In Sections 9.1 and 9.2, we present some results that follow easily
from the techniques that we developed in the previous chapters. On the other hand, we
describe in Sections 9.3 and 9.4 two results that do not fall into the scope of these methods.
Finally, we briefly describe in Section 9.5 some further developments.
9.1. The nonlinear Schrodinger equation with a magnetic field. Throughout
this section, we assume that Ω = R3. We study the nonlinear Schrodinger equation in
presence of an external, constant magnetic field. Given b ∈ R, b 6= 0, we consider the
(vector valued) potential Φ defined by
Φ(x) =b
2(−x2, x1, 0), for x = (x1, x2, x3) ∈ R3,
which is the vector potential of the (constant) magnetic field ~B =−−→curl(A), that is
~B ≡ (0, 0, b).
We define the operator A on L2(R3) by
D(A) = u ∈ L2(R3); ∇u+ iΦu ∈ L2(R3) and 4u+ 2iΦ · ∇u− |Φ|2u ∈ L2(R3),
Au = 4u+ 2iΦ · ∇u− |Φ|2u, for u ∈ D(A).
We consider the nonlinear Schrodinger equation
iut + Au+ g(u) = 0,
u(0) = ϕ;(9.1.1)
and we refer to Avron, Herbst and Simon [1,2,3], Combes, Schrader and Seiler [1], Eboli
and Marques [1], Kato [2], Reed and Simon [1] and B. Simon [1] for its physical relevance.
We begin with the following observation.
Lemma 9.1.1. A is a self-adjoint, ≤ 0 operator on L2(R3).
Proof. Note that D(R3) ⊂ D(A); and so D(A) is a dense subset of L2(R3). Furthermore,
given u, v ∈ D(A), one has 〈Au, v〉L2 = −〈∇u + iΦu,∇v + iΦv〉L2 . It follows easily that
A is ≤ 0 and symmetric. Therefore, it remains to solve equation Au − λu = f , for every
f ∈ L2(R3) and λ > 0. This follows easily by applying Lax-Milgram’s lemma in the
196
Hilbert space H = u ∈ L2(R3); ∇u + iΦu ∈ L2(R3), equipped with the scalar product
〈u, v〉H = 〈∇u+ iΦu,∇v + iΦv〉L2 + λ〈u, v〉L2.
Therefore, we can apply the results of Section 2.5. In particular, D(A) is a Hilbert
space when equipped with the norm
‖u‖2D(A) = ‖Au‖2
L2 + ‖u‖L2 ,
and iA generates a group if isometries (T (t))t∈R on the Hilbert space (D(A))′. (T (t))t∈R
restricted to either of the spaces D(A), XA, L2(R3), X ′A is a group of isometries, where
XA is defined by
XA = u ∈ L2(R3); ∇u+ iΦu ∈ L2(R3),
and
‖u‖2XA
= ‖∇u+ iΦu‖2L2 + ‖u‖2
L2 .
In addition, A can be extended to a self-adjoint, ≤ 0 operator on (D(A))′ (which we still
denote by A), and A is bounded XA → X ′A and L2(R3) → (D(A))′. Furthermore, we have
the following result.
Lemma 9.1.2. The following properties hold.
(i) XA → Lp(R3), for every 2 ≤ p ≤ 6;
(ii) Lq(R3) → X ′A, for every
6
5≤ q ≤ 2;
(iii) D(A) → Lp(R3), for every 2 ≤ p ≤ ∞.
Proof. Let u ∈ XA. We have
|∇(|u|)| =∣∣∣∣Re
(u
|u|(∇u+ iΦu)
)∣∣∣∣ ,
almost everywhere on the set x ∈ R3; u(x) 6= 0. It follows that
|∇(|u|)| ≤ |∇u+ iΦu|, almost everywhere. (9.1.2)
Therefore, ‖ |u| ‖H1 ≤ ‖u‖XA. Hence (i). Note also that D(Rn) ⊂ XA, from which it
follows that the embedding XA → Lp(R3) is dense; and so, (ii) follows from (i) by duality.
197
Finally, let u ∈ D(A) and set f = Au ∈ L2(R3). For every j ∈ 1, 2, 3, let vj = ∂ju+iΦju.
We have
Avj − vj = −(∂j − iΦj)f − 2i(∇u+ iΦu) · (∂jΦ−∇Φj)− vj . (9.1.3)
Next, observe that |∂jΦ − ∇Φj | ≤ b. Furthermore, ∇ + iΦ is by definition a bounded
operator XA → L2(R3); and so, by duality, ∇ − iΦ is bounded L2(R3) → X ′A. In
particular, the right hand side of (9.1.3) belongs to X ′A and ‖Avj − vj‖X′
A≤ C‖u‖D(A).
It follows easily that vj ∈ XA and that ‖vj‖XA≤ C‖u‖D(A). Taking successively j =
1, 2, 3 we obtain the inequality ‖∇u + iΦu‖XA≤ C‖u‖D(A). Applying (i), it follows that
‖∇u+ iΦu‖L6 ≤ C‖u‖D(A). Therefore, by (9.1.2), ‖ |∇(|u|)| ‖L6 ≤ C‖u‖D(A). (iii) follows,
by Sobolev’s embedding theorem.
Lemma 9.1.3. If ε > 0 and 1 ≤ p <∞, then (I−εA)−1 is continuous Lp(R3) → Lp(R3)
and ‖(I − εA)−1‖L(Lp,Lp) ≤ 1.
Proof. Let θ ∈ C1(R+,R+) be such that both θ and θ′ are bounded, θ ≥ 0, θ′ ≥ 0, and
θ(0) = 0. By applying the method of proof of Proposition 2.4.1, it is sufficient to show
that
〈Au, θ(|u|2)u〉L2 ≤ 0, for all u ∈ D(A). (9.1.4)
Consider ρ ∈ D(R3) such that 0 ≤ ρ ≤ 1 and ρ(x) = 1 for |x| ≤ 1, and set ρm(x) = ρ(x
m)
for m ≥ 1. Let u ∈ D(A). We have
〈Au, θ(|u|2)u〉L2 = limm→∞
〈Au, ρmθ(|u|2)u〉L2 . (9.1.5)
In addition, since ρmθ(|u|2)u has compact support,
〈Au, ρmθ(|u|2)u〉L2 = −Re
(∫
R3
∇u · ∇(ρmθ(|u|2)u))
−2Im
(∫
R3
ρmθ(|u|2)uΦ · ∇u)−∫
R3
ρm|Φ|2θ(|u|2)|u|2.
It follows from Cauchy-Schwarz inequality that
−2Im
(∫
R3
ρmθ(|u|2)uΦ · ∇u)≤∫
R3
ρm|Φ|2θ(|u|2)|u|2 +
∫
R3
ρmθ(|u|2)|∇u|2;
and so,
〈Au, ρmθ(|u|2)u〉L2 ≤ −Re
(∫
R3
∇u · ∇(ρmθ(|u|2)u))
+
∫
R3
ρmθ(|u|2)|∇u|2.
198
An elementary calculation shows that, almost everywhere,
−Re(∇u · ∇(ρmθ(|u|2)u)) + ρmθ(|u|2)|∇u|2 =
−ρmθ′(|u|2)(|u|2|∇u|2 − Re(u2∇u2))− 1
2∇ρm · ∇Θ(|u|2) ≤ −1
2∇ρm · ∇Θ(|u|2),
where Θ(s) =
∫ s
0
θ(σ) dσ. It follows that
〈Au, ρmθ(|u|2)u〉L2 ≤ 1
2
∫
R3
Θ(|u|2)4ρm −→m→∞
0.
Applying (9.1.5), we obtain (9.1.4).
Finally, we have the following estimate of (T (t))t∈R.
Lemma 9.1.4. There exist δ > 0 and C < ∞ such that T (t) is continuous L1(R3) →L∞(R3) for every t ∈ (−δ, δ) and t 6= 0. Moreover,
‖T (t)u‖L∞ ≤ C
|t|3/2‖u‖L1 ,
for every u ∈ L1(R3) and t ∈ (−δ, δ), t 6= 0.
Proof. For every t such that sin(bt) 6= 0, we have the following formula (see Avron, Herbst
and Simon [1]).
T (t)u(x) =b
4π(4πit)1/2sin(bt)
∫
R3
e−iF (x,y,t)u(y) dy,
where
F (x, y, t) =(x3 − y3)
2
4t+b
4
((x1 − y1)
2 + (x2 − y2)2)cotg(bt)− b
2(x1y2 − x2y1).
It follows that
‖T (t)‖L(L1,L∞) ≤|b|
|t|1/2|sin(bt)| .
The result follows easily.
Consider now a real valued potential V : R3 → R such that V ∈ Lp(R3) + L∞(R3)
for some p > 3/2.
199
Consider a function f : R3× [0,∞) → R. Assume that f(x, u) is measurable in x and
continuous in u, and that f(x, 0) = 0, almost everywhere on R3. Assume that there exist
constants C and α ∈ [0, 4) such that
|f(x, v)− f(x, u)| ≤ C(1 + |u|α + |v|α)|v − u|, for almost all x ∈ R3 and all u, v ∈ R.
Extend f to R3 ×C by setting
f(x, z) =z
|z|f(x, |z|), for all z ∈ C, z 6= 0, and almost all x ∈ R3.
Set
F (x, z) =
∫ |z|
0
f(x, s) ds, for all z ∈ C and almost all x ∈ R3.
Finally, consider an even, real-valued potential W : R3 → R. Assume that W ∈L1(R3) + L∞(R3).
Set
g(u) = V u+ f(·, u(·)) + (W ∗ |u|2)u,
G(u) =
∫
R3
1
2V (x)|u(x)|2 + F (x, u(x)) +
1
4(W ∗ |u|2)(x)|u(x)|2
dx,
and
E(u) =
∫
R3
|∇u+ iΦu|2 dx−G(u).
We have the following result (see Cazenave and Esteban [1]).
Theorem 9.1.5. If g is as above, then the following properties hold.
(i) For every ϕ ∈ XA, there exists T∗(ϕ), T ∗(ϕ) > 0 and there exists a unique, maximal
solution u ∈ C((−T∗(ϕ), T ∗(ϕ)), XA)∩C1((−T∗(ϕ), T ∗(ϕ)), X ′A) of problem (9.1.1). u
is maximal in the sense that if T ∗(ϕ) <∞ (respectively, T∗(ϕ) <∞), then ‖u(t)‖A →∞, as t ↑ T ∗(ϕ) (respectively, as t ↓ −T∗(ϕ));
(ii) we have conservation of charge and energy, that is
‖u(t)‖L2 = ‖ϕ‖L2 , and E(u(t)) = E(ϕ),
for all t ∈ (−T∗(ϕ), T ∗(ϕ));
(iii) we have continuous dependence of the solution on the initial datum in the sense that
both functions T∗(ϕ) and T ∗(ϕ) are lower semicontinuous, and that if ϕm → ϕ in XA
200
and if [−T1, T2] ⊂ (−T∗(ϕ), T ∗(ϕ)), then um → u in C([−T1, T2], XA), where um is
the maximal solution of (9.1.1) with initial datum ϕm;
(iv) if ϕ ∈ D(A), then u ∈ C((−T∗(ϕ), T ∗(ϕ)), D(A)) ∩ C1((−T∗(ϕ), T ∗(ϕ)), L2(R3)).
Proof. It follows from Lemmas 9.1.1 to 9.1.4 that A and g verify the assumptions of
Theorems 4.6.5 and 5.5.1 (cf. Remarks 4.3.2 and 5.2.9, concerning the assumptions of g).
Remark 9.1.6. It follows easily from conservation of energy and Lemma 9.1.2 (i) that
there exists δ > 0 such that if ‖ϕ‖XA≤ δ, then the maximal solution u of (9.1.1) is global
and sup‖u(t)‖XA; t ∈ R <∞ (compare the proof of Theorem 6.2.1 and Example 1, page
99).
Remark 9.1.7. In addition to the assumptions of Theorem 9.1.5, suppose that W+ ∈Lq(R3)+L∞(R3) for some q > 3/2, and that exists 0 ≤ δ < 4/3 such that F (x, u) ≤ C(1+
|u|δ)|u|2 for all u ∈ C. Then, for every ϕ ∈ XA, the maximal solution u of (9.1.1) is global
and sup‖u(t)‖XA; t ∈ R <∞ (compare the proof of Theorem 6.1.1 and Example 1, page
94).
Concerning the existence of solutions of (9.1.1) for initial data in L2(R3), we have the
following result (see Cazenave and Esteban [1]).
Theorem 9.1.8. Let g be as in Theorem 9.1.5, and assume further that α < 4/3 and
that W ∈ Lq(R3) + L∞(R3) for some q > 3/2. Let r = maxα+ 2,2p
p− 1,
2q
q − 1 and let
(q, r) be the corresponding admissible pair. Then, for every ϕ ∈ L2(R3), there exists a
unique function u ∈ C(R, L2(R3))∩Lqloc(R, L
r(R3)) with ut ∈ Lqloc(R, (D(A))′), solution
of (9.1.1). In addition, we have ‖u(t)‖L2 = ‖ϕ‖L2 , for all t ∈ R, and u ∈ Laloc(R, L
b(R3)),
for every admissible pair (a, b). Furthermore, if ϕm → ϕ in L2(R3) and if um denotes the
solution of (6.3.4) with initial datum ϕ, then um → u in u ∈ Laloc(R, L
b(R3)), for every
admissible pair (a, b).
Proof. One adapts easily the proofs of Theorems 6.3.1 and 6.3.2 (cf. Remark 6.2.4,
concerning the assumptions of g).
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Remark 9.1.9. Under certain assumptions on g, one can adapt the methods of Sec-
tion 6.4 and show that some solutions of (9.1.1) blow up in finite time (cf. Goncalves
Ribeiro [1]).
Remark 9.1.10. For a certain class of nonlinearities, equation (9.1.1) has stationary
states of the form u(t, x) = eiωtϕ(x) (cf. Esteban and Lions [1]). One obtains stability
results that are similar to those of Sections 8.2 and 8.3. For some nonlinearities, the ground
states are stable (cf. Cazenave and Esteban [1]), and for other nonlinearities, the ground
states are instable (cf. Goncalves Ribeiro [1]).
9.2. The nonlinear Schrodinger equation with a quadratic potential. Through-
out this section, we assume that Ω = Rn. We already studied the nonlinear Schrodinger
equation with an external potential V , with V ∈ Lp(Rn) + L∞(Rn) for some p ≥ 1,
p > n/2. Here, we extend these results to the case of potentials U that are not localized,
but have at most a quadratic growth at infinity, the model case being U(x) = |x|2. More
precisely, consider a real valued potential U ∈ C∞(Rn) such that
U ≥ 0,
and
DαU ∈ L∞(Rn), for all a ∈ Nn such that |α| ≥ 2.
We define the operator A on L2(Rn) byD(A) = u ∈ H1(Rn); U |u|2 ∈ L1(Rn) and 4u− Uu ∈ L2(Rn),
Au = 4u− Uu, for u ∈ D(A).
We consider the nonlinear Schrodinger equationiut + Au+ g(u) = 0,
u(0) = ϕ;(9.2.1)
We begin with the following observation.
Lemma 9.2.1. A is a self-adjoint, ≤ 0 operator on L2(Rn).
Proof. Note that D(Rn) ⊂ D(A); and so D(A) is a dense subset of L2(Rn). Furthermore,
given u, v ∈ D(A), one has
〈Au, v〉L2 = −Re
(∫
Rn
∇u · ∇v + Uuv
).
202
It follows easily that A is ≤ 0 and symmetric. Therefore, it remains to solve equation
Au − λu = f , for every f ∈ L2(Rn) and λ > 0. This follows easily by applying Lax-
Milgram’s lemma in the Hilbert space H = u ∈ H1(Rn); U |u|2 ∈ L1(Rn), equipped
with the norm defined by
‖u‖2H = ‖∇u‖2
L2 +
∫
Rn
U |u|2 + λ‖u‖2L2 ,
for all u ∈ H.
Therefore, we can apply the results of Section 2.5. In particular, D(A) is a Hilbert
space when equipped with the norm
‖u‖2D(A) = ‖Au‖2
L2 + ‖u‖L2 ,
and iA generates a group if isometries (T (t))t∈R on the Hilbert space (D(A))′. (T (t))t∈R
restricted to either of the spaces D(A), XA, L2(Rn), X ′A is a group of isometries, where
XA is defined by
XA = u ∈ H1(Rn); U |u|2 ∈ L1(Rn),
and
‖u‖2XA
= ‖∇u‖2L2 + ‖u‖2
L2 +
∫
Rn
U |u|2.
In addition, A can be extended to a self-adjoint, ≤ 0 operator on (D(A))′ (which we still
denote by A), and A is bounded XA → X ′A and L2(Rn) → (D(A))′. Furthermore, we
have the following result.
Lemma 9.2.2. The following properties hold.
(i) XA → H1(Rn);
(ii) H−1(Rn) → X ′A;
(iii) D(A) → Lp(Rn), for every 2 ≤ p <∞ such that1
p>
1
2− 2
n.
Proof. (i) follows from the definition of XA, then (ii) follows by duality. We prove (iii)
for n ≥ 3, the proof for n = 1, 2 being easily adapted. Let u ∈ D(A) and let f = Au.
Consider p > 2 and take the L2 scalar product of equation 4u− Uu = f with |u|p−2u (in
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fact, a rigorous proof would require a regularization, see the proof of Lemma 9.2.3 below).
One obtains easily ∫
Rn
|u|p−2|∇u|2 ≤ ‖f‖L2‖u‖p−1L2(p−1) .
Since p2|u|p−2|∇u|2 = 4|∇(|u|p/2)|2, it follows from Sobolev’s inequality that
‖u‖p
Lnp
n−2≤ C‖f‖L2‖u‖p−1
L2(p−1) .
(iii) follows by taking p =2n− 4
n− 4if n ≥ 5 and any p <∞ if n ≤ 4 then applying Holder’s
inequality.
Lemma 9.2.3. If ε > 0 and 1 ≤ p <∞, then (I−εA)−1 is continuous Lp(Rn) → Lp(Rn),
and ‖(I − εA)−1‖L(Lp,Lp) ≤ 1.
Proof. Let θ ∈ C1(R+,R+) be such that both θ and θ′ are bounded, θ ≥ 0, θ′ ≥ 0, and
θ(0) = 0. By applying the method of proof of Proposition 2.4.1, it is sufficient to show
that
〈Au, θ(|u|2)u〉L2 ≤ 0, for all u ∈ D(A).
We have
〈Au, θ(|u|2)u〉L2 = 〈4u, θ(|u|2)u〉L2 −∫
Rn
Uθ(|u|2)|u|2 ≤ 〈4u, θ(|u|2)u〉L2 ,
and we already know that (see the proof of Proposition 2.4.1) 〈4u, θ(|u|2)u〉L2 ≤ 0. The
result follows.
Finally, we have the following estimate of (T (t))t∈R.
Lemma 9.2.4. There exist δ > 0 and C < ∞ such that T (t) is continuous L1(Rn) →L∞(Rn), for every t ∈ (−δ, δ), t 6= 0, and
‖T (t)u‖L∞ ≤ C
|t|n/2‖u‖L1 ,
for every u ∈ L1(Rn) and t ∈ (−δ, δ), t 6= 0.
Proof. See Oh [1], Proposition 2.2.
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Consider now a real valued potential V : Rn → R such that V ∈ Lp(Rn) + L∞(Rn)
for some p ≥ 1, p > n/2.
Consider a function f : Rn × [0,∞) → R. Assume that f(x, u) is measurable in x
and continuous in u, and that f(x, 0) = 0, almost everywhere on Rn. Assume that there
exist constants C and α ∈ [0,4
n− 2) (α ∈ [0,∞) if n = 2) such that
|f(x, v)− f(x, u)| ≤ C(1 + |u|α + |v|α)|v − u|, for almost all x ∈ Rn and all u, v ∈ R.
If n = 1, assume instead that for every M , there exists L(M) such that
|f(x, v)− f(x, u)| ≤ L(M)|v − u|, for almost all x ∈ Rn
and all u, v ∈ R such that |u|+ |v| ≤M.
Extend f to Rn ×C by setting
f(x, z) =z
|z|f(x, |z|), for all z ∈ C, z 6= 0, and almost all x ∈ Rn.
Set
F (x, z) =
∫ |z|
0
f(x, s) ds, for all z ∈ C and almost all x ∈ Rn.
Finally, consider an even, real-valued potential W : Rn → R. Assume that W ∈Lp(Rn) + L∞(Rn), for some p ≥ 1, p > n/4.
Set
g(u) = V u+ f(·, u(·)) + (W ∗ |u|2)u,
G(u) =
∫
Rn
1
2V (x)|u(x)|2 + F (x, u(x)) +
1
4(W ∗ |u|2)(x)|u(x)|2
dx,
and
E(u) =
∫
Rn
|∇u+ iΦu|2 dx−G(u).
We have the following result (see Oh [1,2] for a similar result in the cas where g(u) =
−λ|u|αu).
Theorem 9.2.5. If g is as above, then the following properties hold.
(i) For every ϕ ∈ XA, there exists T∗(ϕ), T ∗(ϕ) > 0 and there exists a unique, maximal
solution u ∈ C((−T∗(ϕ), T ∗(ϕ)), XA)∩C1((−T∗(ϕ), T ∗(ϕ)), X ′A) of problem (9.2.1). u
is maximal in the sense that if T ∗(ϕ) <∞ (respectively, T∗(ϕ) <∞), then ‖u(t)‖A →∞, as t ↑ T ∗(ϕ) (respectively, as t ↓ −T∗(ϕ));
205
(ii) we have conservation of charge and energy, that is
‖u(t)‖L2 = ‖ϕ‖L2 , and E(u(t)) = E(ϕ),
for all t ∈ (−T∗(ϕ), T ∗(ϕ));
(iii) we have continuous dependence of the solution on the initial datum in the sense that
both functions T∗(ϕ) and T ∗(ϕ) are lower semicontinuous, and that if ϕm → ϕ in XA
and if [−T1, T2] ⊂ (−T∗(ϕ), T ∗(ϕ)), then um → u in C([−T1, T2], XA), where um is
the maximal solution of (9.2.1) with initial datum ϕm;
(iv) if ϕ ∈ D(A), then u ∈ C((−T∗(ϕ), T ∗(ϕ)), D(A)) ∩ C1((−T∗(ϕ), T ∗(ϕ)), L2(Rn))
Proof. It follows from Lemmas 9.2.1 to 9.2.4 that A and g verify the assumptions of
Theorems 4.6.5 and 5.5.1 (cf. Remarks 4.3.2 and 5.2.9, concerning the assumptions of g).
Remark 9.2.6. It follows easily from conservation of energy and Lemma 9.2.2 (i) that
there exists δ > 0 such that if ‖ϕ‖XA≤ δ, then the maximal solution u of (9.2.1) is global
and sup‖u(t)‖XA; t ∈ R <∞ (compare the proof of Theorem 6.2.1 and Example 1, page
99).
Remark 9.2.7. In addition to the assumptions of Theorem 9.2.5, suppose that W+ ∈Lq(Rn) + L∞(Rn) for some q ≥ 1, q > n/2, and that exists 0 ≤ δ < 4/n such that
F (x, u) ≤ C(1 + |u|δ)|u|2 for all u ∈ C. Then, for every ϕ ∈ XA, the maximal solution u
of (9.2.1) is global and sup‖u(t)‖XA; t ∈ R < ∞ (compare the proof of Theorem 6.1.1
and Example 1, page 94).
Concerning the existence of solutions of (9.2.1) for initial data in L2(Rn), we have the
following result.
Theorem 9.2.8. Let g be as in Theorem 9.2.5, and assume further that α < 4/n and that
W ∈ Lq(Rn) + L∞(Rn) for some q ≥ 1, q > n/2. Let r = maxα+ 2,2p
p− 1,
2q
q − 1 and
let (q, r) be the corresponding admissible pair. Then, for every ϕ ∈ L2(Rn), there exists a
unique function u ∈ C(R, L2(Rn))∩Lqloc(R, L
r(Rn)) with ut ∈ Lqloc(R, (D(A))′), solution
of (9.2.1). In addition, we have ‖u(t)‖L2 = ‖ϕ‖L2 , for all t ∈ R, and u ∈ Laloc(R, L
b(Rn)),
206
for every admissible pair (a, b). Furthermore, if ϕm → ϕ in L2(Rn) and if um denotes the
solution of (6.3.4) with initial datum ϕ, then um → u in u ∈ Laloc(R, L
b(Rn)), for every
admissible pair (a, b).
Proof. One adapts easily the proofs of Theorems 6.3.1 and 6.3.2 (cf. Remark 6.2.4,
concerning the assumptions of g).
Remark 9.2.9. Under certain assumptions on g, one can adapt the methods of Sec-
tion 6.4 and show that some solutions of (9.2.1) blow up in finite time. More precisely,
if we assume that |x · ∇U | ≤ C(|x|2 + U) and that g verifies the assumptions of Proposi-
tion 6.4.2, one can show that (with the notation of Proposition 6.4.2)
f ′′(t) = 16E(ϕ) +
∫
Rn
(8(n+ 2)F (u)− 4nRe(f(u)u)) dx
+ 8
∫
Rn
(V +1
2x · ∇V )|u|2 dx+ 4
∫
Rn
((W +1
2x · ∇W ) ∗ |u|2)|u|2 dx
− 8
∫
Rn
(U +1
2x · ∇U)|u|2 dx.
The proof of the above inequality is similar to that of Proposition 6.4.2. Assume further
g verifies (6.4.17), (6.4.18) and (6.4.19), and that
U +1
2x · ∇U ≥ 0.
Then, if ϕ ∈ XA is such that | · |ϕ(·) ∈ L2(Rn) and E(ϕ) < 0, we have T ∗(ϕ) < ∞ and
T∗(ϕ) <∞ (compare the proof of Theorem 6.4.7).
Remark 9.2.10. Take U(x) ≡ |x|2 and g(u) = λ|u|αu, where λ > 0 and4
n≤ α <
4
n− 2
(4
n≤ α < ∞, if n = 1, 2). It follows from Remark 9.2.9 that if ϕ ∈ XA is such that
E(ϕ) < 0, we have T ∗(ϕ) <∞ and T∗(ϕ) <∞.
9.3. The logarithmic Schrodinger equation. Let Ω ⊂ Rn be an open domain. We
consider the following nonlinear Schrodinger equation.
iut +4u+ V u+ uLog(|u|2) = 0,
u(0) = ϕ,(9.3.1)
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where V is some real valued potential. Equation (9.3.1) arises in a model of nonlinear
wave mechanics (see Bialinycki-Birula and Mycielski [1]). We cannot apply the results of
Section 4.2 for solving problem (9.3.1) because the function z 7→ zLog(|z|2) is not Lipschitz
continuous at z = 0, due to the singularity of the Log at the origin. Furthermore, it is
not always clear in what space the nonlinearity makes sense. For example, if Ω = Rn
and u ∈ H1(Rn), then uLog(|u|2) does not in general belong to any Lp for p ≤ 2, nor to
H−1(Rn) (this is again due to the singularity of the Log at the origin). However, we will
solve problem (9.3.1) by a compactness method, but before stating the precise existence
result, we need to introduce some notation. Let us define
F (z) = |z|2Log(|z|2), for every z ∈ C.
Furthermore, let the functions A,B, a, b be defined by
A(s) =
− s2Log(s2), if 0 ≤ s ≤ e−3,
3s2 + 4e−3s− e−3, if s ≥ e−3;B(s) = F (s) +A(s);
and
a(s) =A(s)
s, b(s) =
B(s)
s.
Extend the functions a and b to the complex plane by setting
a(z) =z
|z|a(|z|), b(z) =z
|z|b(|z|), for z ∈ C, z 6= 0.
It follows in particular that A is a convex C1 function, which is C2 and positive except
at the origin. Let A∗ be the convex conjugate function of A (see Brezis [2]). A∗ is also a
convex C1 function, which is positive except at the origin. Define the sets X and X ′ by
X = u ∈ L1loc(Ω); A(|u|) ∈ L1(Ω), and X ′ = u ∈ L1
loc(Ω); A∗(|u|) ∈ L1(Ω).
Finally, set
‖u‖X = infk > 0;
∫
Ω
A
( |u|k
)≤ 1, for u ∈ X,
and
‖u‖X′ = infk > 0;
∫
Ω
A∗( |u|k
)≤ 1, for u ∈ X ′.
Then, we have the following results (see Cazenave [2], Lemmas 2.1 and 2.5, and Kra-
nosel’skii and Rutickii [1]).
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Lemma 9.3.1. X andX ′ are linear spaces. (X, ‖ ‖X) and (X ′, ‖ ‖X′) are reflexive Banach
spaces and X ′ is the topological dual of X. Furthermore, the following properties hold.
(i) If um −→m→∞
u in X, then A(|um|) −→m→∞
A(|u|) in L1(Ω);
(ii) If um −→m→∞
u almost everywhere and if
∫
Ω
A(|um|) −→m→∞
∫
Ω
A(|u|) <∞,
then um −→m→∞
u in X.
Lemma 9.3.2. The operator u 7→ a(u) maps continuously X → X ′. The image by a of
a bounded subset of X is a bounded subset of X ′.
Finally, consider the Banach space
W = H10 (Ω) ∩X,
equipped with the usual norm. It follows from Proposition 2.1.8 that
W ′ = H−1(Ω) +X ′,
and define
E(u) =1
2
∫
Ω
|∇u|2 − 1
2
∫
Ω
V |u|2 − 1
2
∫
Ω
|u|2Log(|u|2),
for every u ∈W , where the potential V ∈ Lp(Ω) +L∞(Ω) for some p ≥ 1, p > n/2. Then,
we have the following result.
Lemma 9.3.3. The operator L : u 7→ 4u+V u+uLog(|u|2) maps continuously W →W ′.
The image by L of a bounded subset of W is a bounded subset of W ′. E is continuous
W → R.
Proof. One verifies easily that for every ε > 0, there exists Cε such that
|b(v)− b(u)| ≤ Cε(|u|ε + |v|ε)|v − u|, for all u, v ∈ C. (9.3.2)
Integrating the inequality (9.1.3) on Ω, and applying Holder’s and Sobolev’s inequalities,
it follows easily that u 7→ b(u) maps continuously H10 (Ω) → H−1(Ω) and that the image
209
by b of a bounded subset of H10 (Ω) is a bounded subset of H−1(Ω). The same holds for
4, and also for u 7→ V u (by Holder’s inequality); and so, the first part of the statement
follows from Lemma 9.3.2. Finally, we have
E(u) =1
2
∫
Ω
|∇u|2 − 1
2
∫
Ω
V |u|2 +1
2
∫
Ω
A(|u|)− 1
2
∫
Ω
B(|u|). (9.3.3)
The first term in the right hand side of (9.3.3) is continuous H10 (Ω) → R, and it follows
from Lemma 9.3.1 (i) that the third term is continuous X → X ′. Furthermore, it follows
from (9.3.2) that
|B(v)−B(u)| ≤ Cε(|u|1+ε + |v|1+ε)|v − u|.
Integrating the above inequality on Ω, and applying Holder’s and Sobolev’s inequalities,
it follows easily that
∫
Ω
|B(v)− B(u)| ≤ C(1 + ‖u‖2H1 + ‖u‖2
H1)‖v − u‖L2 , for all u, v ∈ H10 (Ω).
Therefore, the fourth term in the right hand side of (9.3.3) is continuous H10 (Ω) → R.
Finally, if V = V1 + V2 with V1 ∈ Lp(Ω) and V2 ∈ L∞(Ω), we have
∫
Ω
|V | |u|2 ≤ ‖V1‖Lp‖u‖2
L2p
p−1+ ‖V2‖L∞‖u‖2
L2 . (9.3.4)
Therefore, the second term in the right hand side of (9.3.3) is continuous H10 (Ω) → R,
which completes the proof.
Our main result of this section is the following (see Cazenave and Haraux [1]).
Theorem 9.3.4. Let V be a real valued potential, such that V ∈ Lp(Ω) + L∞(Ω) for
some p ≥ 1, p > n/2. The following properties hold.
(i) For every ϕ ∈ W , there exists a unique, maximal solution u ∈ C(R,W ) ∩ C1(R,W ′)
of problem (9.3.1). Furthermore, supt∈R
‖u(t)‖W <∞;
(ii) we have conservation of charge and energy, that is
‖u(t)‖L2 = ‖ϕ‖L2 , and E(u(t)) = E(ϕ),
for all t ∈ R;
210
(iii) we have continuous dependence of the solution on the initial datum in the sense that
if ϕm → ϕ in W , then um → u in W , uniformly on bounded intervals, where um is
the maximal solution of (9.3.1) with initial datum ϕm.
Before proceeding to the proof, we need the following two lemmas.
Lemma 9.3.5. We have
∣∣Im((vLog(|v|2)− uLog(|u|2))(v − u)
)∣∣ ≤ 4|v − u|2,
for all u, v ∈ C.
Proof. We have
Im((vLog(|v|2)− uLog(|u|2))(v − u)
)= 2 (Log(|v|)− Log(|u|)) Im(vu− uv).
Without loss of generality, we may assume that 0 < |v| ≤ |u|. In this case,
|Log(|v|)− Log(|u|)| ≤ |v| − |u||v| ≤ |v − u|
|v| ,
and
|Im(vu− uv)| = |v(u− v) + v(v − u)| ≤ 2|v| |v − u|.
Hence the result.
Lemma 9.3.6. Given k ∈ N, let Ωk = Ω∩x ∈ Ω; |x| < k. Let (um)m∈N be bounded in
L∞(R, H10 (Ω)). If for every k ∈ N, um
|Ωkis bounded in W 1,∞(R, H−1(Ωk)), then there ex-
ists a subsequence, which we still denote by (um)m∈N, and there exists u ∈ L∞(R, H10 (Ω))
such that the following properties hold.
(i) u|Ωk∈W 1,∞(R, H−1(Ωk)), for every k ∈ N;
(ii) um(t) u(t) in H10 (Ω) as m→∞, for every t ∈ R;
(iii) for every t ∈ R, there exists a subsequence mj such that umj (t, x) → u(t, x) as k →∞for almost all x ∈ Ω;
(iv) um(t, x) → u(t, x) as m→∞ for almost all (t, x) ∈ R× Ω.
211
Proof. Let k ∈ N. um|Ωk
is bounded in L∞(−k, k;H1(Ωk)) ∩ W 1,∞(−k, k; )H−1(Ωk).
Therefore (by Proposition 2.1.7), there exists a subsequence (which we still denote by
(um)m∈N) and u ∈ L∞(−k, k;H1(Ωk)) such that um(t)|Ωk u(t) in H1(Ω)k. Letting
k → ∞ and considering a diagonal sequence, it follows that there exists a subsequence
(which we still denote by (um)m∈N) and u ∈ L∞(R, H1(Ω)) such that um(t)|Ωk u(t) in
H1(Ωk), for every k ∈ N and every t ∈ R. This implies in particular that um(t) u(t)
in H1(Ω). Therefore, u ∈ L∞(R, H10 (Ω)), and (ii) holds. In addition, since the embedding
H1(Ωk) → L2(Ωk) is compact, we have um(t)|Ωk→ u(t)|Ωk
in L2(Ωk), for every k ∈ N
and every t ∈ R. Applying the dominated convergence theorem, it follows that for every
k ∈ N, ∫ k
−k
∫
Ωk
|um − u|2 −→m→∞
0.
In particular, there exists a subsequence mj for which umj → u almost everywhere on
(−k, k) × Ωk, as j → ∞. Letting k → ∞ and considering a diagonal sequence, it follows
that (iv) holds. Furthermore, given t ∈ R and k ∈ N, there exists a subsequence mj
for which umj (t) → u(t) almost everywhere on Ωk, as j → ∞. Letting k → ∞ and
considering a diagonal sequence, it follows that (iii) holds. Finally, it follows from (i) and
Remark 2.3.12 (i) that u|Ωk∈W 1,∞(R, H−1(Ωk)). Hence (i).
Proof of Theorem 9.3.4. We apply a compactness method, and we proceed in four steps.
Consider ϕ ∈W .
Step 1. Construction of approxiamte solutions. We have V = V1 + V2 with V1 ∈ Lp(Ω)
and V2 ∈ L∞(Ω). Given m ∈ N, define the potentials V m1 and V m
2 by
V mj (x) =
Vj(x), if |Vj(x)| ≤ m,
0, if |Vj(x)| > m,
for j = 1, 2. Define the functions am and bm by
am(z) =
a(z), if |z| ≥ 1
m,
mza(1
m), if |z| ≤ 1
m,
bm(z) =
b(z), if |z| ≤ m,
z
mb(m), if |z| ≥ m,
Finally, set
gm(u) = V m1 u+ V m
2 u− am(u) + bm(u), for u ∈ H10 (Ω).
212
Since V m1 , V m
2 ∈ L∞(Ω) and both am and bm are (globally) Lipschitz continuous C → C,
it follows that gm is Lipschitz continuous L2(Ω) → L2(Ω). It follows from Corollary 4.2.12
that there exists a unique solution um ∈ C(R, H10 (Ω)) ∩ C1(R, H−1(Ω)) of problem
ium
t +4um + gm(um) = 0,
um(0) = ϕ.(9.3.5)
In addition,
‖um(t)‖L2 = ‖ϕ‖L2 , and Em(um(t)) = Em(ϕ), (9.3.6)
and for all t ∈ R, where
Em(u) =1
2
∫
Ω
|∇u|2 − 1
2
∫
Ω
V m1 |u|2 − 1
2
∫
Ω
V m2 |u|2 +
1
2
∫
Ω
Φm(|u|)− 1
2
∫
Ω
Ψm(|u|),
and the functions Φm and Ψm are defined by
Φm(z) =1
2
∫ |z|
0
am(s) ds, and Ψm(z) =1
2
∫ |z|
0
bm(s) ds,
for all z ∈ C.
Step 2. Estimates of the approxiamte solutions. It follows from (9.3.6) that um is bound-
ed in L∞(R, L2(Ω)). Note that, by the dominated convergence theorem,
Em(ϕ) −→m→∞
E(ϕ). (9.3.7)
In particular, it follows also from (9.3.6) and (9.3.7) that ( note that Φm ≥ 0 and compare
(9.3.4))
‖um(t)‖H1 ≤ C + C‖V m1 ‖Lp‖um(t)‖2
H1 + ‖Ψm(um(t))‖L1 .
Note that ‖V m1 ‖Lp ≤ ‖V1‖Lp . Note also that we may assume that ‖V1‖Lp is arbitrarily
small, by modifyng V2. In particular, we may assume that C‖V m1 ‖Lp ≤ 1/4. Finally, one
verifies easily (see the proof of Lemma 9.3.3) that there exists C such that
‖Ψm(um(t))‖L1 ≤ 1
4‖um(t)‖2
H1 + C‖um(t)‖2L2 .
Therefore,
um is bounded in L∞(R, H10 (Ω)) (9.3.8)
213
Finally, it follows from elementary calculations that for every ε > 0, there exists Cε such
that
|gm(u)| ≤ |V m1 | |u|+ |V m
2 | |u|+ Cε(|u|1−ε + |u|1+ε).
It follows easily from Holder’s and Sobolev’s inequalities and (9.3.8) that, given k ∈ N,
gm(um) is bounded in L∞(R, L2p
p−1 (Ωk)), (9.3.9)
where Ωk = Ω ∩ x ∈ Ω; |x| < k. In particular, gm(um) is bounded in L∞(R, H−1(Ωk)),
and it follows from (9.3.5) that um|Ωk
is bounded in W 1,∞(R, H−1(Ωk)).
Step 3. Passage to the limit. It follows from Step 2 that um verifies the assumptions of
Lemma 9.3.6. Let u be its limit. It follows from (9.3.5) that for every ψ ∈ D(Ω) and every
φ ∈ D(R), we have
∫
R
〈iumt +4um + gm(um), ψ〉D′,Dφ(t) dt = 0.
This means that
∫
R
(−〈ium, ψ〉φ′(t) + 〈um,4ψ〉φ(t)) dt+
∫
R
∫
Ω
gm(um)ψφdxdt = 0. (9.3.10)
It follows easily from (9.3.8) and from property (ii) of Lemma 9.4.6 that
∫
R
(−〈ium, ψ〉φ′(t) + 〈um,4ψ〉φ(t)) dt −→m→∞
∫
R
(−〈iu, ψ〉φ′(t) + 〈u,4ψ〉φ(t)) dt.
(9.3.11)
Furthermore, the function hm(t, x) = gm(um)ψ(x)φ(t) has compact support. Therefore, it
follows from (9.3.9) that hm is bounded in L2p
p−1 (R×Ω). By property (iv) of Lemma 9.4.6,
hm → (V u+ uLog(|u|2))ψφ almost everywhere on R×Ω. Since hm has compact support,
it follows from Proposition 2.2.1 that hm → (V u+uLog(|u|2))ψφ in L1(R×Ω). Applying
(9.3.10) and (9.3.11), we thus obtain
∫
R
(−〈iu, ψ〉φ′(t) + 〈u,4ψ〉φ(t)) dt+
∫
R
∫
Ω
(V u+ uLog(|u|2))ψφdxdt = 0,
which implies that
∫
R
〈iut +4u+ V u+ uLog(|u|2), ψ〉D′,Dφ(t) dt = 0.
214
It follows that for all t ∈ R, we have
iut +4u+ V u+ uLog(|u|2) = 0, (9.3.12)
in H−1(Ωk), for every k ∈ N. In addition, we have by property (ii) of Lemma 9.3.6,
u(0) = ϕ. Finally, it follows easily from (9.3.6), (9.3.7) and (9.3.8) that ‖Φm(um(t))‖L1 is
bounded (see Step 2). Applying property (iii) of Lemma 9.4.6 and Fatou’s lemma, it follows
that u(t) ∈ X for all t ∈ R and that supt∈R
‖u(t)‖X ≤ ∞. Since X is reflexive, it follows
from Corollary 2.1.4 that u is weakly continuous R → X. In particular, u ∈ L∞(R, X)
(see Remark 2.2.2 (i)). Therefore, u ∈ L∞(R,W ), and it follows from equation (9.3.12)
and Lemma 9.3.3 that u ∈W 1,∞(R,W ′). Therefore, equation (9.3.12) makes sense in W ′
for all t ∈ R. Therefore, we can take the W −W ′ duality product of it with iu. It follows
easily that
〈ut, u〉W ′,W = 0, for all t ∈ R,
which means that the function t 7→ ‖u(t)‖2L2 is constant; and so, we have conservation
of charge. This implies that for every t ∈ R, we have ‖um(t)‖L2 → ‖u(t)‖L2; and so,
um(t) → u(t) in L2(Ω). Therefore, by boundedness of um in H10 (Ω) and Holder’s and
Sobolev’s inequalities, um(t) → u(t) in Lq(Ω), for every 2 ≤ q < 2nn−2
(2 ≤ q < ∞ if
n = 1, 2). We now can pass to the limit in (9.3.6). We apply the weak lower semicontinuity
of the H1 norm for the gradient term, we apply property (iii) of Lemma 9.3.6 and Fatou’s
lemma to the term Φm, and we apply Holder’s inequality to the other two terms. Taking
(9.3.7) in account, we finally obtain
E(u(t)) ≤ E(ϕ), for all t ∈ R. (9.3.13)
In conclusion, we have obtained the existence of a function u ∈ L∞(R,W )∩W 1,∞(R,W ′)
that solves problem (9.3.1) and for which we have conservation of charge and energy
inequality (9.3.13).
Step 4. Conclusion. Let us first prove uniqueness in the class L∞(R,W )∩W 1,∞(R,W ′).
Let u and v be two solutions of (9.3.1) in that class. On computing the difference of the
two equations and taking the W −W ′ duality product with i(v − u), it follows that
〈vt − ut, v − u〉W ′,W = −Im
(∫
Ω
(vLog(|v|2)− uLog(|u|2))(v − u)
).
215
In view of Lemma 9.3.5, this implies that
‖v(t)− u(t)‖2L2 ≤ 8
∫ t
0
‖v(s)− u(s)‖2L2 ds.
Uniqueness follows by Gronwall’s lemma. Next, let u be a solution of (9.3.1). Considering
the reverse equation and applying uniqueness (see Step 1 of the proof of Theorem 4.2.8),
it follows that u enjoys conservation of energy. Furthermore, by weak L2 continuity and
conservation of charge, u ∈ C(R, L2(Ω)). Since u is bounded in H10 (Ω), it follows easily
that the terms ∫
Ω
V |u|2 and
∫
Ω
B(|u|)
are continuous R → R. It follows from conservation of energy that∫
Ω
|∇u|2 +
∫
Ω
A(|u|), (9.3.14)
is continuous R → R. Since both terms in (9.3.14) are lower semicontinuous R → R (the
second one by Fatou’s lemma), if follows easily (see Cazenave and Haraux [1], Lemma 2.4.4)
that they are in fact continuous R → R. In particular, u ∈ C(R, H10 (Ω)) and u ∈ C(R, X)
(by Lemma 9.3.1 (ii)). Therefore, u ∈ C(R,W ), and by the equation and Lemma 9.3.3,
u ∈ C1(R,W ′). Finally, one proves continuous dependence by a similar argument (compare
Step 3 of the proof of Theorem 4.2.8). This completes the proof.
Remark 9.3.7. Strangely enough, one can apply the theory of maximal monotone op-
erators to equation (9.3.1). In particular, one can obtain stronger regularity if the initial
datum is smoother, and one can construct solutions of (9.3.1) for initial data in L2(Ω) (see
Cazenave and Haraux [1] and Haraux [1]). Note that one does not know whether or not
the L2 solutions are unique.
Remark 9.3.8. At least in the case where Ω = Rn and V ≡ 0, equation (9.3.1) has
standing waves of the form u(t, x) = eiωtϕ(x), for every ω ∈ R. The ground state, which
is unique modulo space translations and rotations (cf. Section 8.1) is explicitely known. It
is given by the formula
ϕ(x) = en+ω
2 e−|x|2
2 ,
and it is stable, in the sense of Section 8.2 (cf. Cazenave [2], Cazenave and Lions [1]).
Equation (9.3.1) has other interesting properties that are unusual concerning Schrodinger
216
equations when Ω = Rn. For example, it follows easily from conservation of energy that
for every solution u of (9.3.1), we have (cf. Cazenave [2], Proposition 4.3)
inft∈R
inf1≤p≤∞
‖u(t)‖Lp > 0.
Another interesting property is that every spherically symmetric (in space) solution of
(9.3.1) has a relatively compact range in L2(Ω) (cf. Cazenave [2], Proposition 4.4).
9.4. Existence of weak solutions for large nonlinearities. Let Ω ⊂ Rn be an open
domain, and let λ > 0 and α > 0. Consider the following problem.
iut +4u− λ|u|αu = 0;
u(0) = ϕ.(9.4.1)
We already know that if α <4
n− 2(α < ∞, if n = 1, 2), problem (9.4.1) has a solution
u ∈ L∞(R, H10 (Ω)) ∩ W 1,∞(R, H−1(Ω)) for every ϕ ∈ H1
0 (Ω) (cf. Remark 4.2.14). In
addition, if Ω = Rn, or if n = 1, or if n = 2 and α ≤ 2 the solution is unique (see
Remarks 4.3.2, 4.4.2 and 4.5.3). However, our results do not apply when α ≥ 4
n− 2. We
present below a result of Strauss [3] (see also Strauss [2]) that applies for arbitrarily large
α’s. Before stating the result, we need some definitions. Let us denote by V the Banach
space
V = H10 (Ω) ∩ Lα+2(Ω),
equipped with the usual norm (see Proposition 2.1.8). Since D(Ω) is dense in both H10 (Ω)
and Lα+2(Ω), we have
V ′ = H−1(Ω) + Lα+2α+1 (Ω),
where the Banach space H−1(Ω) + Lα+2α+1 (Ω) is equipped with its usual norm (see Propo-
sition 2.1.8). Since 4 is continuous H10 (Ω) → H−1(Ω) and u 7→ |u|αu is continuous
Lα+2(Ω) → Lα+2α+1 (Ω), it follows that the operator
V → V ′
u 7→ 4u− λ|u|αu
is continuous. Therefore, if u ∈ L∞(R, V )∩W 1,∞(R, V ′), equation (9.4.1) makes sense in
V ′. Finally, we define
E(u) =1
2
∫
Ω
|∇u|2 − 1
α+ 2
∫
Ω
|u|α+2,
217
for all u ∈ V . We have the following result (see Strauss [3]).
Theorem 9.4.1. Let λ > 0 and α > 0. Then, for every ϕ ∈ V , there exists a solution
u ∈ L∞(R, V ) ∩W 1,∞(R, V ′) of equation (9.4.1) that verifies
‖u(t)‖L2 = ‖ϕ‖L2 , (9.4.2)
and
E(u(t)) ≤ E(ϕ), (9.4.3)
for all t ∈ R.
Remark 9.4.2. Note that in particular, u ∈ C(R, V ′); and so, u is weakly continuous
R → H10 (Ω) and R → Lα+2(Ω). In particular, u(t) ∈ V for all t ∈ R . Therefore, u(0)
makes sense (in V ) and E(u(t)) is well defined for all t ∈ R.
Remark 9.4.3. Note that when α ≤ 4
n− 2(α < ∞, if n = 1, 2), then H1
0 (Ω) →Lα+2(Ω). Therefore, V = H1
0 (Ω).
Remark 9.4.4. Note that when α <4
n− 2(α < ∞, if n = 1, 2), Theorem 9.4.1 follows
from Remark 4.2.14.
Before proceeding to the proof of Theorem 9.4.1, we need to establish the following
two lemmas.
Lemma 9.4.5. V and V ′ are reflexive.
Proof. We only need to show that V is reflexive. By Eberlein-Smulian’s Theorem, we
need to show that, given any bounded sequence (um)m∈N ⊂ V , there exists a subsequence
mk and u ∈ V such that umk u in V , as k →∞. Let ρ = α+2. We recall that if u ∈ V
and ϕ ∈ V ′, we have
〈u, ϕ〉V,V ′ = 〈u, ϕ1〉H10 ,H−1 + 〈u, ϕ2〉Lρ,Lρ′ ,
where ϕ = ϕ1 +ϕ2 with ϕ1 ∈ H−1(Ω) and ϕ2 ∈ Lρ(Ω) (see Bergh and Lofstrom [1], proof
of Theorem 2.7.1). Note that there no ambiguity concerning the possible decompositions
of ϕ, since if ψ ∈ H−1(Ω) ∩ Lρ′(Ω), we have 〈u, ψ〉H10 ,H−1 = 〈u, ψ〉Lρ,Lρ′ .
218
If (um)m∈N is bounded in V , (um)m∈N is bounded in particular in H10 (Ω) and in
Lα+2(Ω). Since both spaces are reflexive, there exists a subsequence mk and there exists
u ∈ H10 (Ω), v ∈ Lα+2(Ω) such that umk
u in H10 (Ω) and umk
v in Lα+2(Ω). In
particular, umk→ u and umk
→ v in D′(Ω). Whence u = v ∈ V . It follows that for every
ϕ1 ∈ H−1(Ω) and ϕ2 ∈ Lρ(Ω), we have
〈umk, ϕ1〉H1
0 ,H−1 + 〈umk, ϕ2〉Lρ,Lρ′ −→
k→∞〈u, ϕ1〉H1
0 ,H−1 + 〈u, ϕ2〉Lρ,Lρ′ .
This implies that umk u in V .
Lemma 9.4.6. Let (um)m∈N be bounded in L∞(R, H10(Ω)) and in W 1,∞(R, V ′). Then,
there exists a subsequence, which we still denote by (um)m∈N, and there exists u ∈L∞(R, H1
0 (Ω)) ∩W 1,∞(R, V ′) such that the following properties hold.
(i) um(t) u(t) in H10 (Ω) as m→∞, for every t ∈ R;
(ii) for every t ∈ R, there exists a subsequence mk such that umk(t, x) → u(t, x) as k →∞for almost all x ∈ Ω;
(iii) um(t, x) → u(t, x) as m→∞ for almost all (t, x) ∈ R× Ω.
Proof. Let k ∈ N and let Ωk = Ω ∩ x ∈ Ω; |x| < k for k ∈ N. Consider an inte-
ger q > n/2. It follows from Sobolev’s embedding theorem that Hq0 (Ωk) → Lα+2(Ωk),
from which we obtain by duality Lα+2α+1 (Ωk) → H−q(Ωk). Therefore, um
|Ωkis bounded
in L∞(−k, k;H1(Ωk)) ∩W 1,∞(−k, k; )H−q(Ωk). Therefore (by Proposition 2.1.7), there
exists a subsequence (which we still denote by (um)m∈N) and u ∈ L∞(−k, k;H1(Ωk))
such that um(t)|Ωk u(t) in H1(Ω)k. Letting k → ∞ and considering a diagonal se-
quence, it follows that there exists a subsequence (which we still denote by (um)m∈N) and
u ∈ L∞(R, H1(Ω)) such that um(t)|Ωk u(t) in H1(Ωk), for every k ∈ N and every t ∈ R.
This implies in particular that um(t) u(t) in H1(Ω). Therefore, u ∈ L∞(R, H10(Ω)),
and (i) holds. In addition, since the embedding H1(Ωk) → L2(Ωk) is compact, we have
um(t)|Ωk→ u(t)|Ωk
in L2(Ωk), for every k ∈ N and every t ∈ R. Applying the dominated
convergence theorem, it follows that for every k ∈ N,
∫ k
−k
∫
Ωk
|um − u|2 −→m→∞
0.
219
In particular, there exists a subsequence mj for which umj → u almost everywhere on
(−k, k) × Ωk, as j → ∞. Letting k → ∞ and considering a diagonal sequence, it follows
that (iii) holds. Furthermore, given t ∈ R and k ∈ N, there exists a subsequence mj
for which umj (t) → u(t) almost everywhere on Ωk, as j → ∞. Letting k → ∞ and
considering a diagonal sequence, it follows that (ii) holds. Finally, it follows from (i) and
Lemma 9.4.5 that um(t) u(t) in V (hence in V ′), for all t ∈ R. Applying Theorem 2.2.4
and Remark 2.3.12 (i), it follows that u ∈ L∞(R, V ) ∩W 1,∞(R, V ′). This completes the
proof.
Proof of Theorem 9.4.2. We construct the solution u by a compactness method, and we
proceed in three steps.
Step 1. Construction of a sequence of approximate solutions. Given an integer m ≥ 1,
let
fm(z) =
− λ|z|αz, if |z| ≤ m;
− λmαz, if |z| ≥ m.
In particular, fm is globally Lipschitz continuous C → C. Let
Gm(z) =
∫ |z|
0
fm(s) ds.
Given u ∈ H10 (Ω), let
gm(u)(x) = fm(u(x)), for almost all x ∈ Ω,
and
Em(u) =1
2
∫
Ω
|∇u|2 +
∫
Ω
Gm(u).
It follows from Corollary 4.2.12 that there exists a unique solution um ∈ C(R, H10 (Ω)) ∩
C1(R, H−1(Ω)) of ium
t +4um + gm(um) = 0,
um(0) = ϕ.(9.4.4)
Furthermore, we have
‖um(t)‖L2 = ‖ϕ‖L2 , (9.4.5)
and
Em(um(t)) = Em(ϕ), (9.4.6)
220
for every t ∈ R.
Step 2. Estimates of um. Note that Gm ≥ 0. Therefore, it follows from (9.4.5) and
(9.4.6) that
um is bounded in L∞(R, H10 (Ω)), (9.4.7)
and
Gm(um) is bounded in L∞(R, L1(Ω)). (9.4.8)
On the other hand, one verifies easily that
|gm(z)|α+2α+1 ≤ (α+ 2)Gm(z), for all z ∈ C and all m ∈ N.
Applying (9.4.8), it follows that
gm(um) is bounded in L∞(R, Lα+2α+1 (Ω)). (9.4.9)
Therefore, it follows from (9.4.5) that
umt is bounded in L∞(R, V ′). (9.4.10)
Step 3. Conclusion. It follows from (9.4.7) and (9.4.10) that we can apply Lemma 9.4.6
to the sequence um. Let u be the limit of um. It follows from properties (i) and (ii)
of Lemma 9.4.6, the weak lower semicontinuity of the H1 norm and Fatou’s lemma that
u(t) ∈ Lα+2(Ω) for every t ∈ R and that (9.4.3) holds. In particular, u ∈ L∞(R, Lα+2(Ω));
and so u ∈ L∞(R, V ). Furthermore, it follows from property (i) that u(0) = ϕ. Finally, it
follows from equation (9.4.4) that for every φ ∈ D(R) and every ψ ∈ D(Ω), we have
∫
R
〈iumt +4um + gm(um), ψ〉D′,Dφ(t) dt = 0.
This means that
∫
R
(−〈ium, ψ〉φ′(t) + 〈um,4ψ〉φ(t)) dt+
∫
R
∫
Ω
gm(um)ψφdxdt = 0. (9.4.11)
It follows easily from (9.4.7) and from property (i) of Lemma 9.4.6 that
∫
R
(−〈ium, ψ〉φ′(t) + 〈um,4ψ〉φ(t)) dt −→m→∞
∫
R
(−〈iu, ψ〉φ′(t) + 〈u,4ψ〉φ(t)) dt.
(9.4.12)
221
Furthermore, the function hm(t, x) = gm(um)ψ(x)φ(t) has compact support. Therefore, it
follows from (9.4.9) that hm is bounded in Lα+2α+1 (R×Ω). By property (iii) of Lemma 9.4.6,
hm → −λ|u|αuψφ almost everywhere on R×Ω. Since hm has compact support, it follows
from Proposition 2.2.1 that hm → −λ|u|αuψφ in L1(R × Ω). Applying (9.4.11) and
(9.4.12), we thus obtain
∫
R
(−〈iu, ψ〉φ′(t) + 〈u,4ψ〉φ(t)) dt− λ
∫
R
∫
Ω
|u|αψφdxdt = 0,
which implies that ∫
R
〈iut +4u− λ|u|αu, ψ〉D′,Dφ(t) dt = 0.
Since u ∈ L∞(R, V ), it follows easily that ut ∈ L∞(R, V ′) and that u satisfies (9.4.1).
It remains to establish conservation of charge. This follows easily by taking the V − V ′
duality product of the equation with iut ∈ V ′. This completes the proof.
Remark 9.4.7. In the case where α >4
n− 2, it is not known whether the solution given
by Theorem 9.4.1 is unique or not, even when Ω = Rn. As well, we do not know whether
or not the energy is conserved.
Remark 9.4.8. Remember that Proposition 4.2.3 applies to the case λ < 0 and α <4
n− 2. On the contrary, in the case α ≥ 4
n− 2, the method of proof of Theorem 9.4.1 does
not apply when λ < 0. We do not know whether or not it is possible to construct (local)
solutions of (9.4.1) in this case.
9.5. Comments. Below are a few references concerning results that we did not present
in these notes.
Consider the nonlinear Schrodinger equation in Rn with the nonlinearity g(u) =
λ|u|αu. It follows from the results of Section 6.3 that for α < 4/n, the (global) initial value
problem is well set in L2(Rn). In the case α = 4/n, the (local) initial value problem is also
well set in L2(Rn) (see Cazenave and Weissler [2], Theorem 1). However, the alternative on
the maximal existence time is such that it is difficult to give sufficient conditions for global
existence. If λ > 0, we know (see Section 6.4) that some solutions blow up in H1(Rn).
One can show that they cannot be continued as L2 solutions either (see Cazenave and
Weissler [2], Remark 1). However, when λ < 0, one would expect that the solution is
222
global for every initial datum. However, this is known only if we assume further that the
initial datum ϕ is small, or if | · |ϕ(·) ∈ L2(Rn) (see Cazenave and Weissler [2], Remarks 4
and 6).
Consider again the nonlinear Schrodinger equation in Rn with the nonlinearity g(u) =
λ|u|αu. It follows from the results of Section 4.3 that for α <4
n− 2, the initial value
problem is well set in H1(Rn). In the case α =4
n− 2(and n ≥ 3), the initial value
problem is also well set in H1(Rn) (see Cazenave and Weissler [2], Theorem 2). However,
it is difficult to give sufficient conditions for global existence. If λ > 0, it follows from the
techniques of Section 6.4 that some solutions blow up in H1(Rn). However, when λ < 0,
one would expect that the solution is global for every initial datum. This is known only if
we assume further that the initial datum ‖∇ϕ‖L2 is small (see Cazenave and Weissler [2],
Remark 7) or if ϕ is sphericaly symmetric (see Soffer [1]).
The nonlinear Schrodinger equation can be considered in other spaces than the energy
space. The difficulty is in general to give sufficient conditions for global existence. Ginibre
and Velo [4] (see also the former results of Reed [1]) considered the local and global initial
value problems in the Sobolev space Hm(Rn), for m > n/2. Cazenave and Weissler [4]
considered the local and global initial value problems in the (fractional order) Sobolev
space Hs(Rn) for s < n/2. In both case, a minimal regularity of the nonlinearity is
required in order to solve the equation by a fixed point argument. One sufficient condition
for global existence is that the initial datum is small.
Nonlinear Schrodinger equations are also considered in exterior domains. When n = 1,
or when n = 2 and under some growth condition on the nonlinearity, local (or global)
existence follows from the results of Sections 4.4 and 4.5. In some other cases, one can still
obtain global solutions for small initial data and study their asymptotic behavior. See for
example Chen [1], M. Tsutsumi [3], Y. Tsutsumi [3] and Yao [1].
The conservation laws that we used in these notes are conservation of charge and
energy, and the pseudo conformal conservation laws. They are related to the invariance of
the equation for some groups of transforms. On this subject, consult Ginibre and Velo [8],
and Olver [1]. When n = 1 and g(u) = λ|u|2u, there are infinitely many conservation
laws (cf. Zakharov and Shabat [1]), while in general, there does not seem to be other
useful conservation laws (cf. Serre [1]). In relation with the invariance properties of
223
nonlinear Schrodinger equations, one can construct families of explicit solutions, for some
nonlinearities (cf. Fuschich and Serov [1,2,3]). Unfortunately, these solutions do not belong
in general to the energy space.
The asymptotic behavior of solutions of the cubic nonlinear Schrodinger equation
in a one dimensional bounded domain, was studied by methods of infinite dimensional
dynamical systems by Lebowitz, Rose and Speer [1].
Nonlinearities of different types were also considered. For example, Baillon, Cazenave
and Figueira [2] (see also Cazenave [1]) studied a nonlinearity of integral form; quasilinear
Schrodinger equations were studied in particular by Klainerman and Ponce [1] (see also
Lange [3,4] for a suggestive numerical study); Stubbe and Vazquez [1,2] considered a
nonlinear Schrodinger equation with a magnetic interaction (different from the one of
Section 9.1), and proved the existence of stationary states.
Systems of Schrodinger equations, or coupled systems with other equations (Klein-
Gordon, for example) are also of a great interest. Zakharov systems were studied by Scho-
chet and Weinstein [1] (See also Lee [1]); coupled systems of Schrodinger and Klein-Gordon
equations were considered in particular by Baillon and Chadam [1], Bachelot [1] and Ozawa
and Tsutsumi [1]. Davey-Stewartson systems were considered inparticular by Ghidaglia
and Saut [1], Cipolatti [1,2] and Ozawa [1]. Note that nonlinear Schrodinger’s equation
can be viewed as a nonrelativistic limit of Klein-Gordon’s equation (see Najman [1]).
224
References.
ADAMS R.A.[1] Sobolev spaces, Academic press, New-York (1975).
AGMON S., DOUGLIS A. and NIRENBERG L.[1] Estimates near the boundary for solutions of elliptic partial differential equations sat-isfying general boundary conditions, Commun. Pure Appl. Math. 17 (1964), 35—92.
AITKENS A.H, HAYES J.N. and ULRICH P.B.[1] Propagation of high energy 10.6µ laser beams through the atmosphere, N.R.L. Report#7293 (1971).
AVRON J., HERBST I. and SIMON B.[1] Schrodinger operators with magnetic fields, I. General interactions, Duke Math. J. 45
(1978), 847—883.[2] Separation of center of mass in homogeneous magnetic fields, Ann. Phys. 114 (1978),431—451.[3] Schrodinger operators with magnetic fields, III. Atoms in homogeneous magnetic fields,Commun. Math. Phys. 79 (1981), 529—572.
BACHELOT A.[1] Probleme de Cauchy pour des systemes hyperboliques semi-lineaires, Ann. Inst. HenriPoincare, Analyse Non Lineaire 1 (1984), 453—478.
BAILLON J.-B., CAZENAVE T. and FIGUEIRA M.[1] Equation de Schrodinger non lineaire, C. Rend. Acad. Sc. Paris 284 (1977), 869—872.[2] Equation de Schrodinger avec non-linearite integrale, C. Rend. Acad. Sc. Paris 284
(1977), 939—942.
BAILLON J.-B. and CHADAM J. M.[1] The Cauchy problem for the coupled Schrodinger-Klein-Gordon equations, in Con-
temporary developments in continuum mechanics and partial differential equations, NorthHolland Math. Studies, North Holland, 1978.
BALABANE M.[1] Decroissance des solutions d’equations de type de Schrodinger, C. R. Acad. Sc. Paris292 (1981), 511-513.[2] On a regularizing effect of Schrodinger type groups, Ann. Inst. Henri Poincare, AnalyseNon Lineaire 6 (1989), 1—14.
BALABANE M. and EMAMI RAD H.A.[1] Smooth distribution group and Schrodinger equation in Lp(Rn), J. Math. Anal. Appl.70 (1979). [2] Lp estimates for Schrodinger evolution equations, Trans. Amer. Math. Soc.292 (1985), 357—373.
BALL J. M.[1] Finite-time blow-up in nonlinear problems, in Nonlinear evolution equations, M GCrandall Ed., Academic Press, New York, 1978, 189—205.[2] Remarks on blow-up and nonexistence theorems for nonlinear evolution equations,Quart. J. Math. Oxford 28 (1977), 473—486.
225
BARAB J. E.[1] Nonexistence of asymptotically free solutions for a nonlinear Schrodinger equation, J.Math. Phys. 25 (1984), 3270—3273.
BEN-ARTZI M. and DEVINATZ A.[1] Local smoothing and convergence properties of Schrodinger type equations, J. Funct.Anal. 101 (1991), 231—254.
BEN-ARTZI M. and KLAINERMAN S.[1] Decay and regularity for the Schrodinger equation, J. Anal. Math. 58 (1992), 25—37.
BERESTYCKI H. and CAZENAVE T.[1] Instabilite des etats stationnaires dans les equations de Schrodinger et de Klein-Gordonnon lineaires, C. Rend. Acad. Sc. Paris 293 (1981), 489—492.
BERESTYCKI H., GALLOUET T. and KAVIAN O.[1] Equations de champs scalaires Euclidiens non-lineaires dans le plan, C. Rend. Acad.Sc. Paris 297 (1983), 307—310.
BERESTYCKI H. and LIONS P.-L.[2] Nonlinear scalar field equations, Arch. Rat. Mech. Anal. 82 (1983), 313—375.
BERESTYCKI H., LIONS P.-L. and PELETIER L. A.[1] An O.D.E. approach to the existence of positive solutions for semilinear problems inRn, Indiana Univ. Math. J. 30 (1981), 141—157.
BERGH J. and LOFSTROM J.[1] Interpolation spaces, Springer, New York, 1976.
BIALINYCKI-BIRULA I. and MYCIELSKI J.[1] Nonlinear wave mechanics, Ann. Phys. 100 (1976), 62—93.[2] Wave equations with Logarithmic nonlinearities, Bull. Acad. Pol. Sc. 23 (1975),461—466.
BLANCHARD P., STUBBE J. and VAZQUEZ L.[1] On the stability of solitary waves for classical scalar fields, Ann. Inst. Henri Poincare,Physique Theorique 47 (1987), 309—336.
BREZIS H.[1] Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de
Hilbert, North-Holland, Amsterdam, London (1972), Lecture Notes #5.[2] Analyse fonctionnelle, theorie et applications, Masson, Paris (1983).
BREZIS H. and CAZENAVE T.[1] Nonlinear evolution equations, in preparation.
BREZIS H. and GALLOUET T.[1] Nonlinear Schrodinger evolution equations, Nonlinear Anal. T.M.A. 4 (1980), 677—681.
CAZENAVE T.[1] Equations de Schrodinger non lineaires en dimension deux, Proc. Royal Soc. Edimburgh84 (1979), 327—346.[2] Stable solutions of the logarithmic Schrodinger equation, Nonlinear Anal. T. M. A. 7
(1983), 1127—1140.
226
[3] Stability and instability of stationary states in nonlinear Schrodinger equations, incontributions to nonlinear partial differential equations I, Research Notes in Mathematics#89, Pitman, London, 1983.[4] Some remarks on the asymptotic behavior of solutions to nonlinear Schrodinger equa-tions, in Bielefeld encounters in mathematics and physics IV and V, trends and develop-
ments in the eighties, World Scientific, Singapore, 1985.[5] Equations d’evolution non lineaires, theorie elementaire et proprietes globales, Publica-tion du Laboratoire d’Analyse Numerique #88001, Universite Pierre et Marie Curie, Paris,1988.
CAZENAVE T., DIAS J.P. and FIGUEIRA M.[1] A remark on the decay of the solutions of some Schrodinger equations, Rend. Sem.Mat. Univ. Polit. Torino 40 (1982), 129—137.
CAZENAVE T. and ESTEBAN M. J.[1] On the stability of stationary states for nonlinear Schrodinger equations with an externalmagnetic field, Mat. Apl. Comp. 7 (1988), 155—168.
CAZENAVE T. and HARAUX A.[1] Equation d’evolution avec non-linearite logarithmique, Ann. Fac. Sc. Toulouse 2
(1980), 21—51.[2] Introduction aus problemes d’evolution semi-lineaires, Mathematiques et Applications#1, Ellipses, Paris, 1990.
CAZENAVE T. and LIONS P.-L.[1] Orbital stability of standing waves for some nonlinear Schrodinger equations, Commun.Math. Phys. 85 (1982), 549—561.
CAZENAVE T. and WEISSLER F. B.[1] Some remarks on the nonlinear Schrodinger equation in the subcritical case, in New
Methods and Results in Nonlinear Field Equations, Ph. Blanchard, J.-P. Dias and J. Stubbe(eds.), Lect. Notes in Phys. # 347, Springer, 1989, 59—69.[2] Some remarks on the nonlinear Schrodinger equation in the critical case, in Nonlinear
Semigroups, Partial Differential Equations, and Attractors, T.L. Gill and W.W. Zachary(eds.), Lect. Notes in Math. # 1394, Springer, 1989, 18—29.[3] The Cauchy problem for the nonlinear Schrodinger equation in H1, Manuscripta Math.61 (1988), 477—494.[4] The Cauchy problem for the critical nonlinear Schrodinger equation in Hs, NonlinearAnal, TMA 14 (1990), 807—836.[5] The structure of solutions to the pseudo-conformally invariant nonlinear Schrodingerequation, Proc. Royal Soc. Edinburgh 117A (1991), 251—273.[6] Rapidly decaying solutions of the nonlinear Schrodinger equation, Commun. Math.Phys. 147 (1992), 75—100.
CHADAM J. M. and GLASSEY R. T.[1] Global existence of solutions to the Cauchy problem for time-dependent Hartree equa-tions, J. Math. Phys. 16 (1975), 1122—1130.
CHEN Y.-M.[1] Global existence to the solution of nonlinear Schrodinger equation in exterior domains,Acta. Math. Appl. Sinica 2 (1985).
227
CIPOLATTI R.[1] On the existence of standing waves for a Davey-Stewartson system, Commun. Part.Differ. Eq. 17 (1992), 967—988.[2] On the instability of ground states for a Davey-Stewartson system, Ann. Inst. HenriPoincare, Physique Theorique 58 (1993), 85—104.
COMBES J.M., SCHRADER R. and SEILER R.[1] Classical bounds and limits for energy distributions of Hamilton operators in electro-magnetic fields, Ann. Phys. 111 (1978), 1—18.
CONSTANTIN P. and SAUT J.-C.[1] Effets regularisant locaux pour des equations dispersives generales, C. Rend. Acad.Sci. Paris 304 (1987), 407—410.[2] Local smoothing properties of dispersive equations, J. Amer. Math. Soc. 1 (1988),413—439.[3] Local smoothing properties of Schrodinger equations, Indiana Univ. Math. J. 38
(1989), 791—810.
DIAS J.-P.[1] Time decay for the solutions of some nonlinear Schrodinger equations, in Bielefeld
encounters in mathematics and physics IV and V, trends and developments in the eighties,World Scientific, Singapore, 1985.
DIAS J.-P. and FIGUEIRA M.[1] Conservation laws and time decay for the solutions to some nonlinear Schrodinger-Hartree equations, J. Math. Anal. Appl. 84 (1981), 486—508.
DIESTEL I. and UHL J.J. Jr.[1] Vector measures, Math. Survey #15, Amer. Math. Soc., Providence, 1977.
DINCULEANU N.[1] Vector measures, Pergamon Press, New York, 1967.
DONG G.-C. and LI S. J.[1] On the initial value problem for a nonlinear Schrodinger equation, J. Diff. Eq. 42
(1981), 353—365.
DUNFORD C. and SCHWARTZ J. T.[1] Linear operators, Interscience Publ., New York, 1958.
EBOLI O.J.P. and MARQUES G.C.[1] Solitons as Newtonian particles, Phys. Rev. B 28 (1983), 689—696.
ESTEBAN M. J. and LIONS P.-L.[1] Stationary solutions of nonlinear Schrodinger equations with an external magnetic field,in Partial differential equations and the calculus of variations, Vol. 1, L. Modica et al.editors, Birkhauser, Boston, 1989, 401—449.
FRISCH H., SULEM C. and SULEM P.-L.[1] Tracing complex singularities with spectral methods, J. Comp. Phys. 50 (1983), 138—161.
228
FUJIWARA D.[1] A construction of the fundamental solution for the Schrodinger equation, J. Anal. Math.35 (1979), 41—96.[2] Remarks on the convergence of Feynaman path integrals, Duke Math. J. 47 (1980),559—600.
FUSHCHICH B. W. I. and SEROV N. I.[1] On some exact solutions of the three-dimensional nonlinear Schrodinger equation, J.Phys. A 20 (1987), L929—L933.[2] On some exact solutions of a system of nonlinear differential equations for spinor andvector fields, J. Phys. A 20 (1987), 4173—4190.[3] On the reduction and some new exact solutions of the nonlinear Dirac and Dirac-Klein-Gordon equations, J. Phys. A 21 (1988), L5—L9.
GHIDAGLIA J-M. and SAUT J-C.[1] Sur le probleme de Cauchy pour les equations de Davey-Stewartson, C. Rend. Acad.Sci. Paris 308 (1989), 115—120.
GIDAS B., NI W.M. and NIRENBERG L.[1] Symmetry of positive solutions of nonlinear elliptic equations in Rn, Advances in Math.Suppl. Studies 7A (1981), 369-402.
GILBARG D. and TRUDINGER N. S.[1] Elliptic partial differential equations of the second order, Springer-Verlag, Berlin (1983).
GINIBRE J. and VELO G.[1] On a class of nonlinear Schrodinger equations, J. Funct. Anal. 32 (1979), 1—71.[2] On a class of nonlinear Schrodinger equations, special theories in dimensions 1, 2 and3, Ann. Inst. Henri Poincare 28 (1978), 287—316.[3] On a class of nonlinear Schrodinger equations with non local interaction, Math. Z. 170
(1980), 109—136.[4] On the global Cauchy problem for some nonlinear Schrodinger equations, Ann. Inst.Henri Poincare, Analyse Non Lineaire 1 (1984), 309—323.[5] The global Cauchy problem for the nonlinear Schrodinger equation revisited, Ann. Inst.Henri Poincare, Analyse Non Lineaire 2 (1985), 309—327.[6] Scattering theory in the energy space for a class of nonlinear Schrodinger equations, J.Math. Pure Appl. 64 (1985), 363—401.[7] Time decay of finite energy solutions of the nonlinear Klein-Gordon and Schrodingerequations, Ann. Inst. Henri Poincare, Physique Theorique 43 (1985), 399—442.[8] Sur une equation de Schrodinger non-lineaire avec interaction non-locale, in Nonlinear
partial differential equations and their applications, College de France Seminar, vol. 2 (H.Brezis & J.L. Lions editors), Research Notes in Math. #60, Pitman (1982), 155—199.[9] Scattering theory in the energy space for a class of Hartree equations, to appear.
GLASSEY R. T.[1] On the blowing up of solutions to the Cauchy problem for nonlinear Schrodinger equa-tions, J. Math. Phys. 18 (1977), 1794—1797.[2] On the asymptotic behavior of nonlinear wave equations, Trans. Amer. Math. Soc.182 (1973), 187—200.
229
GOGNY D. and LIONS P.-L.[1] Hartree-Fock theory in nuclear physics, RAIRO, Modelisation Math. Anal. Numer. 20
(1986), 571—637.
GONCALVES RIBEIRO J.M.[1] Instability of symmetric stationary states for some nonlinear Schrodinger equations withan external magnetic field, Ann. Inst. Henri Poincare, Physique Theorique 54 (1991),403—433.
GRILLAKIS M.[1] Linearized instability for nonlinear Schrodinger and Klein-Gordon equations, Commun.Pure Appl. Math. 41 (1988), 747—774.
GRILLAKIS M., SHATAH J. and STRAUSS W. A.[1] Stability theory of solitary waves in the presence of symmetry, I. J. Funct. Anal. 74
(1987), 160—197.
HARAUX A.[1] Nonlinear evolution equations: Global behavior of solutions, Lecture Notes in Math.#841, Springer (1981).[2] Linear semi-groups in Banach spaces, in Semigroups, theory and applications, H. Brezis,M.G. Crandall and F. Kappel (Eds.), Pitman research notes in Mathematics series #152,Longman, 1986, 93—135.
HAYASHI N.[1] Asymptotic behavior of solutions to time-dependent Hartree equations, Arch. Rat.Mech. Anal. 100 (1988), 191—206.[2] Time decay of solutions to the Schrodinger equation in exterior domains, Ann. Inst.Henri Poincare, Physique Theorique 50 (1989), 71—93.[3] Smoothing effect for nonlinear Schrodinger equations in exterior domains, J. Funct.Anal. 89 (1990), 444—458.[4] Global existence of small analytic solutions to nonlinear Schrodinger equations, DukeMath. J. 60 (1990), 717—727.[5] Smoothing effect of small analytic solutions to nonlinear Schrodinger equations, AnnInst. Henri Poincare, Physique Theorique 57 (1992), 385—394.
HAYASHI N. and KATO K.[1] Regularity in time of solutions to nonlinear Schrodinger equations, J. Funct. Anal. 128
(1995), 253—277.
HAYASHI N., NAKAMITSU K. and TSUTSUMI M.[1] On solutions of the initial value problem for the nonlinear Schrodinger equations in onespace dimension, Math. Z. 192 (1986), 637—650.[2] On solutions of the initial value problem for the nonlinear Schrodinger equations, J.Funct. Anal. 71 (1987), 218—245.[3] Nonlinear Schrodinger equations in weighted Sobolev spaces, Funk. Ekva 31 (1988),363—381.
HAYASHI N. and OZAWA T.[1] Time decay of solutions to the Cauchy problem for time-dependent Schrodinger-Hartreeequations, Commun. Math. Phys. 110 (1987), 467—478.
230
[2] Smoothing effect for some Schrodinger equations, J. Funct. Anal. 85 (1989), 307—348.[3] Time decay for some Schrodinger equations, J. Funct. Anal. 85 (1989), 307—348.[4] Scattering theory in the weighted L2(Rn) spaces for some Schrodinger equations, Ann.Inst. Henri Poincare, Physique Theorique 48 (1988), 17—37.[5] Lower bounds for order of decay or of growth in time for solutions to linear and nonlinearSchrodinger equations, Publ. Res. Inst. Math. Sci. (Kyoto Univ.) #560, Kyoto Univ,1986.
HAYASHI N. and SAITOH S.[1] Analiticity and global existence of small solutions to some nonlinear Schrodinger equa-tions, Commun. Math. Phys. 129 (1990), 27—42.[2] Analiticity and smoothing effect for the Schrodinger equation, Ann. Inst. HenriPoincare, Physique Theorique 52 (1990), 163—173.
HAYASHI N. and TSUTSUMI Y.[1] Scattering theory for Hartree type equations, Ann. Inst. Henri Poincare, PhysiqueTheorique 46 (1987), 187—213.
JENSEN A. and KATO T.[1] Spectral properties of Schrodinger operators and time decay of wave functions, DukeMath. J. 46 (1979), 583—611.
JONES C. K. R. T.[1] An instability mechanism for radially symmetric standing waves of a nonlinear Schro-dinger equation, J. Diff. Eq. 71 (1988), 34—62.
JONES C. and KUPPER T.[1] On the infinitely many solutions of a semilinear elliptic equation, SIAM J. Math. Anal.17 (1986), 803—836.
JOURNE J.-L., SOFFER A. and SOGGE C.[1] Decay estimates for Schrodinger equations, Commun. Pure Appl. Math 64 (1991),572—604.
KATO T.[1] On nonlinear Schrodinger equations, Ann. Inst. Henri Poincare, Physique Theorique46 (1987), 113—129.[2] Perturbation theory for linear operators, Springer, New York, 1966.[3] Nonlinear Schrodinger equations, in Schrodinger Operators, Lect. Notes in Phys. #345,Springer, 1989, 218—263.
KATO T. and YAJIMA K.[1] Some examples of smooth operators and the associated smoothing effect, Rev. Math.Phys. 1 (1989), 481—496.
KAVIAN O.[1] A remark on the blowing-up of solutions to the Cauchy problem for nonlinear Schro-dinger equations, Trans. Amer. Math. Soc. 299 (1987), 193—205.
KENIG C., PONCE G. and VEGA L.[1] Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J. 40
(1991), 33—69.
231
[2] Small solutions of the nonlinear Schrodinger equation, Ann. Inst. Henri Poincare,Analyse Non Lineaire 10 (1993), 255—288.
KLAINERMAN S. and PONCE G.
[1] Global, small amplitude solutions to non-linear evolution equations, Commun. PureAppl. Math. 36 (1983), 133—141.
KRANOSEL’SKII M. A. and RUTICKII Y. B.
[1] Convex functions and Orlicz spaces, Noordhoff, 1961.
KWONG M.K.
[1] Uniqueness of positive solutions of 4u − u + up = 0 in Rn, Arch. Rat Mech. Anal.105 (1989), 243—266.
LAM J. F., LIPPMANN B. and TAPPERT F.
[1] Self trapped laser beams in plasma, Phys. Fluids 20 (1977), 1176—1179.
LANGE H.
[1] On nonlinear Schrodinger equations in the theory of quantum mechanics dissipativesystems, preprint.
[2] On some nonlinear Schrodinger equations of Hartree type, Math. Verf. Math. PhysikBa. 21 (1980), 99—111.
[3] Asymptotical behaviour of solutions to quasilinear Schrodinger equations, preprint.
[4] Blow up of solutions to singular nonlinear Schrodinger equations and colliding solitarywaves, preprint.
LEBOWITZ J. L., ROSE H. A. and SPEER E. R.
[1] Statistical mechanics of the nonlinear Schrodinger equation, J. Stat. Phys. 50 (1988),657—687.
LEE Jyh-Hao
[1] Global solvability of the derivative nonlinear Schrodinger equation, Trans. Amer. Math.Soc. 314 (1989), 107—118.
LE MESURIER B., PAPANICOLAOU G., SULEM C. and SULEM P.-L.
[1] The focusing singularity of the nonlinear Schrodinger equation, in Directions in partial
differential equations, M.G. Crandall, P.H. Rabinowitz and R.E. Turner editors, AcademicPress, New York, 1987, 159—201.
[2] Local structure of the self-focusing singularity of the nonlinear Schrodinger equation,Physica D 32 (1988), 210—266.
[3] Focusing and multi-focusing solution of the nonlinear Schrodinger equation, Physica D
31 (1988), 78—102.
LIEB E. H. and SIMON B.
[1] The Hartree-Fock theory for coulomb systems, Commun. Math. Phys. 53 (1974),185—194.
LIN J. E. and STRAUSS W. A.
[1] Decay and scattering of solutions of a nonlinear Schrodinger equation, J. Funct. Anal.30 (1978), 245—263.
232
LIONS J.-L.[1] Quelques methodes de resolution des problemes aux limites non lineaires, Dunod etGauthiers-Villars, Paris, 1969.
LIONS J.-L. and MAGENES E.[1] Problemes aux limites non homogenes et applications, Dunod et Gauthiers-Villars, Paris,1968.
LIONS P.-L.[1] The Choquard equation and related questions, Nonlinear Anal. TMA 4 (1980), 1063—1073.[2] Some remarks on Hartree equation, Nonlinear Anal. TMA 5 (1981), 1245—1256.[3] The concentration-compactness principle in the calculus of variations. The locallycompact case. Part 1, Ann. Inst. Henri Poincare, Analyse non lineaire 1 (1984), 109—145.[4] The concentration-compactness principle in the calculus of variations. The locallycompact case. Part 2, Ann. Inst. Henri Poincare, Analyse non lineaire 1 (1984), 223—283.
MARCUS M. and MIZEL V. J.[1] Nemitsky operators on Sobolev spaces, Arch. Rational Mech. Anal. 51 (1973), 347—370.[2] Complete characterization of functions which act, via superposition, on Sobolev spaces,Trans. Amer. Math. Soc. 251 (1979), 187—218.[3] Every superposition operator mapping one Sobolev space into another is continuous, J.Funct. Anal. 33 (1979), 217—229.
MC LAUGHLIN D., PAPANICOLAOU G. C., SULEM C. and SULEM P.-L.[1] The focusing singularity of the cubic Schrodinger equation, Phys. Rev. A (1986).
MERLE F.[1] Construction of solutions with exactly k blow-up points for the Schrodinger equationwith critical nonlinearity, Commun. Math. Phys. 129 (1990), 223—240.[2] On uniqueness and continuation properties after blowup time of self-similar solutions ofnonlinear Schrodinger equations with critical exponent and critical mass, Commun. PureAppl. Math. 65 (1992), 203—254.[3] Determination of blow-up solutions with minimal mass for nonlinear Schrodinger equa-tions with critical power, Duke Math. J. (1993), in press.
MERLE F. and TSUTSUMI Y.[1] L2-concentration of blow-up solutions for the nonlinear Schrodinger equation with thecritical power nonlinearity, J. Differ. Eq. 84 (1990), 205—214.
MURATA M.[1] Rate of decay of local energy and spectral properties of elliptic operators, Japan J.Math. 6 (1980), 77—127.
NAJMAN B.[1] The nonrelativistic limit of the nonlinear Klein-Gordon equation, Nonlinear Anal., inDifferential equations with applications in biology, physics and engineering, F. Kappel etal (Eds), Lect. Notes Pure Appl. Math. #133, Marcel Dekker, 1991, 291—299.
233
NAWA H.[1] “Mass concentration” phenomenon for the nonlinear Schrodinger equation with criticalpower nonlinearity II, Kodai Math. J. 13 (1990), 333—348.[2] “Mass concentration” phenomenon for the nonlinear Schrodinger equation with criticalpower nonlinearity, Funk. Ekva. 35 (1992), 1—18.
NAWA H. and TSUTSUMI M.[1] On blow up for the pseudo-conformally invariant nonlinear Schrodinger equation, Funk.Ekva 32 (1989), 417—428.
NIRENBERG L.[1] On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa 13 (1959),115—162.
OGAWA T.[1] A proof of Trudinger’s inequality and its application to nonlinear Schrodinger’s equation,Nonlinear Anal. T.M.A. 14 (1990), 765—769.
OGAWA T. and OZAWA T.[1] Trudinger type inequalities and uniqueness of weak solutions for the nonlinear Schro-dinger equations, J. Math. Anal. Appl. 155 (1991), 531—540.
OGAWA T. and TSUTSUMI Y.[1] Blow-up of H1 solutions for the nonlinear Schrodinger equation, J. Differ. Eq. 92
(1991), 317—330.[2] Blow-up of H1 solutions for the one dimensional nonlinear Schrodinger equation withcritical power nonlinearity, Proc. Amer. Math. Soc. 111 (1991), 487—496.
OH Y.-G.[1] Existence of semiclassical bound states of nonlinear Schrodinger equations with poten-tials of the class (V )a, Commun. Part. Differ. Eq. 13 (1988), 1499—1519.[2] Lyapunov stability of semiclassical bound states of nonlinear Schrodinger equationswith potentials, Commun. Math. Phys. 121 (1989), 11—33.
OLVER P. J.[1] Applications of Lie groups to differential equations, Graduate Texts in Math., Springer,New York, 1986.
OZAWA T.[1] Exact blowup solutions to the Cauchy problem for the Davey-Stewartson systems, Publ.RIMS, Kyoto Univ. #741, 1991.
OZAWA T. and TSUTSUMI Y.[1] Asymptotic behavior of solutions for the coupled Klein–Gordon–Schrodinger equations,Publ. RIMS Kyoto Univ. #775 (1991).
PATERA A., SULEM C. and SULEM P.-L.[1] Numerical simulation of singular solutions to the two dimensional cubic Schrodingerequation, Commun. Pure Appl. Math. 37 (1984), 755—778.
PAZY A.[1] Semi-groups of linear operators and applications to partial differential equations Sprin-ger, New-York, 1983.
234
PECHER H. and VON WAHL W.[1] Time dependent nonlinear Schrodinger equations, Manuscripta Math. 27 (1979), 125—157.
REED M.[1] Abstract nonlinear wave equations, Lecture Notes in Math. #507, Springer, New York,1976.
REED M. and SIMON B.[1] Methods of modern mathematical physics, II, Academic Press, 1975.
ROSE H. A. and WEINSTEIN M. I.[1] On the bound states of the nonlinear Schrodinger equation with a linear potential,Physica D 30 (1988), 207—218.
RUIZ A. and VEGA L.[1] On local regularity of Schrodinger equations, Duke Math. J. International MathematicsResearch Notices, 1993, 13—27.
SCHOCHET H. and WEINSTEIN M. I.[1] The nonlinear Schrodinger limit of the Zakharov equations governing Langmuir turbu-lence, Commun. Math. Phys. 106 (1986), 569—580.
SCHONBEK T.[1] Decay of solutions of Schroedinger equations, Duke Math. J. 46 (1979), 203—213.
SEGAL I. E.[1] Nonlinear semi-groups, Ann. of Math. 2 (1963), 339—364.
SERRE D.[1] A propos des invariants de diverses equations d’evolution, in Nonlinear partial differ-
ential equations and their applications, College de France Seminar, vol. 3, (H. Brezis &J.L. Lions editors), Research Notes in Math. #70, Pitman, 1981, 326—336.
SHATAH J. and STRAUSS W. A.[1] Instability of nonlinear bound states, Commun. Math. Phys. 100 (1985), 173—190.
SIMON B.[1] Functional integration and quantum physics, Academic Press, New York, 1979.
SIMON J.[1] Compact sets in the space Lp(0, T ;B), Annali Mat. Pura Appl. 146 (1987), 65—96.
SJOLIN P.[1] Regularity of solutions to Schrodinger equations, Duke Math. J. 55 (1987), 699—715.
SOFFER A.[1] Phase-space analysis of some nonlinear waves and global existence, to appear.
SOFFER A. and WEINSTEIN M.I.[1] Multichannel nonlinear scattering theory for nonintegrable equations, Commun. Math.Phys. 133 (1990), 119—146.[2] Multichannel nonlinear scattering theory for nonintegrable equations II. The case ofanisotropic potentials and data, J. Differ. Eq. 98 (1992), 376—390.
235
STEIN E.M.[1] Singular integrals and differentiability of functions, Princeton University Press, Prince-ton, 1970.
STRAUSS W. A.[1] On the continuity of functions with values in various Banach spaces, Pacific J. Math19 (1966), 543—551.[2] On weak solutions of semilinear hyperbolic equations, An. Acad. Brasil. Cien. 42
(1970), 645—651.[3] Nonlinear invariant wave equations, Lecture Notes in Physics #73, Springer, 1978.[4] Existence of solitary waves in higher dimensions, Commun. Math. Phys. 55 (1977),149—162.[5] Nonlinear scattering theory, in Scattering Theory in Mathematical Physics, J. A. Lavitaand J.-P. Planchard (eds.), Reidel, 1974, 53—78.[6] Nonlinear scattering theory at low energy, J. Funct. Anal., 41 (1981), 110—133.
STRICHARTZ[1] Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of waveequations, Duke Math. J. 44 (1977), 705—714.
STUBBE J. and VAZQUEZ L.[1] Nonlinear Schrodinger equations with magnetic field effect: existence and stability ofsolitary waves, Lecture Notes in Phys. #342, M. Balabane, P. Lochak and C. Sulem (eds.),Springer, 1989, 336—342.[2] A nonlinear Schrodinger equation with magnetic field effect: solitary waves with internalrotation, Portugaliae Math. 46 (1989), 493—499.
SUYDAM B. R.[1] Self focusing of very powerful laser beams, IEEE J. Q.E. 10 (1974), 837—843.
TSUTSUMI M.[1] Nonexistence of global solutions to the Cauchy problem for the damped nonlinearSchrodinger equation, SIAM. J. Math. Anal. 15 (1984), 357—366.[2] On smooth solutions to the initial-boundary value problem for the nonlinear Schrodingerequation in two space dimensions, Nonlinear Anal. T.M.A. 13 (1989), 1051—1056.[3] On global solutions to the initial-boundary value problem for the nonlinear Schrodingerequation in exterior domains, Commun. Part. Differ. Eq. 16 (1991), 885—907.
TSUTSUMI Y.[1] Scattering problem for nonlinear Schrodinger equations, Ann. Inst. Henri Poincare,Physique Theorique 43 (1985), 321—347.[2] Global strong solutions for nonlinear Schrodinger equations, Nonlinear Anal., T. M. A.11 (1987), 1143—1154.[3] Global solutions of the nonlinear Schrodinger equation in exterior domains, Commun.Part. Differ. Eq. 8 (1984), 1337—1374.[4] L2-solutions for nonlinear Schrodinger equations and nonlinear groups, Funk. Ekva.30 (1987), 115—125.[5] Rate of L2-concentration of blow-up solutions for the nonlinear Schrodinger equationwith the critical nonlinearity, Nonlinear Anal. T.M.A. 15 (1990), 719—724.
236
TSUTSUMI Y. and YAJIMA K.[1] The asymptotic behavior of nonlinear Schrodinger equations, Bull. Amer. Math. Soc.11 (1984), 186—188.
VEGA L.[1] Schrodinger equations: pointwise convergence to the initial data, Proc. Amer. Math.Soc. 102 (1988), 874—878.
VLADIMIROV M. V.[1] On the solvability of mixed problem for a nonlinear equation of Schrodinger type, Sov.Math. Dokl. 29 (1984), 281—284.
WEINSTEIN A.[1] A symbol class for Schrodinger equations on Rn, Amer. J. Math. 107 (1985), 1—23.
WEINSTEIN M. I.[1] Nonlinear Schrodinger equations and sharp interpolation estimates, Commun. Math.Phys. 87 (1983), 567—576.[2] Lyapunov stability of ground states of nonlinear distersive evolution equations, Com-mun. Pure Appl. Math. 29 (1986), 51—68.[3] Modulational stability of ground states of nonlinear Schrodinger equations, SIAM J.Math. Anal. 16 (1985), 567—576.[4] On the structure and formation of singularities of solutions to nonlinear dispersiveevolution equations, Commun. Part. Differ. Eq. 11 (1986), 545—565.
YAJIMA K.[1] Existence of solutions for Schrodinger evolution equations, Commun. Math. Phys. 110
(1987), 415—426.
YAO J.-Q.[1] Comportement a l’infini des solutions d’une equation de Schrodinger non lineaire dansun domaine exterieur, C. Rend. Acad. Sc. Paris. 294 (1982), 163—166.
YOSIDA K.[1] Functional Analysis, Springer, New York, 1965.
ZAKHAROV V.E. and SHABAT A.B.[1] Exact theory of two dimensional self focusing and one dimensional self modulation ofwaves in nonlinear media, Sov. Phys. J.E.T.P. 34 (1972), 62-69.
ZELDITCH S.[1] Reconstruction of singularities for solutions of Schrodinger equations, Commun. Math.Phys. 90 (1983), 1—26.
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Index of subjects.
Admissible pair, 35.
Blow up, 113—117.
Bounded solution, cf. Estimates (uniform).
Bound state, 176.
Charge, cf. Conservation of charge.
Compact embedding, 21.
Compactness lemma, 11, 25, 211, 219.
Complete metric space, 130, 148.
Conservation laws, cf. Charge, Energy, Pseudo-conformal.
Conservation of charge, 49, 54, 61, 66, 69, 71, 74, 76, 102, 104, 200, 201, 206, 210, 218.
Conservation of energy, 49, 61, 66, 69, 71, 74, 76, 200, 206, 210.
Continuous dependence, 61, 66, 69, 71, 74, 76, 102, 104, 200, 201, 206, 210.
Differential equation (equivalence with the integral eq.), 29—31.
Energy, cf. Conservation of energy.
Estimates, cf. Uniform estimates.
Existence (local), 54, 61, 66, 69, 71, 74, 76, 200, 201, 205.
Existence (global), 49, 65, 76, 95, 99, 102, 104, 201, 206, 210, 218.
Fixed point, 131, 149.
Functional analysis, 10—31.
Gagliardo-Nirenberg’s inequality, 21.
Global solution, cf. Existence (global).
Ground state, 176.
Group of isometries, 28—31, 32.
Hilbert space, 28.
Integral equation (equivalence with the differential eq.), 29—31.
Intersection of Banach spaces, 13.
Local solution, cf. Existence (local).
Localized solution, cf. Standing wave.
Maximal solution, 61, 65, 69, 71, 74, 76, 200, 205.
Maximum principle, 171.
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Morawetz’ estimate, 136.
Pseudo-conformal conservation law, 119.
Regularity, 77, 79, 201, 206.
Scattering operator, 132, 150.
Smoothing effect, 33—443, 46, 82, 84, 87, 93,105.
Sobolev’s inequality, 20–22.
Sobolev’s embedding, 21.
Sobolev space, 17—25.
Standing wave, 157.
Stationary state, cf. Standing wave.
Sum of Banach spaces, 13.
Uniform estimates, 49, 66, 76, 94, 95, 99, 102, 104, 200, 201, 206, 210, 218.
Uniqueness, 66, 69, 71, 76, 102, 104, 200, 201, 205, 206, 210.
Variation of the parameters formula, 29.
Wave operator, 132, 149.
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