An introduction to semilinear evolution equations

An Introduction to Semilinear Evolution Equations Revised edition '

nz 1NIVERSIDAD COMPLUTENSE ^.^,f ,? I I) I I ^ ^^^ I I I^ I ) ^^^I ^II I ^I I I^I I^I I

5311910045

Thierry Cazenave CNRS and University of Paris VI, France

and

Alain Haraux CNRS and University of Paris VI, France

Translated by Yvan Martel University of Cergy-Pontoise, France

k 7

0R . 5/.??0

r

CLARENDON PRESS • OXFORD 1998

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Oxford University Press, Great Clarendon Street, Oxford 012 6DP Oxford New York Athens Auckland Bangkok Bogota Buenos Aires Calcutta Cape Town Chennai Dares Salaam Delhi Florence HongKong Istanbul Karachi KualaLumpur Madrid Melbourne Mexico City Mumbai Nairobi Paris Sao Paolo Singapore Taipei Tokyo Toronto Warsaw and associated companies in Berlin Ibadan Oxford is a trade mark of Oxford University Press Published in the United States by Oxford University Press Inc., New York Introduction aux problemes devolution semi-lineaires © Edition Marketing SA, 1990 First published by Ellipses

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I

Translation © Oxford University Press, 1998 Aide par le ministere francais charge de la culture All rights reserved. No part of thispublication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press. Within the UK, exceptions are allowed in respect of any fair dealing for the purpose of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act, 1988, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms and in other countries should be sent to the Rights Department, Oxford University Press, at the address above. This book is sold subject to the condition that it shall not, by way of trade or otherwise, be lent, re-sold, hired out, or otherwise circulated without the publisher's prior consent in any form of binding or cover other than that in which it is published and without a similar condition including this condition being imposed on the subsequent purchaser. A catalogue record for this book is available from the British Library Library of Congress Cataloging in Publication Data (Data available)

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1

ISBNO 19 850277 X(Hbk) Typeset by Yvan Martel Printed in Great Britain by Bookcraft (Bath) Ltd, Midsomer Norton, Avon

Preface This book is an expanded version of a post-graduate course taught for several years at the Laboratoire d'Analyse Numerique of the Universite Pierre et Marie Curie in Paris. The purpose of this course was to give a self-contained presentation of some recent results concerning the fundamental properties of solutions of semilinear evolution partial differential equations, with special emphasis on the asymptotic behaviour of the solutions. We begin with a brief description of the abstract theory of semilinear evolution equations, in order to provide the reader with a sufficient background. In particular, we recall the basic results of vector integration (Chapter 1) and linear semigroup theory in Banach spaces (Chapters 2 and 3). Chapter 4 concerns the local existence, uniqueness, and regularity of solutions of abstract semilinear problems. In Nature, many propagation phenomena are described by evolution equations or evolution systems which may include non-linear interaction or selfinteraction terms. In Chapters 5, 6, and 7, we apply some general methods to the following three problems. (1) The heat equation ut =

Au,

(0.1)

which models the thermal energy transfer in a homogeneous medium, is the simplest example of a diffusion equation. This equation, as well as the selfinteraction problem Ut = Au

+

f(u),

(0.2)

can be considered on the entire space RN or on various domains S1 (bounded or not) of R N . In the case in which ci # RN, we need to specify a boundary condition on I' = 852. It can be, for example, a homogeneous Dirichlet condition u=0

onr,

(0.3)

or a homogeneous Neumann condition

au _ an 0 our,

(0.4)

vi Preface

Chapter 5 studies in detail the properties of the solutions of (0.2)—(0.3) when Sl is bounded. In this problem, the maximum principle plays an important role. This is the reason for studying equation (0.2) in the space of continuous functions. Vector-valued generalizations of the form aui

ac

= czAui

+ fi(ul, ... , uk),

i = 1, ... , k,

(0.5)

called reaction—diffusion systems, often arise in chemistry and biology. One of the main tools in the study of these systems (and in particular of their nonnegative solutions) is the maximum principle, which gives a priori estimates in L 0° (5l) k for the trajectories. We thus develop Co methods rather than L 2 methods, which are easier but less suitable in this framework. (2) The wave equation (also called the Klein—Gordon equation)

= Au — mu,

Utt

(0.6)

with m > 0, models the propagation of different kinds of waves (for example light waves) in homogeneous media. Non-linear models of conservative type arise in quantum mechanics, whereas variants of the form Utt = Au — f (u, Ut)

(0.7)

appear in the study of vibrating systems with or without damping, and with or without forcing terms. Other perturbations of the wave equation arise in electronics (the telegraph equation, semi-conductors, etc.). The basic method for studying (0.6) with suitable boundary conditions (for example (0.3)) consists of introducing the associated isometry group in the energy space H l x L 2 . Local existence and uniqueness of solutions is established in this space. However, in general, the solutions are differentiable only in the sense of the larger space L 2 x H -1 . These local questions are considered in Chapter 6. (3) The Schrodinger equation iUt = Au,

(0.8)

possesses a combination of the properties described in (1) and (2). Primarily a simplified model for some problems of optics, this equation also arises in quantum field theory, possibly coupled with the Klein—Gordon equation. Various non-linear perturbations of (0.7) have appeared recently in the study of laser beams when the characteristics of the medium depend upon the temperature; for example, focusing phenomena in some solids (where the medium can break down if the temperature reaches a critical point) and contrastingly, defocusing in a gas medium which weakens the transmitted signal according to the distance

vii

Preface

from the source. A close examination of sharp properties of solutions of the non-linear Schrodinger equation is delicate, since this problem has a mixed or degenerate nature (neither parabolic nor hyperbolic). In Chapter 7, which is devoted to Schrodinger's equation, it becomes clear that even the local theory requires very elaborate techniques. The choice of these three problems as model examples is somewhat arbitrary. This selection was motivated by the limited experience of the authors, as well as by the desire to present the easiest models (in particular, semilinear models) for a first approach to the theory of evolution equations. We do not address several other equally worthy problems, such as transport equations, vibrating plates, and fundamental equations of fluid mechanics (such as Boltzmann's equation, the Navier—Stokes equation, etc.). Such complicated systems require many specific methods which could not be covered or even approached in a work of this kind. Chapters 8, 9, and 10 are devoted to some techniques and results concerning the global behaviour of solutions of semilinear evolution problems as the time variable converges to infinity. In Chapter 8, we establish that, for several kinds of evolution equations, the solutions either blow up in finite time in the original space or they are uniformly bounded in this space for all t >_ 0. This is the case for the heat equation and the Klein—Gordon equation with attractive nonlinearity, as well as for non-autonomous problems with dissipation. No such alternative is presently known for Schrodinger's equation. Chapter 9 is devoted to some basic notions of the theory of dynamical systems and its application to models (1) and (2) in an open, bounded domain of R N We restrict ourselves to the basic properties, and we give an extensive bibliography for the interested reader. In Chapter 10, we study the asymptotic stability of equilibria. We also discuss the connection between stability and positivity in the case of the heat equation. Finally, in the notes at the end of each chapter there are various bibliographical comments which provide the reader with a larger overview of the theories discussed. Moreover, the limited character of the examples studied is compensated for by a rather detailed bibliography that refers to similar works. We hope that this bibliography will serve our goal of a sufficient yet comprehensible introduction to the available theory of evolution problems. At the time of publication, new results will have made some parts of this book obsolete. However, we think that the methods presented are, and will continue to be for some years, an indispensable basis for anyone wanting a global view of evolution problems. .

Paris

1998

T. C. A. H.

U

Contents 1.

Notation . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Preliminary results . . . . . . .. . . . . . . . . . . . . . . . 1 1.1. 1.2. 1.3. 1.4.

Some abstract tools The exponential of a linear continuous operator Sobolev spaces Vector-valued functions 1.4.1. Measurable functions 1.4.2. Integrable functions ... 1.4.3. The spaces LP(I,X) 1.4.4. Vector-valued distributions 1.4.5. The spaces W 1, P(I,X) .

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1 1 2 4 4 7

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10 13

m-dissipative operators . . . . . . . . . . . . . . . . . . .

18

2.1. 2.2. 2.3. 2.4. 2.5. 2.6.

18 19 21 22 25 26 26 27

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2.

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Unbounded operators in Banach spaces Definition and main properties of m-dissipative operators Extrapolation Unbounded operators in Hilbert spaces Complex Hilbert spaces Examples in the theory of partial differential equations 2.6:1. The Laplacian in an open subset of R N L 2 theory 2.6.2. The Laplacian in an open subset of R N Co theory 2.6.3. The wave operator (or the Klein—Gordon operator) in Ha (1l) x L 2 (1l) 2.6.4. The wave operator (or the Klein—Gordon operator) in L 2 (1) x H '(Il) 2.6.5. The Schrodinger operator .

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8

29 30 31

3. The Hille—Yosida—Phillips Theorem and applications . . . . 33 3.1. 3.2. 3.3. 3.4. 3.5.

The semigroup generated by an m-dissipative operator Two important special cases Extrapolation and weak solutions Contraction semigroups and their generators Examples in the theory of partial differential equations .

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33 35 38 39 42



x Contents

3.5.1. The heat equation . . . . . . . . . . . . . . . . . 42 3.5.2. The wave equation (or the Klein—Gordon equation) . . . 47 3.5.3. The Schrodinger equation . . . . . . . . . . . . . . 47 3.5.4. The Schrodinger equation in Rr` . . . . . . . . . . . 48 4.

5.

6.

Inhomogeneous equations and abstract semilinear problems . . . . . . . . . . . . . . . . . . . . . . . . . .

50

4.1. Inhomogeneous equations . . . . . . . . . . . . . . . . . 4.2. Gronwall's lemma . . . . . . . . . . . . . . . . . . . . 4.3. Semilinear problems . . . . . . . . . . . . . . . . . . . 4.3.1. A result of local existence . . . . . . . . . . . . . . 4.3.2. Continuous dependence on initial data . . . . . . . . 4.3.3. Regularity . . . . . . . . . . . . . . . . . . . . 4.4. Isometry groups . . . . . . . . . . . . . . . . . . . . .

50 54 55 56 59 60 61

The heat equation . . . . . . . . . . . . . . . . . . . . .

62

5.1. 5.2. 5.3. 5.4. 5.5.

62 64 65 72 76

Preliminaries Local existence Global existence Blow-up in finite time Application to a model case .

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The Klein—Gordon equation . . . . . . . . . . . . . . . .

78

6.1. Preliminaries 6.1.1. An abstract result 6.1.2. Functionals on Ho (S2) 6.2. Local existence 6.3. Global existence 6.4. Blow-up in finite time 6.5. Application to a model case

78 78 79 82 84 87 89

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The Schrodinger equation . . . . . . . . . . . . . . . . .

91

7.1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . 91 7.2. A general result . . . . . . . . . . . . . . . . . . . 92 7.3. The linear Schrodinger equation in RN . . . . . . . . . . . 95 7.4. The non-linear Schrodinger equation in R N local existence . . 100 7.4.1. Some estimates . . . . . . . . . . . . . . . . . . 101 7.4.2. Proof of Theorem 7.4.1 . . . . . . . . . . . . . . . 106 7.5. The non-linear Schrodinger equation in R N global existence . . 112 :

:

Contents xi

7.6. The non-linear Schrodinger equation in R N blow up in finite time 7.7. A remark concerning behaviour at infinity 7.8. Application to a model case :

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Bounds on global solutions

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8.1. The heat equation 8.1.1. A singular Gronwall's lemma: application to the heat equation 8.1.2. Uniform estimates 8.2. The Klein—Gordon equation 8.3. The non-autonomous heat equation 8.3.1. The Cauchy problem for the non-autonomous heat equation 8.3.2. A priori estimates 8.4. The dissipative non-autonomous Klein—Gordon equation .

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The invariance principle and some applications

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9.1. Abstract dynamical systems 9.2. Liapunov functions and the invariance principle 9.3. A dynamical system associated with a semilinear evolution equation 9.4. Application to the non-linear heat equation 9.5. Application to a dissipative Klein—Gordon equation .

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10. Stability of stationary solutions

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10.1. Definitions and simple examples 10.2. A simple general result 10.3. Exponentially stable systems governed by PDE 10.4. Stability and positivity 10.4.1. The one-dimensional case 10.4.2. The multidimensional case

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Bibliography

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Index

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114 120 121 124 124 125 129 130 134 134 135 137 142 142 143 145 146 149 154 154 156 158 164 165 167 169 185

I I

Notation the space of linear, continuous mappings from X to Y the space of linear, continuous mappings from X to X the topological dual of the vector space X the Banach space (D(A) , (I II D(A)) with II uIID(A) = II uiI + IIAuII, when A is a linear operator with a closed graph the space of C°° (real-valued or complex valued) functions with compact support in S2

= C°°(^) = D( l) the space of continuous functions with compact support in S2 the space of functions of C(S2) which are zero on 011 the space of distributions on 11 the space of measurable functions on 11 such that I uI P is integrable (1 < p< oo)

= (fn u')', for u E Lp(1)

the space of measurable functions u on SZ such that there exists C such that I u(x) I < C for almost every x E 11 = Inf{C > 0, Iu(x)I < C almost everywhere}, for u E L(1) the conjugate exponent of p, i.e. p' = p/(p — 1) for 1 < p < oo =

0 kI

N

a = (ai, ... , aN), IaI = E a^

_ { f E LP (St), Da f E LP (S2) for all a E N' such that I al < m}

_ .Ijcj<m II D"uII LP for u E Wm , r(SZ) the closure of D(fl) with respect to the norm II Ilwm , p = Wm,2(fl ) = (EIaI< m (IID - uIIL 2 )

)

2 1/2

for u E H m(1l)

xiv Notation

H)

= Wo

' 2 (fl)

D(I, X) the space of C°° functions with compact support from I to X u '

= ut

= du/dt, for u E D'(I, X)

C,(I,X) the space of continuous functions with compact support from I to

X Cb(I, X) the space of continuous and bounded functions from I to X Cb, w (I,X) the space of uniformly continuous and bounded functions from I

to X LP(I, X) the space of measurable functions u on I with values in X and such

that IIuIIP is integrable (1 < p < oo)

= (Ii IuIP)1 /P, for u E LP(I,X) L(I,X) the space of measurable functions u on I such that there exists C such that IIu(x)II < C for almost every x E I IIUIILP

IIuIIL°°

= Inf{C > 0, Iu(x)I < C almost everywhere}, for u E LO°(I, X)

IIuIIw

= IIUIILP + IIn' IILP for u E W1,P(I,X)

W 1 'P(I,X) = {u E LP(I,X), u' E LP(I,X), in the sense of D'(I,X)} i, P

u 1 Preliminary results

1

1.1. Some abstract tools We recall here some classical theorems of functional analysis that are necessary for the study of semilinear evolution equations. The proofs can be found in Brezis [2]. Theorem 1.1.1. (The Banach Fixed Point Theorem) Let (E, d) be a complete metric space and let f : E –+ E be a mapping such that there exists k E [0, 1) satisfying d(f (x), f (y)) < kd(x, y) for all (x, y) E E x E. Then there exists a unique point 10 E E such that f (xo) = xp. Theorem 1.1.2. (The Closed Graph Theorem) Let X and Y be Banach spaces and let A: X — Y be a linear mapping. Then A E L(X,Y) if and only if the graph of A is a closed subspace of X x Y. Remark 1.1.3. We recall that the graph of A is G(A) = {( x, y) E X x Y; y = Ax}. Theorem 1.1.4. (The Lax—Milgram Theorem) Let H be a Hilbert space and let a : H x H – IR be a bilinear functional. Assume that there exist two constants C < oo, a > 0 such that: (i) Ia(u,v)I ClIull

IMI for all (u, v) E H x H (continuity);

aIIuII 2 for all u E H (coerciveness). Then, for every f E H* (the dual space of H), there exists a unique u E H such (ii) a(u, u) >

that a(u, v) = (f, v) for ally E H. 1.2. The exponential of a linear continuous operator Let X be a Banach space and let A E C(X). Definition 1.2.1. We denote by e A the sum of the series

E n, An.

n>0

It is clear that the series is norm convergent in C(X) and that Ile A ll < e lIA1I Furthermore, it is well known that if A and B commute, then e A+B = eAeB.

.

I I

2 Preliminary results

In addition, for fixed A, the function t '-4 e tA belongs to C°° (R, £(X)) and we have tA = e1AA = Ae IA dt e for all t E R. Finally, we have the following classical result. Proposition 1.2.2. Let A E £(X). For all T> 0 and all x E X, there exists

a unique solution u E C 1 ([0, TI, X) of the following problem: u (t) = Au(t), for alit E [0,T]; u(0) = x. This solution is given by u(t) = e tA x, for all t E [0, T]. Proof. It is clear that e tA x is a solution therefore, we need only show uniqueness. Let v be another solution and let z(t) = e -tA V(t). We have

z '( t ) = e -tA (Av(t)) - A(e -tA V(t)) = 0. Therefore, z(t) __ z(0) = x; and so v(t) = ex.

❑

1.3. Sobolev spaces We refer to Adams [1] for the proofs of the results given below. Consider an open subset S2 of R N . A distribution T E D'(S2) is said to belong to LP(11) (1

= fn

f (x)cp(x) dx,

for all E D(52). In that case, it is well known that let p E {1, cc]. Define Wm'()

f

is unique. Let m E N and

= { f E Lp(1l), D& f E LP(Sl) for all a E N tm such that Ial < m }.

Wm , P(cl) is a Banach space when equipped with the norm defined by

IIfIlwm,p =

IID`fIILp,

for all f E W'r(Il). For all m,p as above, we denote by Wo '(S2) the closure of D(1l) in W'P(f ). If p = 2, one sets Wt ,2 (f) = H-(l), Wo ' 2 (1) = Ho (1l) and one equips Ht(SZ) with the following equivalent norm:

11A

_

IIDc'UIIL2 JaI<m

Sobolev spaces 3

Then Hm(1l) is a Hilbert space with the scalar product v)Hm

(u,

=

f

D a uD a v dx

IIm

If S2 is bounded, there exists a constant C(S2) such that

uIIL2 <_ C(cl)IIVUIIL2, for all u E Ho (S2) (this is Poincare's inequality). It may be more convenient to equip Ho (1l) with the following scalar product

(u, v) =

in

Vu • Vv dx,

which defines an equivalent norm to on the closed space HH(SZ). The following two results are essential in the theory of partial differential equations. Theorem 1.3.1. If S2 is open and has a Lipschitz continuous boundary, then: (i) if 1 < p < N, then W 1 'P(IZ)

Lq(fl), for every q E [p,p *], where p* _

Npl (N — p); (ii) if p = N, then W 1 'P(S2) y L 9 (1l), for every q E [p, oo); (iii) if p> N, then Wr'r(cl) ---> L(c) fl C°' 11 (52), where a = (p — N)/p.

Theorem 1.3.2. In addition, if 11 is bounded, embeddings (ii) and (iii) of Theorem 1.3.1 are compact. Embedding (i) is compact for q E [p,p*). Remark 1.3.3. The conclusions of Theorems 1.3.1 and 1.3.2 remain valid without any smoothness assumption on 52, if one replaces W 1 'P(1l) by WW'P(c) We also recall the following result (see Friedman [1], Theorem 9.3, p. 24). Theorem 1.3.4. Let q, r be such that 1 < q, r < oo, and let j, m be integers, 0 < j < m. Leta E [j/m,1] (a <1 if m — j — N/r is an integer >_ 0), and let p be given by

p

\1 m /

q

Then there exists C(q, r, j, m, a, n) such that

a

IIDauIIL1 IcI=7

for every u E D(II81`')


IkI='m

IIDauIIL-

IIuIIL9a,

4 Preliminary results

Finally, we recall the following composition rule (see Marcus and Mizel [11). Proposition 1.3.5. Let F : R - IR be a Lipschitz continuous function, and let 1

F'(u)Vu, if u V N; if u E N;

0,

almost everywhere in Q. In particular, we have the following result. Corollary 1.3.6. Let 1 < p < oo. For all u E W 1 'P(Sl), we have u+, u - , Jul E W 1 'P(S2). Moreover, V(u+) _Vu, if u > 0; {0,

ifu<0;

almost everywhere. 1.4. Vector-valued functions We present here some results on vector integration and vector-valued distributions that will be useful throughout this book. We consider a Banach space X and an open interval I C R. 1.4.1. Measurable functions Definition 1.4.1. A function f : I ---> X is measurable if there exists a set E C I of measure 0 and a sequence (f n ) n,>o C Cc (I, X) such that f(t) —> f (t) asn—^oo, for alit EI\E. Remark 1.4.2. If f : I —> X is measurable, then Iii f 11 : I --# R is measurable. Proposition 1.4.3. Let (ff )>o be a sequence of measurable functions I —> X and let f : I --+ X be such that ff (t) —> f (t) as n —> oo, for almost all t E I. Then f is measurable. Proof. f,,, —> f on I \E with JEl = 0. Let (fn,k)k>o be a sequence of continuous functions with compact supports such that fn ,k —> f, almost everywhere as k —> oo. By applying Egorov's theorem to the sequence li fn ,k — fn iI, we obtain on the existence of E^,, C I with IEj < 2', such that fn ,k -# f

Vector-valued functions 5

I \En . Let k(n) be such that II fn,k(n) fnII < 1/n on I \En and let g n = fk(n). Take F = E U (f u En ) then IFI = 0. Let t E I \ F. We have f, (t) —* f (t); m>O n>m

on the other hand, for n large enough, t E I \E n . It follows that IIg n — fn I <1/n. Therefore, g(t) —> f (t) and so f is measurable. ❑ Remark 1.4.4. If f : I -- X and cp : I -4 R are measurable, then cp f : I --> X is measurable. Remark 1.4.5. If (xn)n>o is a family of elements of X and if (w n ) _ > 0 is a family of measurable subsets of I such that w i n w^ _ 0 for i j, then i n>o x1 is measurable. Proposition 1.4.6. (Pettis' Theorem) Consider f : I —i X. Then f is measurable if and only if the following two conditions are satisfied: (i) f is weakly measurable (i.e. for every x' E X*, the function t H (x', f (t)) is measurable); (ii) there exists a set N C I of measure 0 such that f (I \ N) is separable. Proof. First, since f is measurable, it is clear that f is weakly measurable. Now let (fn)n>o C Cc (I,X) be a sequence such that fn -i f on I\N as n --> oo, where INI = 0. It is clear that fn (I \ N) is separable, and then so is f (I \ N). Conversely, we may assume that f (I) is separable, so that X is separable (by possibly replacing X by the smallest closed subspace of X containing f (I)). We need the following lemma (see Yosida [1], p. 132).

Lemma 1.4.7. Let X be a separable Banach space, let X* be its dual, and let S* be the unit ball of X. There exists a sequence (x) > of of S* such that, for every x' E S*, there exists a subsequence (x'nk )k> o of (x)> with with xnk (x)—>x'(x) for all x c X. Proof. Let

(xn)n>o

be dense in X. For all n> 0, define Fn : S* -* 2 2 (n), by Fn(x ) = (x (x1) ...,x 1xn)), '

'

'

for all x' E X*. Since t 2 (n) is separable, there exists a sequence (xn k )k>o of S* such that Fn ((xn k )k> o ) is dense in Fn (S*). In particular, for all x' E X*, there exists xn E S* such that x' ( x .7) — x lk(,. ( x 7)I

n

1

6 Preliminary results

for 1 <j < n. It follows that x'nk(.) (xi) --^ x'(xi) as n --> oo, for all j E N. Since (x n ) n >o is dense in X, we deduce easily that xn k(n) (x) —* x'(x) as n --f oo, for all x E X. The result follows. End of the proof of Proposition 1.4.6. Let x E X. Then t H 1 f(t) — xli is

measurable. Indeed, for all a > 0, {t, IIf(t)—x11
Ilx'll<1 and it follows from Lemma 1.4.7 that

{t, 111(t) — x11 < a} = (1 {t, Ixn'(.f (t) — x)I < a}. n>O

Since the set on the right-hand side is clearly measurable, t F-4 11 1(t) — xli is measurable. Now consider n >_ 0. The set f (I) (which is separable) can be covered by a countable union of balls B(x) of centres xj and radius 1/n. Consider fn : I — X, defined by fn=Y"x,j1(✓;^

where wo = {t; f (t) E B(xo)} and wj = {t; f (t) E B(x 3 )} \ U B(xi). By

O
Remark 1.4.5, fn is measurable, and it is clear by construction that ii fn (t) — f (t)!1 < 1/n for all t E I. Therefore, it follows from Proposition 1.4.3 that f is measurable. Corollary 1.4.8. Let f : I —+ X be weakly continuous (i.e. if t o --> t, then f (t n ) — f (t) weakly in X). Then f is measurable.

Proof. By Proposition 1.4.6, it is sufficient to prove that f (I) is separable. Let A be the convex hull of f (I n Q) and let A be the weak closure of A. We have f (I) c A. On the other hand, A is also the strong closure of A and so A is separable. D Corollary 1.4.9. Let (fn ) n >o be a sequence of measurable functions I --p X and let f: I --> X. If, for almost every t E I, fn (t) — f (t) as n —> oo, then f is measurable.

Proof. Let x' E X*. Since (x', f(t)) ---* (x', f (t)) almost everywhere, the function t f--* (x', f (t)) is measurable, and so f is weakly measurable. On the other hand, for every n E N, there exists a set E n of measure 0 such that fn (I \ En ) is separable. Consider the set E = U E, IEI = 0. Let A n>1

be the convex hull of U f,,(I \ E) and let A be the weak closure of A. We n>1

_

have f (I \ E) C A. Furthermore, A is also the strong closure of A and so A is El separable. It follows that f is measurable.

Vector-valued functions 7

1.4.2.

Integrable functions

Definition 1.4.10. A measurable function f : I -> X is integrable if there exists a sequence (fn ) n >o C Cc (I, X) such that

f Ilfn(t) — f(t)II dt --, 0 asn ->oo. Remark 1.4.11. III f - f II is non-negative and measurable, and so fI IIfn - f I makes sense. Proposition 1.4.12. Let f : I - X be integrable. There exists x E X such that if a sequence (fn ) n > o C C,(I, X) satisfies fI IIfn - f II —4 0, as n - then one has fI fn —* x as n -* oo. Proof.. We have

f fn—^fP ) < f IIfn—fII+jIIf—ftI. Therefore, fl fn is a Cauchy sequence that converges to some element x E X. Consider another sequence (9 n ) n >o that satisfies f II9n - f II —> 0 as n --> oo. We have

II J7 9n Therefore,

—

xl

fl gn , x

as

II9n—A+LIIfn—fAI +I n -.

X. ❑

oo.

Definition 1.4.13 The element x constructed above is denoted by f f, or fI f. If I = (a, b), it is also denoted by fa f. As for real-valued functions, it is convenient to set lb a f= - f ab f. Proposition 1.4.14. (Bochner's Theorem) Let f : I -4 X be measurable. Then f is integrable if and only if 111 11 is integrable. In addition, we have

If

_ fII. .

Proof. Assume that f is integrable and consider a sequence (fn)n>o C C^(I, X)

such that f If — I ll --> 0. We have II f 1I < IlfniI + Ilfn — f 1I; and so II f I1 is

integrable.



8 Preliminary results

Conversely, suppose that II f I is integrable. Let gn E C,(I, R) be a sequence such that gn — J^f 1I in L'(I) and such that g, <_ g almost everywhere, for some g E L 1 (I). Let (fn ) n >o C C,(I, X) be a sequence such that fn —* f almost everywhere. Finally, let

I f'

un

I(n+ n fn

We have (u n I1 <_ g E L'(I) and u n — f almost everywhere. Therefore, we have fl Ilan — fII — 0 and so f is integrable. Finally, it follows from Fatou's lemma that

YI f I < lim J u, < lim J IIun i1 < f n_00

n-.cx

LI

which completes the proof.

Corollary 1.4.15. (The Dominated Convergence Theorem) Let (fn ), >o be a sequence of integrable functions I --> X, let g : I —> IR be an integrable function, and let f : I — X. If

for all n E N, 1 fn II < g, almost everywhere on I, f(t)—f(t) as n —> oo, for almost all t E I, then f is integrable and f1f = lim fI fn . n o0

1.4.3.

The spaces LP(I,X)

Let p E [1,J. One denotes by LP(I,X) the set of Definition 1.4.16. (equivalence classes of) measurable functions f : I —* X such that t --> I ff (t)II belongs to LP(I). For f E LP(I, X), one defines

If (IL

if p < oo; if p= cc. P = NsSUPtEIII A0111

^I f (t)jjP dt) ,

Proposition 1.4.17. (LT'(I,X)j I • ^^LP) is a Banach space. If p < oo, then D(I,X) is dense in LP(I,X).

Proof. The proof is similar to that of the real-valued case (in particular, the density of D(I, X) is obtained by truncation and convolution). ❑ Remark 1.4.18. Let f E LP(I,X) and let g E LP (I,X*). Then t

(g(t),f(t))x•,x

Vector-valued functions 9

is integrable and

I

J I (g(t), f (t))X*,X I

(If IILpIIJIILp'

The following result is related to the preceding remark. The proof is much more difficult than that for real-valued functions. Theorem 1.4.19. If 1 < p < oo and if X is reflexive or if X* is separable, then (LP(I,X))* Lp (I, X*). In addition, if 1

❑

Remark 1.4.20. If I is bounded and if 1 < q

L9 (I, X).

Definition 1.4.21. Let 1 < p < oo. We denote by L , (I, X) the set of measurable functions f : I —* X such that for every compact interval J C I,

fj,^ E LP(J,X).

Proposition 1.4.22.

Let X and Y be Banach spaces and let A E L(X,Y). if f E LP(I,X), then Af E LP(I,Y) and IlA.fIlLp C HAIjc(x,y)Ilf1ILP. If.f E L'(I,X), then A(fi f) = f1 Af•

1 I

Proof.

First, assume that p < oo. The result is well known for f E D(I, X), and the general case follows from a density argument (Proposition 1.4.17). If p = oc, it is clear that Af is measurable, and that, for almost all t E I,

IIAf(t)II

(jAIlL(x ,Y)Af(t)II C tAUL(X,Y)MfjjL° l 0

hence the result.

❑

1

Corollary 1.4.23. If X yY and if f E L 1 (I,X), then f E L'(I,Y) and the integrals of f in the sense of X and Y coincide. Proposition 1.4.24. Let 1 < p < oo. Let (f),,,>o be a bounded sequence of LP(I,X) and let f : I —; X be such that f(t) — f (t) weakly in X as n --> for almost alit El. Then f E LP(I,X), and I f II LP < lim of 1lfn11LN. Proof.

By Corollary 1.4.9, f is measurable. We define g,,, and g by g(t) = inf Ifk(t)I) k>n g(t) = lim g(t) g(t) = l im in nf I f,,, (t) II

almost everywhere.

I

10 Preliminary results

Since gn (t) <_ f.(t) 1 almost everywhere, it follows from the monotone convergence theorem that g is integrable and that j glILP = lim 11gn,jILP. By weak lower semicontinuity of the norm, we have 119r.JLP
IIfIILT(I.a) < .p,] L1 = u

El

hence the result. 1.4.4.

Vector-valued distributions

Definition 1.4.25. We denote by D'((I, X)) the space L(D(I), X). It is called the space of X-valued distributions on I. Remark 1.4.26. For the definition of D(I) and its topology, see Schwartz [1]. Remark 1.4.27. Let f E Li, C (I, X). Then

(Tf,) = ff for p E D(I), defines a distribution Tf E D'(I,X). We will sometimes denote by f the distribution Tf. Definition 1.4.28. Let T E D'(I, X). AVe define the distributional derivative

ofT,T'ED'(I,X),by (T', v) _

—

(T,'),

for cp E D(I). Proposition 1.4.29. Let 1 < p < oo and let f E L'(R, X). Let

1 Th.f ( t )

/' t+h

T1/t

f ( s ) d s,

for t E R and h 0. Then Ta f E LP(R, X) f Cb(R, X) and Th f — f in L"" (R, X) and almost everywhere as h —i 0. Proof. By the dominated convergence theorem, it is clear that Th f E C(R, X).

Furthermore, by Holder's inequality, we have 1 t+h

jITaf(t)II P < f

JIf(s)jj''ds.

Vector-valued functions 11

Hence

1

t}h

11f(s)IIIdsdt

j IIThf(t) IIPdt < f Jt h

Ilf(t)IIIdtds<

f Ilf(t)II P dt.

It follows that Th E C(LP(R,X)) and that IlThjI <_ 1. Let Ah = Th - I. One has I Ah II L(LP) < 2. Let (fn ),I> i C D(R, X) be a sequence such that fn --> f of -> ac in LP (R, X) and on IR \ E, with IEI = 0 (such a sequence exists by Proposition 1.4.17). Let t E R \ E. We have

IAhf(t)II < IIAh(f(t) — f^(t))II + IJAhfn(t)II 1

/ t+h

I11(t) — f(t)II + h J

^

t

IIf(s) — fn(s)IJ ds + IIAhfn(t)[I

Let : > 0. For n large enough, one has II f (t) - f,(t)II < E/4. On the other hand, since II f O - fn(•)II E L o^(R), by the theory of Lebesgue )Dints (see Dunford and Schwartz [1], p. 217, Theorem 8) we know that 1

IIf(s) — f(s)IIds—> Ilf(t)

j

fn(t)II,

for almost all t E R, as h -> 0. Therefore, for almost all t, n being fixed so that f (t) - f(t)II < E/4, and if h is small enough, we have

J

1 f t}h t

Ilf(s) - fn(s)II ds <e/2.

Finally-, since ff E D(R, X), for h small enough, we find that IIAhfn(t)IILP < E/4.

Therefore, for almost all t and for h sufficiently small, we have IIAhf(t)IILP < E._ Taking E = 1/n, we obtain IIAhf(t)IILP ;< 1/n if h is small enough, for all t E JR \ En , where the measure of E n is 0. It follows that Thf --4 f as h -3 0, for all t e R \ U E n , i.e. almost everywhere. Furthermore, we have n>1

II Ahf II LP(R,X) <_ 211f — fn II LP(R,X) + I [Ahf^.Il LP(R,X)'

Given any n, it is well known that II Ah ffl II r P ( R , x ) -' 0 as h -*0; it follows that II A h f IIL (i,x) -+ 0, which completes the proof. ❑ „

Corollary 1.4.30. Let f E Li ;(I, X) be such that f = 0 in D'(I, X). Then f = 0 almost everywhere.



12 Preliminary results

Proof. First, we remark that if J is a bounded subinterval of I, we have f, f 0. Indeed, let (n)n>1 C D(I), V,, < 1, and con --> 1j almost everywhere. We have

f

J

f = lim

f

fw = lim(f, ca) = 0.

Then fix a bounded subinterval J C I and consider

{

f E L i (IR, X), defined by

J(t) (t) = f (t), if t E J, J(t)=0, ift¢J.

It follows that Th f = 0 for all h > 0. By Proposition 1.4.29, we obtain f = 0 almost everywhere. Therefore, f = 0 almost everywhere on J. Since J is arbitrary, we have f = 0 almost everywhere. ❑ Corollary 1.4.31. Let g E L (I, X), to E I and = ft. 9(s) ds. Then:

1(t)

let f E C(I, X) be given by

(i) f` = gin D'(I, X);

(ii) f is differentiable almost everywhere and f' = g almost everywhere. Proof. Reasoning as before, we may restrict ourselves to the case I = IR and gEL 1 (R,X). We have Thg(t) = f(t+h)

-

f(t)

h

By Proposition 1.4.29, we deduce (ii). Now consider cp E D(R). We have

(f',

w) _ -(f, co') = -

f

f (t)w'(t) dt.

Furthermore,

p( t + h) — P(t)

cp'(t) uniformly on R, as h —. 0.

Therefore,

`,

_ — lim. = — li o

f

f (t) c(t + h) — co(t) h co(t) f (t(t)

by Proposition 1.4.29; hence (i).

_ i

o

JR T hgco = (g, co),

Vector-valued functions 13

Proposition 1.4.32. Let T E D'(I, X) be such that T' = 0. Then there exists z,o E X such that T = xo, i.e. (T, ^) = xof, I

for all tpED(I). Proof. Let 0 E D(I) be such that f6 = 1, and let xo = (T, 6). ' Let (a, b) be the support of 0 and let to E I, to < a. Now consider cp E D(I). We define

D(I)by (t)

(V(s) - 6(s) f

=

) da) ds,

I

to

for alltEI. We have \ I

).

Hence

/'

0 = (T,') _ (T,) — xo

Jr

.

It follows that (T,) _ xo

J ^; I

hence the result. 1.4.5.

❑

The spaces W' P(I, X)

Definition 1.4.33. Let 1

sflnse of D'(I, X). For f E W' P(I, X), we s et ILf Ilwl,p = II fJI LP + If IILp,

Proposition 1.4.34. (W 1 °P(I,X),II • (I w f,P) is a Banach space. Proof. The proof is similar to that of the real case. Theorem 1.4.35. Let 1

(I) f E W1,P(I X);,.'. (ii) there exists g E LP(I, X) such that for almost all to, t E I, we have f (t) =;

f(to)+f o g(s)ds; iii) there exists g E LP(I,X), xo E X, and to E I such that we have f (t) = xo + fro g(s) ds, for almost all t E I;

14 Preliminary results

(iv) f is absolutely continuous, differentiable almost everywhere, and f' belongs to LP(I,X); (v) f is weakly absolutely continuous, weakly differentiable almost everywhere, and f' belongs to LP(I, X). Proof. (i) .(ii)` Let t o E I. For any t E I, set

w(t) _ .f (t) — .f (to) — Lt .f'(s) ds. We have w E C(I,X) and w' = 0 in D'(I, X) by Corollary 1.4.31. Therefore, there exists X o E X such that w = xo in D'(I„X) (Proposition 1.4.32). Since w(0) = 0, it follows from Corollary 1.4.30 that w = 0 almost everywhere; hence (ii). (ii)(iii) is immediate. (iii)^(iv). By possibly modifying f on a set of measure 0, we may assume that

1(t) = xo + f g(s) ds,

to for all t E I, and we apply Corollary 1.4.31. (iv)(v) is immediate. (v)=(i). Let g be the weak derivative of f. Let t o E I, and set t

(t) = f(t) — f(to) - f g(s) ds, to

for all t E I. By Corollary 1.4.31, cp is differentiable almost everywhere and its derivative is 0 almost everywhere. Let x' E X* and let zb be defined by ib(t) = (x', p(t)). Ali is absolutely continuous, differentiable almost everywhere and ,b'(t) = 0 almost everywhere. Since 0(to) = 0, we obtain 0(t) - 0. Since x' is arbitrary, we conclude that co(t) - 0. Hence, (i) follows from Corollary 1.4.31. ❑

w Corollary 1.4.36. Let 1 < p < oo. Then W I"P(I, X)

I

Proof. We have

Ilf(t) - f(s)I1

I

1''

sf

s

t

11f'(o )II da, ,

for all s, t E I. Hence we have uniform continuity. Furthermore, if we set h(.) = IIf( . )II, we have !h(t) — h(s)I : Ilf(t) — f(s)II.

Cb, v, (I, X ).

Vector-valued functions 15

/^ t

It follows that I h(t) – h(s)I <

J3 IIf'(^)II dam,

for all s, t E I. Therefore, h is absolutely continuous and we have Ih' < II f'II E which completes L 5 (I) almost everywhere. We obtain h E W 1 'P(I) ^-+ L°O(I), the proof. Corollary 1.4.37. III is bounded, then C°° (7, X) is dense in W 1' ' (I, X ). be such that g, ---> f' in Proof. Let f E W''r'(I,X) and let (g n ),,>1 C D(I,X) LP(I,X). Let to el, and set fn(t) = .f (to)

+ f tgn(s) ds. t.

in W 1 'T'(I,X), as It is now easy to verify that fn E C °O (I,X) and that f,, —f T1 - 4 00.

a (I,X), with Corollary 1.4.38. Let 1 < P < oo. Then W 1 'P(I,X)` ----> C°'

a = (p – 1 )/p Proof. By Holder's inequality, we have

IIf(t+h) -f(t)jj
(

j ,

t+h

1p

jjf'(s)jjPds {< ha ^If'fILr; / 0

hence the result.

< p < oo and let f E Corollary 1.4.39. Assume that I = (a, b). Let 1 we have W1'P(I,X). Then, for all CE I,

f( .+h) -f( .)

f^

h

in LP((a,c),X), as h 1 0. and we apply PropoProof. We extend f on lib by a function f E W r 'P (R, X) sition 1.4.28 and Theorem 1.4.35.

11

p < oo and let f E Theorem 1.4.40. Suppose that X is reflexive. Let 1 '< exists cp E Lr(I,IR) such if there only if and LP(I,X). Then f E W l 'P(I,X) that T

IIf(T) - f(t)1I <— t f p(s)ds for almost all t,T E I. In that case, we have II f'II LP(r,x) II F^ILn(I,n2)

(1.1)

16

Preliminary

results

Proof. It is clear that this condition is necessary (for example, take p() _ Conversely, since X is complete, we easily verify that we can modify f in a set of measure 0 so that (1.1) holds for all t -r E I. In particular, f is continuous and we may consider only the case in which I = R. Since f is continuous, it is clear that f (R) is separable. By possibly considering the smallest closed subspace of X, we restrict ourselves to the case in which X is reflexive and separable, and so X'' is separable. For all h > 0, set

fh(t) = f(t + h) - f(t) h If p = oc, it is clear that fh is bounded in LP(IE€,X). If p < oe, by Holder's inequality, we have t+h

(s)I'ds.

lifh(t)lip < T I

Integrating on R, we find that 1r ]

R

rt+h

fh(t) <-J J h

t

((s)I r dsdt=

1r r5 J j

h

a-h

w(t)V'dtds

(1.2)

_ f(s)ds. Therefore fh is bounded in LP(R, X). Let (xn ) be dense in X*. For all n, the function yhn (t) = (x; , f (t)) satisfies ^ n( T ) — n ( t ) <

xn (

Jt

p(s) ds .

is absolutely continuous, and so differentiable on H \ E n , with In particular, =0. Let E= U E. We have = 0, and for all t e H\E and nEN, n>1

we have (fh(t), x') --> n(t), as h -* 0. Let F be the complement of the set of Lebesgue's points of (we know Fl = 0 ). By (1.1), for all t E H \ F, we have fh(t)M < K(t) < oo, if Ih is small enough. We claim that for all t E hR \ (E U F), there exists w(t) e X such that fh(t) — w(t) as h -> 0. Indeed, ^l fh(t) ( is bounded, and since X is reflexive, there exists a sequence h„ — 0 and w(t) E X, such that fh„ (t) — w(t) weakly in X as n -- x. In particular, we have (w(t), x tn ) = V'(t), for all n E N. Since the sequence (x)>i is dense in X*, w(t) does not depend on the sequence h n ,

and so fh(t) — w(t). By Proposition 1.4.24, we have w E LP(R,X), and (W( LP ( I.X ) < 'p( LP(I,&) . By Theorem 1.4.35(v) and Corollary 1.4.31, we have = w. ❑ f E W"P(R. X) and f'

Vector-valued functions 17

From this, we immediately deduce the following result Corollary 1.4.41. Assume that X is reflexive. Let f : I —^ X be bounded, Lipschitz continuous with Lipschitz constant L. Then, f E W(I,X) and

< L.

I

Corollary 1.4.42. Assume that X is reflexive and that 1 < p < co. Let (fn ),a >o be a bounded sequence in W"r(I, X) and let f : I —^ X be such that fn (t) — f (t) as n —> oc, for almost every t E I. Then f E W i ^P(I, X), and

M.f'M Lr(r,.x) < liiminf (f,'ALF (1,X). ,

I

Proof. By Proposition 1.4.24, we have f E LP(I, X). Let, E be a set of measure , for all t E I \ E. For all t, r C I \ E, we 0 such that fn (t) — f (t) as n

have

< 1 im of ))fn(t) - fn( T )11 < u rninf f Ji f(s)^I ds.

)J ( t ) - f()

Consider = )f). y?n is bounded in LP(I) and so there exists a subsequence (k )k>1 and p E LP(I) such that :p nk — ^p weakly in LP(I) (weak-* if p = oc) L In particular, we have as k — oo, and litn inf ^^ ^p,^ k ^1 Ln — lim inf n--oo

k—oc

t

(1.3)

t

J

4

k

(s) ds —>

J p(s) ds, as k --^ oo,

(1.4)

t



for all t, r E I, and ((^F))Lv < n—+oo liminf))fn(ILr,.

(1.5)

It follows from (1.3) and (1.4) that §f(t)—f(T)11 <_i p(s)ds, "

t

for all t, r E I; therefore f E W l 'P(I, X) (Theorem 1.4.40). We complete the proof by applying (1.5). ❑ Notes. For §1.1 and §1.2, consult Brezis [2], Dunford and Schwartz [2], and Yosida [1]. For §1.3, see Adams [1], Bergh and Lofstrom [1], Brezis [2], and Gilbarg and Trudinger [1], and for §1.4, see Dinculeanu [1], Dunford and Schwartz [1], and the appendix of Brezis [1].

2 m-dissipative operators Throughout this chapter, X is a Banach space, endowed with the norm

II • II

2.1. Unbounded operators in Banach spaces Definition 2.1.1. A linear unbounded operator in X is a pair (D, A), where D is a linear subspace of X and A is a linear mapping D — X. We say that A is bounded if there exists c > 0 such that IIAu for all u E {x E D.

II <_ c,

IIrII < 1}. Otherwise, A is not bounded

Remark 2.1.2. Note that a linear unbounded operator can be either bounded or not hounded. This somewhat strange terminology is in general use and should not lead to misunderstanding in our applications. Remark 2.1.3. If A is bounded, A is the restriction to D of an operator A E (Y, X), where Y is a closed linear subspace of X, such that D C Y. If A is not bounded, there exists no operator A E G(Y,X) with Y closed in X and D C Y, such that AID = A. Definition 2.1.4. Let (D, A) be a linear operator in X. The graph G(A) of A and the range R(A) of A are defined by

G(A)={(u,f)EXxX;uED and f=Au}, R(A) = A(D). G(A) is a linear subspace of X x X, and R(A) is a linear subspace of X Remark 2.1.5. In this chapter, a linear unbounded operator is just called an operator where there is no risk of confusion. As usual, we denote the pair (D, A) by A with D(A) = D, meaning the domain of A is D. Note, however, that when one defines an operator, it is absolutely necessary to define its domain. Remark 2.1.6. When D(A) = X, it follows from Theorem 1.1.2 that A E G(X) if and only if G(A) is closed in X. More generally, for not bounded operators, it is very useful to know whether or not the graph is closed.

Definition and main properties of m-dissipative operators 19

2.2. Definition and main properties of m dissipative operators -

Definition 2.2.1. An operator A in X is dissipative if

I^u - AAull > Ilull, for all u e D(A) and all A > 0. Definition 2.2.2. An operator A in X is m-dissipative if (i) A is dissipative; (ii) for all A > 0 and all f E X, there exists u E D(A) such that u — AAu = f. Remark 2.2.3. If A is in-dissipative in X, it is clear, from Definitions 2.2.1 and 2.2.2, that for all f E X and all A > 0, there exists a unique solution u of the equation u — AAu = f. In addition, one has I ull < I f 1I . Definition 2.2.4. Let A be an m-dissipative operator in X and A > 0. For all f E X, we denote by Jaj or by (I — AA) 'f the solution a of the equation u — AAu= f. -

Remark 2.2.5. By Remark 2.2.3, one has Ja E L(X) and IJajI c ( x ) < 1. Proposition 2.2.6. Let A be a dissipative operator in X. The following properties are equivalent. (i) A is m-dissipative in X; (ii) there exists Ao > 0 such that for all f E X, there exists a solution u E D(A) ofu—A0Au= f. Proof. It is clear that (i)^(ii). Let us show that (ii)^(i). Let A > 0. Note that the equation u — AAu = f is equivalent to

U-A O Au=

A°f+(1- A° )u.

Since A is dissipative and R(I — \ o A) = X, the operator Ja 0 = (I — A0A) -1 can be defined as in Definition 2.2.4. This operator is a contraction on X. Next, note that the preceding equation is also equivalent to Ao u=J50 L0 f+(1_ )u) (

.

If 2A > A 0 , this last equation is u = F(u), where F is Lipschitz continuous x--X, with a Lipschitz constant k = I(A — A0)/AI < 1. Applying Theorem 1.1.1, there exists a solution u of u — AAu = f, for all A E (A0/2,).

20 m-dissipative operators Iterating this argument n times, there exists a solution for all A E (2 "`Ao, oo), ❑ n > 1. Since n is arbitrary there exists a solution for all A > 0. -

Proposition 2.2.7. If A is m- dissipative, then G(A) is closed in X.

Proof. Since J1 E £(X), G(J1) is closed. It follows that G(I - A) is closed, and so G(A) is closed.

❑

Corollary 2.2.8. Let A be an m- dissipative operator. For every u E D(A), let IILIID(A) = IIulI + IIAull. Then (D(A), II - II D(A)) is a Banach space, and

A E 12(D(A), X).

Remark 2.2.9. In what follows, and in particular in Chapters 3 and 4, D(A) means the Banach space (D(A), II - IID(A))• Proposition 2.2.10. If A is m-dissipative, then li

o II

Jau - ull = 0 for all

uED(A). Proof. We have IIJ) -

III <2, and by density we need only consider the case

u E D(A). We have

Jau-u=Ja(u-(I-AA)u); and so lIJ\ u — ulI


❑

Definition 2.2.11. Let A be an m-dissipative operator. For A > 0, we denote by A), the operator defined by Aa=AJ)=

Ja-I

We have Aa E G(X) and IIAaIIc(x) 5 2/A. Proposition 2.2.12. If A is m-dissipative and if D(A) = X, then A ),u ---> Au as A 0 for all u E D(A).

Proof. Let u E D(A). By Proposition 2.2.10, one has JAAu-Au—*0 asAJ0. On the other hand, it follows easily from Definition 2.2.11 that

Aau = J)Au. Thus, hence the result.

IIAau - AuII = IIJ)Au - AuII --30 as A j 0; ❑

Extrapolation 21

2.3. Extrapolation In this section, we show that, given an m-dissipative operator A on X with dense domain, one can extend it to an m-dissipative operator A on a larger space X. This result will be very useful for characterizing the weak solutions in Chapters 3 and 4. Proposition 2.3.1. Let A be an m-dissipative operator in X with dense domain. There exists a Banach space X, and an m-dissipative operator A in X,

such that

(i) X '—

with dense embedding;

(ii) for all u E X, the norm of u in X is equal to liJlulI; (iii) D(A) = X, with equivalent norms; (iv) Au = Au, for all u E D(A). In addition, X and A satisfying (i)-(iv) are unique, up to isomorphism. Proof. For u E X, we define IMuIIt = I^JluII. It is clear that III - III is a norm on X. Let X be the completion of X for the norm . X is unique, up to an isomorphism, and X '— X, with dense embedding. On the other hand, observe

that J1Au = Jlu — u, du E D(A). Thus,

IIIAuIII <- IMuIII + huh <- 211u11, Vu E D(A).

Hence, A can be extended to an operator A E £(X, Y). We define the linear operator A on X by

D(4)=X, Au=Au, VuED(A). It is clear that A satisfies (iii) and (iv). Now, let us show that A is dissipative. Take X> 0. Let u E D(A) and let v = Jl u. One has

v — .XAv = Ji(u — ,Au) Since A is dissipative, it follows that IIIu — AuIII = liv — AAvII > IIvli = IIIuIIl )

By continuity of A, we deduce that ^IIv — AAuJIi ? lhIUJI1, `du E X;

22 m-dissipative operators and so A is dissipative. Finally, let f E X, and (f>o C X, with f,,, ---> f in X as n , oo. Set u n = J1 f,. Since (fn ) n,> o is a Cauchy sequence in X, (un,)n >o is also a Cauchy sequence in X; and so there exists u E X, such that u n --p u inXasn --*oo. We have

f = u n — Aun = un — Au n . ..

Since A E £(X,Y); it follows that f = u — Au = u — Au. Hence A is mc dissipative. The uniqueness of A follows from the uniqueness of A. Corollary 2.3.2. Ifs E X is such that Ax E X, then x E D(A) and Ax = Ax.

Proof. Let f = x — Ax E X. Since A is m- dissipative, there exists y E D(A) such that y — Ay = f. By Proposition 2.3.1(iii), we have (x — y) — A(x — y) = 0, ❑ = y. and since A is dissipative, we obtain x 2.4. Unbounded operators in Hilbert spaces Throughout this section, we assume that X is a Hilbert space, and we denote by (•, •) its scalar product. If A is a linear operator in X with dense domain, then

G(A*) = {(v, go) E X x X; (cp, u) = (v, f) for all (u, f) E G(A)}, defines a linear operator A* (the adjoint of A). The domain of A* is

D(A*) = {v E X, 3C < oo, ((Au, v)I < and A* satisfies

du E D(A)},

(A*v, u) = (v, Au), `d E D(A),

Indeed, the linear mapping u H (v, Au), defined on D(A) for all v E D(A*), can be extended to a unique linear mapping cp E X' X, denoted by cp = A*v. It is clear that G(A*) is systematically closed. Finally, it follows easily that if B E £(X), then (A + B)* = A* + B *. 1

Proposition 2.4.1. (R(A)) = {v E D(A*); A*v = 0}.

(0,v) E G(A*). (v,Au) = 0,Vu E D(A) This last property is equivalent to v E D(A*) and A*v = 0; hence the result. ❑ Proof. One has v E (R(A))

Proposition 2.4.2. A is dissipative in X if and only if (Au, u) < 0, for all

u E D(A).

Unbounded operators in Hilbert spaces 23

Proof. If A is dissipative, one has —2A(Au, u) + A 2 IIAu1I 2 = IIu — AAuII 2 — IIUII 2 > 0, VA > 0, Vu E D(A). Dividing by A and letting A j 0, we obtain (Au, u) < 0, for all u E D(A). Conversely, if the last property is satisfied, then for all A > 0 and u E D(A) we have IIu — AAuII 2 = IIuII 2 — 2A(Au, u) + A 2 IIAuII 2 > IIull 2

,

and then A is dissipative.

❑

Corollary 2.4.3. If A is m-dissipative in X, then D(A) is dense in X.

Proof. Let z E (D(A)) l , and let u = Jjz

E

D(A). We have

0 = (z, u) = (u — Au, u). Hence,

IIuII2 = ( Au, u) < 0.

It follows that u = z = 0; and so D(A) is dense in X.

❑

Corollary 2.4.4. If A is m-dissipative in X, then

J,u -- u as A j 0, for all u E X and AAu --* Au as A j 0, for all u E D(A). Proof. We apply Corollary 2.2.3 and Propositions 2.2.10 and 2.2.12.

❑

Theorem 2.4.5. Let A be a linear dissipative operator in X with dense domain. Then A is m-dissipative if and only if A* is dissipative and G(A) is

closed. Proof. If A is m-dissipative, then G(A) is closed, by Proposition 2.2.7. Let us show that A* is dissipative. Let v E D(A*). We have (A*v, Jay) _ (v, AJav) = (v, AAv) _ (v,JA v—v)= {(v,Jav)—

IIvII2} <0.

24 m-dissipative operators Since (A*v, Jav) -->(A*v, v) as A J 0, it follows that A* is dissipative. Conversely, since A is dissipative and G(A) is closed, it is clear that R(I — A) is closed in X. On the other hand, by Proposition 2.4.1, one has (R(I — A)) = {v E D(A*); v — A*v = 0} = {0}, since A* is dissipative. Therefore R(I — A) = X, and A is m- dissipative, by 0 Proposition 2.2.6. Definition 2.4.6. Let A be a linear operator in X with dense domain. We say that A is self-adjoint (respectively skew-adjoint) if A* = A (respectively A* = —A). Remark 2.4.7. The equality A* = +A has to be taken in the sense of operators. It means that D(A) = D(A*) and A*u = ±Au, for all u E D(A). Corollary 2.4.8. If A is a self-adjoint operator in X, and if A < 0 (i.e. (Au, u) <0, for all u E D(A)), then A is m- dissipative. Proof. By Proposition 2.4.2, A is dissipative. Since A* = A, A* is dissipative. Finally, G(A*) is closed, so that G(A) is closed. We finish the proof by applying Theorem 2.4.5. 0 Corollary 2.4.9. If A is a skew-adjoint operator in X, then A and —A are m- dissipative.

Proof. Let u E D(A). One has (Au, u) = (u, A*u) = —(u, Au). Hence (Au, u) = 0. It follows from Proposition 2.4.2, that A and —A are dissipative. We conclude as in Corollary 2.4.8. El Corollary 2.4.10. Let A be a linear operator in X with dense domain, such that G(A) C G(A*) and A < 0. Then A is m- dissipative if and only if A is self-adjoint. Proof. Applying Corollary 2.4.8, we need only show that if A is m- dissipative then A is self-adjoint. Let (u, f) E G(A*), and let g = u — A*u = u — f. Since A is m- dissipative, there exists v E D(A) such that g = v — Av, and since G(A) C G(A*), we have v E D(A*) and g = v—A*v. Therefore (v—u)—A*(v—u) = 0 and since A* is dissipative (Theorem 2.4.5), we obtain u = v. Thus, (u, f) E G(A); and

soA=A*.

El

Complex Hilbert spaces 25 Corollary 2.4.11. Let A be a linear operator in X with dense domain. Then A and —A are m-dissipative if and only if A is skew-adjoint.

I

Proof. Applying Corollary 2.4.9, it suffices to show that if A and —A are mdissipative, then A is skew-adjoint. Applying Proposition 2.4.2 to A and —A, we obtain (Au, u) = 0, Vu E D(A). For all u, v E D(A), we obtain (Au, v) + (Av, u) = (A(u + v), u + v) — (Au, u) — (Av, v) = 0. Therefore G(—A) C G(A *). It remains to show that G(AC *) G(—A). Consider (u, f) E G(A*) and let g = u — A*u = u — f. Since —A is m-dissipative, there exists v E D(A) such that g = v + Av, and since G(—A) C G(A*), we have v E D(A*) and f = v — A*v. Hence (v — u) — A*(v — u) = 0 and since —A* is dissipative (Theorem 2.4.5), we obtain u = v. Therefore, (u, f) E G(A*); and so A = —A*. ❑ 2.5.

I

Complex Hilbert spaces

In this section, we assume that X is a complex Hilbert space. Recall that by definition X is a complex Hilbert space provided that there exists a continuous R-bilinear mapping b: X x X —> C satisfying the following properties: b(iu, v) = ib(u, v),

V(u, v) E X x X;

b(v, u) = b(u, v),

V(u, v) E X x X;

b(u, u) = ^lull 2 ,

1

Vu E X.

In that case (u, v) = Re(b(u,_ v)) defines a (real) scalar product on X. Equipped with this scalar product, X is a real Hilbert space. In what follows, we consider X as a real Hilbert space. Let A be a linear operator on the real Hilbert space X. If A is C-linear, we can define iA as a linear operator on the real Hilbert space X. Proposition 2.5.1. Assume that D(A) is dense and that A is C-linear. Then A* is C-linear, and (iA)* = —iA*. Proof.

Let v E D(A), f = A*v and let z E C. For all u E D(A), we have

I

(zf, u) = (f, u) = (v, A(u)) = (v, zAu) = (zv, Au). Therefore zv E D(A*) and zf = A(zv). Hence A* is C-linear. In addition, (—if, u) = (v, A(iu)) = (v, iAu),

I

aw

1

26 m-dissipative operators for all (v, f) E G(A*) and all u E D(A); and so G(—iA*) C G ((iA)*). Applying this result to iA, we obtain G(—i(iA)*) C G ((i • iA)*) = G(—A*). It follows that G ((iA)*) C G(—iA*), and so G ((iA)*) = G(—iA*).

❑

Corollary 2.5.2. If A is self-adjoint, then iA is skew-adjoint. Proof. (iA)* = —iA* = —iA.

❑

2.6. Examples in the theory of partial differential equations

1

2.6.1. The Laplacian in an open subset of RN: L 2 theory Let Sl be any open subset of R N , and let Y = L 2 (S2). We can consider either realvalued functions or complex-valued functions, but in both cases, Y is considered as a real Hilbert space (see §2.5). We define the linear operator B in Y by D(B) = {u E Ho(S2); Au E L 2 (cl)}; {

Bu = Au, du E D(B).

Proposition 2.6.1. B is m- dissipative with dense domain. More precisely, B is self-adjoint and B < 0. We need the following lemma. Lemma 2.6.2. We have

I

f

Vu•Vvdx. vAudx=— n Js z i

I

(2.1)

for alluED(B) andvEHo(5l). Proof. (2.1) is satisfied by v E D(l). The lemma follows by density, since both ❑ terms of (2.1) are continuous in v on Ho (S2). Proof of Proposition 2.6.1. First, D(S2) C D(B), and so D(B) is dense in Y. Let u E D(B). Applying (2.1) with v = u, we obtain (Bu, u) < 0, so that B is dissipative (Proposition 2.4.2). The bilinear continuous mapping b(u, v)

=J

(uv + Vu • Vv)dx

Examples in the theory of partial differential equations 27

is coercive in Ho(f ). It follows from Theorem 1.1.4 that, for all f E L 2 (Sl), there exists u E Ho (Sl) such that

J (uv+Du•Vv)dx = J fvdx, by E Ho(Q). We obtain u — Du= f,

in the sense of distributions. Since, in addition u E Ho (S2), we obtain u E D(B) and u — Bu = 1. Therefore B is m-dissipative. Finally, for all u, v E D(B), we have, by (2.1), (Bu, v) = (u, Bv). Therefore G(B) C G(B*), and by Corollary 2.4.10, it follows that B is selfadjoint.

o

Remark 2.6.3. If Sl has a bounded boundary of class C 2 , then D(B) = H 2 (1) fl Ho(cl), with equivalent norms (see Brezis [2], Theorem IX.25, p. 187, or Friedman [1], Theorem 17.2, p. 67). 2.6.2. The Laplacian in an open subset of RN: Co theory Let Sl be a bounded open subset of IR N , and let Z = L°°(Sl). We define the linear operator C in Z by D(C) = {u E Ho (cl) n Z, Au E Z}, Cu = Au, b'u E D(C). Proposition 2.6.4. C is m-dissipative in Z. Proof. First, let us show that C is dissipative. Let ,A > 0, f E Z, and let M = I f ^I L^ . Let u E Ho (f l) be a solution of

u — AAu= f, in D'(S2). In particular, this equation is satisfied in L 2 (S2), and we have

(u—M)—AA(u—M)=f—M, in L 2 (S1). On the other hand, v = (u — M)+ E Ho(1l), with Vv = 1 {1uI>M}Du (Corollary 1.3.6). Applying Lemma 2.6.2, we obtain

Jv

2

dx+w foul>M)

^DuI2dx= J(f—M)vdx_<0.

28 m-dissipative operators Therefore f v 2 dx < 0, and so v = 0. We conclude that u < M a.e. on Q. Similarly, we show that u >_ —M a.e. on Q. Hence u E L°°(f ), and 11 U 11 L°° < II f 1I L— • It follows that C is dissipative. Now let f E L°°(1) C L 2 (c ). By §2.6.1, there exists u E Ho(Sl), with Du E L 2 (1l), a solution of u — Au = f, in L 2 (S2). We already know that u E L(1l), so that u E D(C), and u—Cu = f. Therefore C is m-dissipative. ❑ Lemma 2.6.5. If S2 has a Lipschitz continuous boundary, then

D(C) C C0(l) = {u E C(S2); u 1 an = 0}. Proof. The proof is difficult, and uses the notion of a barrier function (see Gilbarg and Trudinger [1], Theorem 8.30, p. 206). ❑

Remark 2.6.6. It follows from Lemma 2.6.5 that in general the domain of C is not dense in Z. The fact that the domain is dense will turn out to be very important (see Chapter 3). This is the reason why we are led to consider another example. We now set X = C o (l), and we define the operator A as follows:

(D(A)= {uEXnH.()),AuEX }, SI Au = Au, Vu E D(A). Proposition 2.6.7. Assume that S2 has a Lipschitz continuous boundary. Then A is m-dissipative, with dense domain.

Proof. D(Sl) is dense in X, and D(SZ) C D(A); and so D(A) is dense in X. On the other hand, X is equipped with the norm of L°°(S2), and so X —+ Z and G(A) C G(C). Since C is dissipative, A is also dissipative. Now let f E X y L°° (f ). Since C is m- dissipative, there exists u E D(C), such that u — Au = 1. By Lemma 2.6.5, we have u E X, and so Au E X. Therefore, u E D(A) and u — Au = f. Hence A is m- dissipative. ❑ Remark 2.6.8. In the three examples of §2.6.1 and §2.6.2, note that the same formula (the Laplacian), corresponds to several operators that enjoy different properties (since they are defined in different domains). In particular, the expression the operator A has a meaning only if we specify the space in which this operator applies and its domain.

Examples in the theory of partial differential equations 29

2.6.3. The wave operator (or the Klein—Gordon operator) in Hp (S2) x L 2 (Q)

Let S2 be any open subset of RN, and let X = Ho (fl) x L 2 (1). We deal either with real-valued functions or with complex-valued functions, but in both cases X is considered as a real Hilbert space (see §2.5). Let A = inf {11VU1IL2,u E Ho(S2),

IIuII L 2 = 1}.

(2.2)

(In the case in which 5l is bounded, we recall that A is the first eigenvalue of —,L in Ho (S2), and that A> 0). Let m> —A. Then X can be equipped with the scalar product ((u, v), (w, z)) _ / (Du • Vv + maw + vz) dx.

1

This scalar product defines a norm on X which is equivalent to the usual norm. We define the linear operator A in X by D(A) = {(u,v) E X, Au E L 2 (S2),v E Ho(S2)}; A(u, v) = (v, Au — mu), V(u, v) E D(A). Proposition 2.6.9. A is skew-adjoint, and in particular A and —A are m-

dissipative with dense domains. Proof. D(1) x V(l) C D(A) and so D(A) is dense in X. On the other hand, for all ((u, v), (w, z)) E D(A)z, and by (2.1), we have (A(u, v), (w, z))

(Ov . Ow + mvw + (Du — mu)z) dx

=J = — (vu •Vz+muz+ (Aw — mw)v) dx

J

_ —((u, v), A(w, z))•

(2.3)

Applying (2.3) with (u, v) _ (w, z), it follows that (A(u, v), (u, v)) = 0. Hence A is dissipative (Proposition 2.4.2). Now let (f, g) E X. The equation (u, v) — A(u, v) _ (1,9) is equivalent to the following system: r2u—iu =f+g;

(2.4)

Slv=u— f.

(2.5)

o

By Proposition 2.6.1, there exists a solution u E H (Sl) of (2.4), satisfying L u E L 2 (1). Next, we solve (2.5) and we obtain v E Ho (52). Therefore (u, v) E D(A) and (u, v) — A(u, v) = ( f, g), so that A is m-dissipative. Similarly, we show that —A is m-dissipative. By (2.3), we have G(A) C G(—A*). Corollary 2.4.11 proves that A is skew-adjoint. ❑

1

30 m-dissipative operators

2.6.4. The wave operator (or the Klein-Gordon operator) in L 2 (Sl) x

H'(2)

I



Let S2 and m be as in §2.6.3. We recall that Ho (Q) ' L 2 (Q) '-+ (Ho (Q))' _ H-1(Q) with dense embeddings. We equip Ho (1) with the scalar product defined in §2.6.3. Theorem 1.1.4 shows that E Ho (s^), o^u — m = u in D'(sf)},

H '(r) = {u E D'(), 5 -

(2.6)

and that we can equip H '(Q) with the scalar product -

(u,v)-i =

J (v

0

.V +

Y = L 2 (Q) x H - '(St). We deal either with real-valued functions or with complex-valued functions, but in both cases X is considered as a real Hilbert space (see X2.5). We define the linear operator B in Y by D(B) = Ho(l) x L 2 (Q)^ {

B(u, v) = (v, Au - inn) E Y,

(u, v) E D(B).

Proposition 2.6.10. B is skew-adjoint. In particular, B and -B are m-

dissipative with dense domains. Proof. D(S2) x D(S2) C D(B) and so D(B) is dense in Y. Let ((u, v), (w, z)) E D(B)2, and consider cp„ and z defined by (2.6). Since v, z E L 2 (t1), we have E L 2 (Q). Applying (2.1), we obtain

v), (w, Z))L2 X H_ =

I

f vwdx+ (Au - mu, z) - i

J vw dx + J (Du • V(p + muc) ) dx = J vw dx - f u(A - m) dx

=

2

z

z

=

I

J vw dx - J uz dx.

Similarly, we have ((u,v),B(w,z)) t,2 XH -1 =

J zudx - J wvdx.

Therefore, (B(u,v),(w,x))L xH-1=

-

((u,v),B(w,z))L2xx-1.

(2.7)

Examples in the theory of partial differential equations 31

Applying (2.7) with (u, v) = (w, z), it follows that (B(u, u), (u, v)) = 0.

Thus, B is dissipative (Proposition 2.4.2). Now let (f, g) E Y. The equation (u, v) — B(u, v) = (f, g) is equivalent to the system (2.4)—(2.5) of §2.6.3. By Theorem 1.1.4 (see the proof of Proposition 2.6.1), there exists a solution u E Ho (f2) of (2.4). Next, we solve (2.5) and we obtain v E L 2 (fl). Therefore (u, v) E D(B) and (u, v) — B(u, v) = (f, g); hence B is m-dissipative. Similarly, we show that —B is m-dissipative. By (2.7), we have G(B) C G(—B`). Corollary 2.4.11 proves that B is skew-adjoint. ❑ Proposition 2.6.11. We use the same notation as in §2.6.4. Then Y and B are the extensions of X and A given by Proposition 2.3.1. Proof. Properties (i), (iii), and (iv) are clearly satisfied. We need only show (ii), i.e. §§(I — A)—'UMMx, VU E X. IIU)r Let U e X and V E D(A) be such that U = (I — A)V. We show that §(I A)Vy §Vf^x. Indeed, since B is skew-adjoint, we have

-

II(I - A)V IIY = ((I - B)V, (I - B)V)' - II^IIY + II BV(Y. Let V = (u, v). We have

JJBV 11 2 = ^w11 + §§AU - mu)

1 - §V L2 + IIuI1 2 , = §§VI1 2 .

hence the result. 2.6.5.

0

The Schrodinger operator

Let f be any open subset of R h', and let Y = L 2 (52,C). Y is considered as a real Hilbert space (see §2.5). We define the linear operator B in Y by

D(B) = {u E Ho (1l. C), L u E Y}; { By = i^u, Vu E D(B). In what follows, we write L 2 (4) and H) instead of L 2 (cl, C) and Ho (St, C). Proposition 2.6.12. B is skew-adjoint, and in particular B and -B are mdissipative with dense domains. Proof. The result follows from Proposition 2.6.1 and Corollary 2.5.2.

32 m-dissipative operators

Remark 2.6.13. As in §2.6.1, if 11 has a bounded boundary of class C 2 , then D(B) = H 2 (S2) fl Ho(fl), with equivalent norms. We now set X = H - '(1, C) and, given u E X, we denote by cp v, E Ho (52, C) the solution of -^cp v, + = u in X. We equip X with the scalar product (u, v)-i = (^P.., ^Pv)H1 =Re J

dx,

for u, v E X. We define the linear operator C in X by D(C) = Ho ( 1l);

Cu=Lu, VuED(C). Proposition 2.6.14. C is self-adjoint < 0.

Proof. We have D(Sl, C) C D(C) so that D(C) is dense in X. Furthermore, for all u, v E D(C), (Cu,v)_i = (Cu —u,v)_1 +(u,v)_1 = (u,cp„)Hi +(u,V)_1

=

-

(u,v)L2 + (u,v)_1.

(2.8)

Taking u = v, it follows that

(Cu,u)-1= —IIkIIL2 + IIUIIH-1 < 0, and so C is dissipative. Theorem 1.1.4 proves that C is m-dissipative. By (2.8), we have (Cu,v)_1 = (u,Cv)_1,

for all u,v E D(C). It follows that G(C) C G(C*), and so C is self-adjoint 0 (Corollary 2.4.10). Finally, consider the operator A in X given by

J D(A) = Ho (l); Au=i/u, VuED(A). Applying Proposition 2.6.14 and Corollary 2.5.2, we obtain the following result. Corollary 2.6.15. A is skew-adjoint, and in particular A and -A are mdissipative with dense domains.

Notes. For more information about §2.6, see Brezis [2), Courant and Hilbert [1], as well as Gilbarg and Trudinger [1].

El 1

3 The Hille—Yosida—Phillips Theorem and applications

3.1.

1 I

The semigroup generated by an m-dissipative operator

Let X be a Banach space and let A be an m-dissipative operator in X, with dense domain. For A > 0, we consider the operators Ja and A A defined in §2.2, and we set TA(t) = eIA,, for t > 0. ,

Theorem 3.1.1. For all x E X, the sequence u(t) = T(t)x converges uniformly on bounded intervals of [0, T] to a function u E C([0, 00),X), as A J 0. We set T(t)x = u(t), for all x E X and t > 0. Then

T(t) E £(X) and I T(t)II < 1, T(0) = I;

Vt > 0;

T(t + s) = T(t)T(s), bs, t > 0.

(3.1) (3.2) (3.3)

In addition, for all x E D(A), u(t) = T(t)x is the unique solution of the problem u E C([0, oo), D(A)) f1 C'([O, oo), X);

(3.4)

u'(t)

(3.5)

= Au(t), u(0) = x.

Vt > 0;

(3.6)

Finally, T(t)Ax = AT(t)x. forallxED(A) andt>0. Proof. Step 1.

We proceed in five steps. By Definition 2.2.11, for all t

> 0 and all A > 0, we have

T. (t) = e *J-1 e zr = e I e *JA ; and so,

(ITa(t)11 S e zetII Jall <1

(3.7)

t I

34 The Hille–Yosida–Phillips Theorem and applications

In particular,

Ilua(t)II

(3.8)

IIxil,

for all A > 0 and all t > 0.

Step 2. Assume that x E D(A). It is clear by construction that AA and A. commute, for all A, p > 0. In particular, for all s, t > 0, we have ds

{Ta(st)T, (t – st)} = tTA(st)TN,(t – st)(AA – A µ ).

It follows that

Ilua(t)

–u µ (t)II = IIT. (t)x–Tµ (t)xII

<

i

ds {T(st)T(t – st)x} dsl < tIlAax – A µ xII.

We deduce (Proposition 2.2.12) that uA is a Cauchy sequence in C([0,T],X), for all T > 0. Let u E C([0, oo), X) be its limit. Step 3. Set u(t) = T(t)x. By (3.8), we have (IT(t)xII

IIxil,

for all t > 0, x E D(A); and so T(t) can be extended to a unique operator T(t) E £(X) satisfying IIT(t)II < 1, for all t >_ 0. Take x E X, and (x n,)>o C D(A), such that x,,, — x as n –^ oo. We have

IITA(t)x –T(t)xII < IITA(t)x –TA(t)x f hj + IITA(t)x n –T(t)xIj + IIT(t)xn

< 2 11x. – xjI + IITa(t)xn

–

–

T(t)x I)

T(t)x,,Il ;

and so TA(t)x --> T (t)x as A j 0 uniformly on [0, T] for all T > 0. Properties (3.1) and (3.2) follow. To show (3.3), it suffices to remark that TA(t)TA(s) _ TA(t + s), and so IIT(t)T(s)x – T(t

+ s)xlI < IIT(t)T(s)x – T(t)TA(s)xjI + (IT(t)TA(s)x – TA(t)TA (s)xjj

+ IITA(t + s)x – T(t + s)xII. It follows that IIT(t)T(s)x – T(t +s)xII —;0 as A J 0. Step 4. Returning to the case in which u E D(A), set va(t) = AATA(t)x =

TA(t)Aax = u'(t). We have llva(t) – T(t)AxII < T(t)Ax – TA(t)AxII + IIAAx – AxII.

Two important special cases 35

Hence, v, --> T (t)Ax as A .. 0, uniformly on [0, T] for all T > 0. Taking ua (t) = x +

JO

t

va (s) ds,

and letting .\ 1 0, it follows that t

u(t) = x +

fo

T(s)Ax dx.

Thus u E C l ([0, oo), X), and u (t) = T(t)Ax,

(3.9)

for all t > 0. Finally, we have v(t) = A(JATA(t)x), and !IJATT(t)x —T(t)xH) < IITA,(t)x —T(t)xjj + IIJaT(t)x — T(t)xjj. Therefore, (Ja Ta (t)x, A(Ja TA (t)x)) --3(T(t)x,T(t)Ax) in X x X as A 1 0. Since G(A) is closed, it follows that T(t)x E D(A) for all t > 0, and AT(t)x = T(t)Ax, hence (3.7). We conclude that u E C([0, oo), D(A)). Putting together (3.7) and (3.9), we obtain (3.5). Step 5. Uniqueness of the solution of (3.4)—(3.6). Let u be a solution, and let T > 0. Set v(t) = T(r — t)u(t), for t E [0,r]. We have v E C([0,t],D(A)) nC l ([0,t],X), and v'(t) = —AT(r — t)u(t) +T(rr — t)u'(t) = T(r — t)[u (t) — Au(t)) = 0, for all t E [0, r]. Hence, v(r) = v(0), and so u(r) = T(T)x. r >_ 0 being arbitrary, the proof is complete. 0 3.2. Two important special cases We assume in this section that X is a real Hilbert space. The following result sharpens the conclusions of Theorem 3.1.1. Theorem 3.2.1. Assume that A is self-adjoint < 0. Let x E X, and let

u(t) = T(t)x. Then u is the unique solution of the following problem: u E C([0, oo), X)

n C((0, oo), D(A)) n C 1 ((0, oo), X);

(3.10)

u'(t) = Au(t), Vt > 0;

(3.11)

u(0) = x.

(3.12)

36 The Hille-Yosida-Phillips Theorem and applications

In addition, we have

II Au(t) II

< - 1 1 xII;

(3.13) (3.14)

- (Au(t), u(t)) < 2t IIxII2•

Finally, Au(t) 112 <

1

2t

(3.15)

(Ax, x),

if x E D(A). Proof. We easily verify that A. is self-adjoint < 0, for all A > 0. If u(t) = T.\(t)x, the functions Jjua(t)II and IIua(t)II are non-increasing with respect to t. In addition, we have

d ua(t)11

2

(3.16)

= 2(Aaua(t),ua(t)),

dt(A,\u),(t),ua(t)) = 2 (Aaua(t),ua(t)) = 2IIua(t)II•

(3.17)

From (3.17), it follows that -(Axua(t), ua(t)) is non-increasing with respect to t. Integrating (3.16) between 0 and t > 0, it follows that - t(Aaua(t),ua(t))

< - f A),ua(s),ua(s))ds < 0

2(Ix112.

(3.18)

Integrating (3.17), we obtain 2tIIua(t)II <2f1jua(s)jI ds = -(Aax,x) + (Aaua(t),ua(t)) < -(Aax,x). (3.19) and t 2 llua(t)II. < 2]

5 Hence, with (3.18),

t

-f

0

sUu'(s)II ds = 2] s a (AauA(s), ua(s)) ds d

t

(Aaua(s),ua(s)) 2t2IIua(t)112 < IIxII 2 .

( 3.20)

On the other hand, the vector subspace G(A) is closed in X x X strong, and so is closed in X x X weak. Furthermore, u(t) = Aaua(t) = AAJaua(t), and Jaua(t) —'u(t) as A 1 0 (see Step 4 of the proof of Theorem 3.1.1). In

Two important special cases 37

addition (by (3.14)), for all t > 0, IIAAJaua(t)I) is bounded as .1 j 0. Therefore

u(t) E D(A), for all t > 0, with Au(t) = 11i a A,\J.,ua(t), in X weak. (3.10), (3.11), and (3.12) now follow from Theorem 3.1.1, and (3.13), (3.14), and (3.15) are obtained by passing to the limit in (3.20), (3.18), and (3.19). It remains to show the uniqueness of u. To do this, take t > 0 and 0 <e < t. It follows from (3.10) and Theorem 3.1.1 that u(t) = T(t - e)u(e), and so lIu(t) - T(t)xI) < IIu(e) - xli + ljT(e)x - xli —*0 as a ^, 0. Therefore u(t) = T(t)x, for all t > 0, which completes the proof.

❑

Remark 3.2.2. Theorem 3.2.1 means that T(t) has a smoothing effect on the initial data. Indeed, even if x ¢ D(A), we have T(t)x E D(A), for all t > 0. This is in contrast with the isometry groups generated by skew-adjoint operators. Theorem 3.2.3. Assume that A is a skew-adjoint operator. Then (T(t))t>o can be extended to a one-parameter group T(t) : R -> £(X) such that T(t)x E C(R,X), Vx E X;

(3.21)

(IT(t)xll = ^lxii, dx E X,t E R;

(3.22)

T(0) = I;

(3.23)

T(s + t) = T(s)T(t), `ds, t E R.

(3.24)

In addition, for all x E D(A), u(t) = T(t)x satisfies u E C(R, D(A)) fl C' (R, X ) and u'(t) = Au(t), (3.25) foralltER. Proof. We denote by (T+(t)) t >o and (T (t)) t >o the semigroups corresponding to A and -A. We set -

(

T(t)={

T(t),

ift>0;

We easily verify (3.21), (3.22), (3.23), and (3.24) for x E D(A), and then for x E X by density. Finally, d u(t) (0= Ax -

dt

)

d+

a

-t) (0) =

.

dt (0)

(3.25) follows from the last identity and Theorem 3.1.1.

❑

38 The Hille—Yosida—Phillips Theorem and applications Remark 3.2.4. It is clear that if x

V D(A), then u(t) g D(A) for all t E R.

Remark 3.2.5. The conclusions of Theorem 3.2.3 may be satisfied without assuming that A is skew-adjoint. Indeed, it suffices (and the proof would the same) that A and —A are m-dissipative (and X may be any Banach space). Corollary 3.2.6. Following the notation of Theorem 3.2.3, we have

T(t)* = T(—t), for all t E R.

Proof. Let x, y E D(A). We have

d

t (T(t)x,T(t)y) = (AT(t)x, T(t)y) + (T(t)x, AT(t)y) = 0.

Therefore

(x, y) = (T(t)x, T(t)y), for all t E JR and all x, y E D(A). Taking y = T(—t)z, we have (x,T(—t)z) _ ❑ (T(t)x, z), for all x, z E D(A); hence the result, by density. 3.3. Extrapolation and weak solutions We know (Theorem 3.1.1) that if x E D(A) then T(t)x is the solution of (3.4)— (3.6). If X is a Hilbert space and A is a self-adjoint operator, then T(t)x is still the solution of (3.10)—(3.12), even for x E X. However, in general, if x ¢ D(A), T(t)x is not differentiable in X and then it cannot satisfy (3.11). We will see that the results of §2.3 allow us to identify T(t)x. We follow the notation introduced in §2.3, and we denote by (T(t)) t >o and (T(t)) 9 >o the semigroups corresponding to A and A. We begin with the following result. Lemma 3.3.1. For all x E X and all t > 0, we have T(t)x = T(t)x.

Proof. The result is clear for x E D(A). The general case follows from an usual density and continuity argument.

❑

Corollary 3.3.2. Let x E X. Then u(t) = T(t)x is the unique solution of

u E C([o, oo),

x) n C

l

([0, oo), X );

u (t) = Au(t), Vt > 0; u(0) = x. Proof. We apply Lemma 3.3.1 and Theorem 3.1.1.

❑

Contraction semigroups and their generators 39 9j

3.4. Contraction semigroups and their generators Definition 3.4.1. A one-parameter family (T(t)) t > o C £(X) is a contraction semigroup in X provided that (i) iiT(t)ii < 1 for all t > 0; (ii) T(0) = I; (iii) T(t + s) = T(t)T(s) for all s, t > 0;

(iv) for all x E X, the function t i-- T(t)x belongs to C([0, cc), X). Definition 3.4.2. The generator of (T(t)) t >o is the linear operator L defined

by

( T(t)x - x 1 D(L)=S` xEX; h as a limit inX ash jO J7, h

and

T(t)x - x

Lx=1h mm h for all x E D(L).

The following proposition justifies the introduction of m-dissipative operators in Chapter 2. Proposition 3.4.3. Let (T(t)) t >o be a contraction semigroup in X and let L be its generator. Then L is m-dissipative and D(L) is dense in X. Proof. We proceed in three steps. Step 1. L is dissipative. For all x E D(L), A > 0, and h> 0, we have

II x - A

T(h)x - x

11 ? 11 (1 + h) x,, - h IIT(h)xil ? JJxJJ;

hence the result, letting h j. 0. Step 2. L is m-dissipative. We define the operator J by Jx =

f0 00 e

-t T(t)x

dt,

for all x E X. It is clear that J E £(X), with liJlJ <_ 1. For x E X and h > 0, we have

Th h f I (

)

-

Jx = I

J 0

e -t (T (t + h)x - T(t)x) dt

=1 f ^

1 '

h

h Jh -

eh

-

h

f

J e -t T(t)xdt 0

1 /

..11o

e -t T (t)x dt

eh -

h

h e-tT(t)xdt

v'^•

40 The Hille-Yosida-Phillips Theorem and applications

Letting h . 0, we obtain

h

urn m T(h -I Jx=Jx-x; and so Jx E D(L), with LJx = Jx - x, i.e. Jx - LJx = x. Step 3. For all x E X and t > 0, we set xt

= 11 r t T(s)x ds.

J

0

It is clear that x t ----> x as t j 0. To show that D(L) is dense, it suffices to prove that x t E D(L), for all t > 0. Now we have, for all h > 0, t T (hh - I

t

xt =

fh r - 1 t it

+hT(s)xds

-J

t

T(s)x ds

h

T(s)x ds.

0

+h T(s)x ds

-1

f

As h 1 0, the term on the right-hand side converges to T(t)x - x, and so x t E ❑ D(L) with tLx t = T(t)x - x. Theorem 3.4.4. (The Hille-Yosida-Phillips Theorem) A linear operator A is the generator of a contraction semigroup in X if and only if A is m- dissipative with dense domain. Proof. If A is the generator of a contraction semigroup in X, Proposition 3.4.3 shows that A is m- dissipative with dense domain. Conversely, assume that A is m,- dissipative with dense domain, and let (T(t)) t >o be the semigroup corresponding to A given by Theorem 3.1.1. Then, (T(t)) t >o is clearly a contraction semigroup. Denote its generator by L and let us show that L = A. For all x E D(A) and h > 0, we have (Theorem 3.1.1)

j

T(h)x = x + T(s)Ax ds, and so x E D(L) with Lu = Au. Consequently, G(A) C G(L). Finally, let y E D(L). Since A is m- dissipative, there exists x E D(A) such that x - Ax = y - Ly; and since G(A) C G(L), we have (x - y) - L(x - y) = 0. L being dissipative, we have x = y, and so G(L) C G(A). It follows that A = L, which completes the proof. ❑

Contraction semigroups and their generators 41

Finally, the following result shows the uniqueness of the semigroup generated by an m-dissipative operator with dense domain.

1 I

Proposition 3.4.5. Let A be an m-dissipative operator with dense domain. Assume that A is the generator of a contraction semigroup (S(t)) t >o. Then (S(t)) t >o is the semigroup corresponding to A given by Theorem 3.1.1. Proof. Let (T(t)) t > o be the semigroup corresponding to A given by Theorem 3.1.1. Let x E D(A), and u(t) = S(t)x. For all t > 0 and h > 0, we have

u(t + ) – u(t)

_ S(h) – I

u(t)

= S(t)1 – x —> S(t)Ax

as h j 0.

We deduce that S(t)x E D(A), for all t > 0, and that AS(t)x

= S(t)Ax =

du (t),

for all t >_ 0. Thus u E C([0, oo), D(A)) f1 C'([O, oo), X) and u'(t) = An(t), for t > 0. Therefore, by Theorem 3.1.1, we have S(t)x = T(t)x; hence the result, by density. ❑ The following definition is related to Theorem 3.2.3. Definition 3.4.6. A one-parameter family (T(t))tE R of linear operators is said to be an isometry group in X provided that (i)

T(t)xJ =

Ijxl! for all x E X and all t E R;

(ii) T(0) (iii)

= I; T(t + s) = T(t)T(s) for all s, t E R;

(iv) for all x E X the function t

I

T(t)x belongs to C(R,X).

Further to Theorem 3.2.3 and Remark 3.2.5, we have the following result. Proposition 3.4.7. Let A be an m-dissipative operator with dense domain, and let (T(t)) t>o be the contraction semigroup generated by A. Then (T(t))t>o is the restriction to R + of an isometry group if and only if –A is m-dissipative. Proof. It is clear by Theorem 3.2.3 and Remark 3.2.5 that the condition –A is m-dissipative is sufficient. Assume that (T(t)) t >o is the restriction to R + of an isometry group (T(t)) tE R, and set U(t) = T(–t), for t > 0. Then (U(t)) t >o is a contraction semigroup. Let B be its generator. For all h > 0 and x E X, we have U(h) – I = T(–h) – I T(h) – I x=–U(h) x. We deduce immediately that B

= –A; hence the result.

❑

42 The Hille—Yosida—Phillips Theorem and applications

3.5. Examples in the theory of partial differential equations 3.5.1. The heat equation We use the notation of §2.6.1, and we denote by (S(t)) t >o the semigroup generated by B in Y. Lemma 3.5.1. The embedding D(B) —* Ho (1) is continuous. Proof. Let u E D(B). In particular, we have u E Ho(S2) and, by Lemma 2.6.2,

IIuIIH1 = IIUIIL2 + IIVUIIL2 = IIujIL2 - (Bu, u). Thus,

IIUIIHI < 2IIuIID(B)• for all u E D(B).

C

Applying Proposition 2.6.1 and Theorem 3.2.1, we obtain the following

proposition. Proposition 3.5.2. Let cp E L 2 (1l) and let u(t) = S(t)cp for t > 0. Then u is the unique solution of the problem u E C([0, oo), L 2 (1l)) n C l ((0, oo), L 2 (Sl)), Lu E C((0, oo), L 2 (S2));

(3.26)

u (t) = Du(t), Vt > 0;

(3.27)

u(0) = V.

(3.28)

In addition, we have u E C((0, oo), Ho (1));

(3.29)

IIoUIIL2 <_ 71 II^2IIL2, Vt > 0 ; (3.30) IIVuIIL2 <_

1

JJVIIL2, Vt > 0.

(3.31)

Assuming more regularity on cp, the solution u is also more regular Proposition 3.5.3. In Proposition 3.5.2, assume further that cp E Ho (fl). Then u E C([0, oo), Ho (St)) and

IIIUIIL2 <_



IIV^PIIL2,

(3.32)

for all t > 0. In addition, if A E L 2 (Q), then u E C 1 ([0,00),L 2 (1)), Du E C([0, oo), L 2 (1)) and (3.27) is satisfied for t = 0.

Examples in the theory of partial differential equations 43

Proof. Assume first that cp E Ho (St) and AW E L Z (Sl). Then the result is a straightforward consequence of Theorem 3.1.1 and (3.15). By density, (3.32) is verified for all cp E Ho (S2). Now we need only show that u E C([0, oo), Ho (52)), i.e. (by (3.26)), that u(t) --> cp in Ho (S2), as t J. 0. We use the notation introduced in the proof of Theorem 3.2.1. Passing to the limit in (3.19) as A J 0, it follows that

f 0

1 Ilou(s)IIi2 ds < 2IIVwIIi2;

Iiou(t)IIL2 - IIo^IILz =21 t II u(s)IIL2 ds.

fo

Thus Ilou(t)IIL2 —p IIV IIL2, and IIu(t)IIx1—; ^I IIE1 as t 10. On the other hand, we know that u(t) —> cp in L 2 as t j 0; hence the result.

❑

Remark 3.5.4. If ci has a bounded boundary of class C 2 , then (see Remark 2.6.3) D(B) = H 2 (cl) n Ho (S2). Therefore, if cp E H 2 (S2) n Ho (S2), then we have u E C([0, oo), H 2 (c)). Proposition 3.5.5. Let A be defined by (2.2). Then

IIS(t)IIc(L2) 5 e

-

at,

(3.33)

for all t > 0. Proof. Let cp E D(B), and let f (t) = (eAt1IS(t)cpll) 2 , for t > 0. We have e -2At f (t) = 2

f u(t)

2

+ 2 I u(t)u'(t)

= 2A f u(t) 2 + 2 f u(t)Au(t)

= 2f u(t)

2

-2

J IDu(t)I

2

<0.

Thus IIS(t)II < e -At II^PII for all t >_ 0 and all cp E D(B). The general result follows by density. ❑ We now assume that S2 has a bounded Lipschitz continuous boundary and we follow the notation of §2.6.2. Let (T(t)) t > o be the semigroup generated by A in X. We have X Y, and G(A) C G(B). We easily deduce the following result.

44 The Hille-Yosida-Phillips Theorem and applications

Lemma 3.5.6. For all cp E X and all t > 0, we have T(t)cp = S(t)cp. Consequently, for all cp E X, u(t) = T(t) cp satisfies the conclusions of Proposition 3.5.2. In addition, the following estimates hold. Proposition 3.5.7. Let 1 < q < p < oo. Then

I[ S ( t )[I L P < (47rt)- N!



'I'II L ,

(3.34)

for allt>0 and all cpEX. The proof requires the following two results. z

Lemma 3.5.8. Fort > 0, we define K(t) E S(RN ) by K(t)x = (4irt) - e- 4i . Let 0 E CC (R N ) and let v(t) = K(t) * 0. Then v E C([0, oo), Cb(R N )) n C°°((0, oo), Cb (R N )) and, for all 1 < p < oo, we have v E C([0, oc), LP(R N )) f1 C 00 ((0, oo), LP(R N )). In addition: (i) vt =0v for allt>0;

(ii) v(0) = ^,b; (iii) [v(t)IILP < (47rt) - a (9 P)JkbIIL9, for 1 < q < p < oo and for all t > 0. Proof. Regularity and properties (i) and (ii) follow from easy calculations.

Property (iii) is a consequence of Young's inequality*, since I K ( t )IIL P

=p ^(47rt)

2( 1 P) < (4t) *@

P), ),

for 10.

❑

Lemma 3.5.9. Let cp E Y, cp > 0 a.e. on Q. Then, for all t > 0, we have S(t)cp > 0 a.e. on Q. Proof. By density, we may assume that cp E D(B). We set u(t) = S(t), and we consider u E C([0, oo), Ho (S2)). By Proposition 3.5.2, we have, for all t > 0, -

d +() 2 = -

J

utu

-

= -f U Au = f -

Vu - Vu = _f IVU - [ 2 < 0.

From this, we deduce that fn(u)2 < 0, for all t > 0, and so u > 0.

❑

* Recall Young's inequality: 11 f * 9II LP C 11f IIL91191[Lr, with 1 < p, q, r < oo, and 1/p= 1/q+1/r- 1



Examples in the theory of partial differential equations 45

1

Proof of Proposition 3.5.7. By density, we may assume that cp E D(A). Let = Icpl. Invoking Lemma 3.5.9, for all t > 0, we have -S(t)( < S(t)cp < S(t)(, almost everywhere on Sl; and so (3.35)

I jS(t)^PI )Lp -< II S(t)CII Ln. We define E C(R N ) by ( on S2; {

0 onR N \S2.

Then we set v(t) = K(t) * , and (3.36)

u(t) = u(t)^ Q - S(t)(.

1

We have u E C((0, oo), C(S2)) n C((0, oo), H l (S2)) n C 1 ((0, oo), L 2 (S2)) and Au E C((0, oo), L 2 (St)). Furthermore, u(t) = v(t) > 0 on 3; u t = Du for t > 0; and u(0) = 0. Thus,

dt f (u

- )2

=-

Juu t

-

_ -

f u Au = f Du • Vu z

-

z

-

_-

J IVu

- I2

_<0,

and so u(t) < 0, for all t > 0. We then deduce from (3.35) and (3.36) that

II S(t) FPII LP C II v(t)II Lp• We conclude by applying Lemma 3.5.8.

I ❑

Corollary 3.5.10. Let A > 0 be given by (2.2), and let M = e \ l g l 2,N / (4 ">

Then II S(t)II L(x) <- Me

-at ,

(3.37)

for allt>0. Proof. LetcpEXandletT>0. For0
II S(t)'PII L°° < IHIL= < e -At e^ n IIIPIIL. For T < t, it follows from (3.34) that II S(t)^PII L- << (47rt) 4 IIS(t - T)cpII L 2 < (47rt)- 4 e-AteATII^GIIL2


❑

46 The Hille-Yosida-Phillips Theorem and applications

Remark. We cannot take M = 1 in (3.37). More precisely, we have

d

{IIS(t)II'C(x)}t-o = 0 ;

and so, if IIS(t)IIc(x) < M'e - µt with > 0, we have M' > 1. Indeed, let cp E D(l) be such that cp - 1 in a neighbourhood of xo E Il and IIVIIx = 1, and let u = S(t)cp. We see that u E C°°([0, cc) x S2). Thus we have u t (0) - 1 in a neighbourhood of X. Consequently, for all e > 0 and for x in a neighbourhood of x0, we have u(t, x) > 1- Et, for t small enough, and so in particular II u(t) II x > 1 - Et, for t small enough; hence the result. Concerning L" inequalities, note that applying (3.37) and (3.34), we verify easily that for all 1 < p < oo, there exists a constant Mp such that (3.37')

II S(t)II e(LP) <- Mpe-\t,

for all t > 0. Once more, we cannot take Mp = 1. Actually, one can see that, for p > 2, one has Oat

II S(t)Il c(LP) <e,,

for all t > 0, and that this inequality is optimum in the following sense:

a

{IIS(t)II^cLp )}t=o =

t

p2'

-

Indeed, for all cp E D(l), letting u(t) = S(t), and multiplying by lulp -2 u, we obtain the equation satisfied by u. Applying (3.33), we obtain

o= 1 d

j lu(t)IP +

p dt l^

Ivu2 I>

4 p

1d

f lu(t)IP + 2 4 f Iu(t)IP;

p dt st

p st

the inequality follows. To show optimality, it suffices to verify that, for all E > 0, there exists 0 E D(1l) such that

fa IvV)2 I < (A+E)

J IV)IP.

To see this, we consider the first eigenfunction cp l of -A in Ho (S2) (see Brezis [21). For e> 0, let 0, : [0, oo) -+ [0, oo) be such that 6 e - 0 in a neighbourhood of 0, OE(x) < 1 and x - 8 E (x) <_ e for all x >_ 0. Set 2/' _ (9( 1 ))2/P . We verify that Ali, E D(f1). Furthermore, Iv 12 , < IV^pil and, last,

V)

—>

Jn

1pi

as E j 0.

Examples in the theory of partial differential equations 47

Consequently,

J M 2 < fn Iv n 11 2 =A f cp 2 =Alim Ei°Jn 3.5.2. The wave equation (or the Klein-Gordon equation) We use the notation introduced in §2.6.3 and §2.6.4, and we denote by (T(t))tER the isometry group generated by A in X, and by (S(t)) tE R the isometry group generated by B in Y. Proposition 3.5.11. Let (cp, V) E X and let u(t) be the first component of

T(t)(p, z/'). Then u is the unique solution of the following problem: E C(R, Ho (fl)) n C 1 (R, L 2 (1)) n C 2 (R, H -1 ( 1l) ); utt-Du+mu=0, for alltER;

(3.38)

u(0) = cp, u t (0) = z/i.

(3.40)

U

(3.39)

In addition, J {Jvu(t)1 2 + mju(t)J 2 + u(t)2} = Jst

{IvcpI 2 + mIcpI 2 + ,0 2 } ,

( 3.41)

for alit E R. Finally, if(cp,0) E D(A), wehaveu E C'(R,Ho(St))nC 2 (R,L 2 (1l)) and Du E C(R, L 2 (1)). Proof. Let u E D'(R, H -1 (S2)) and set U = (u, u t ). Then U E C(R, D(B)) fl C'(R, Y) if and only if U E C(R, Ho (52)) fl C(R, L 2 (SZ)) fl C 2 (R, H -1 (1)). Furthermore, in that case, (3.39)-(3.40) is equivalent to the equation

U'(t) = BU(t), for all t E R. The result then follows from Propositions 2.6.9, 2.6.10, and 2.6.11, Theorem 3.2.3, and Corollary 3.3.2. Note that (3.41) is equivalent to (3.22). ❑ 3.5.3. The Schrodinger equation We use the notation introduced in §2.6.5, and we denote by (S(t)) tE 1 and (T(t)) tE R the isometry groups generated by B and A. We have G(B) c G(A), and from this it is easy to deduce the following result. Lemma 3.5.12. For all cp E Y, we have S(t)co = T(t)cp, for all t E R. Then we have the following.

48 The Hille-Yosida-Phillips Theorem and applications Proposition 3.5.13. Let cp E H0'(1) and let u(t) = T(t)cp. Then u is the unique solution of the problem u E C(R, Ho (l)) n C l (R, H-1(l)); ju t +,Lu = 0, for all t E R;

(3.42)

u(0) = W.

(3.44)

(3.43)

In addition,

i

f

Iu(t) 2 =

n IVu(t)I 2 =

f IcpI , for all t E R,

(3.45)

2

J IV I 2 ,

(3.46)

for all t E R.

Finally, if Lcp E L Z (cl), then u E C 1 (R, L 2 (S2)) and Au E C(R, L 2 (fl)). Proof. We use Theorem 3.2.3, Corollary 3.3.2, and Lemma 3.5.12. (3.45) is equivalent to (3.22). On the other hand, invoking (3.7) and (3.22), we obtain

(3.47)

IIAu(t)IIx = IIA(PIIx, for all t E R. But, for all V E Ho (S2), we have

II AvII x = II(Av - v) + vIIi = f IVvI 2 -

JM

2

+ IIvIIX

(3.48)

By (3.22), IIu(t)IIx = Ik IIx for all t E R; and so (3.46) follows by putting 0 together (3.47), (3.48), and (3.45). 3.5.4. The Schrodinger equation in R n' We use the notation of §3.5.3, and we assume that Sl = 1[8N. We can state estimates in the spirit of (3.34). Proposition 3.5.14. For all p E [2, oo] and t 54 0. Then T(t) can be extended to an operator belonging to £(LT (RN), L°(R")). In addition, we have

IIT(t)II,C(Lp ,LP) C for all

(4^rItI)N(2-p)'

(3.49)

0.

Proof. Let cp E S(IRN) and let u(t) E C 00 (IR,S(R N )) be defined by Fu(t)(^) = e-'IE12t.F'G(^),

(3.50)

Examples in the theory of partial differential equations 49

forall^ER' andtER. We have idt

P a(t)(6)

- I6I 2 Fu(t)(C) =0 in R N ,

for all t E R; and so iut + 0u = 0 in R N , for all t E R. Therefore we have (Proposition 3.5.13) u(t) = T(t), for all t E R. Now we know that .-1{e-;1e12t}(x) =

f



2

N e t := K(t)x,

(47rt) 2

for all x E R N and t # 0. It follows from (3.50) that u(t) = K(t) * cp for all t 0. We deduce that

T(t) W II L- <

1 (47rt) 2

II(P11t1,

for all t # 0 and cp E S(R N ). Thus, one can extend T(t) to an operator of L(L l (R' ), L°°(1R ')) such that T(t)1I,c(L1, L c) < (4irItj) - . Furthermore, T(t) E G(L 2 (R N ),L 2 (1R N )), with T(t)IIc(L2,L2) = 1. The general case follows from the Riesz interpolation Theorem (see Dunford and Schwartz [1], p. 525, or Bergh and Lofstrom [1], p. 2, Theorem 1.1.1). ❑ Notes. Theorem 3.2.1 can be generalized to the case of the generators of analytic semigroups; see Goldstein [1], Haraux [3], Pazy [1]. One can build semigroups for some classes of operators, rn-dissipative operators (non-linear), and maximal monotone operators. These two classes coincide in Hilbert spaces. See Brezis [1], Crandall and Liggett [1], Crandall and Pazy [1], and Haraux [1, 2].

Id

Inhomogeneous equations and abstract semilinear problems Throughout this chapter, we assume that X is a Banach space and that A is an m-dissipative operator with dense domain. We denote by (T(t)) t >o the contraction semigroup generated by A. 4.1. Inhomogeneous equations Let T> 0. Given x E X and f: [0,T] --> X, our aim is to solve the problem (4.1) u E C([0,T],D(A)) nC'([O,T],X); (4.2)

u'(t) = Au(t) + f (t), `dt E [0, T];

(4.3)

t. u(0)=x.

As in the case of ordinary differential equations, we have the following result (the variation of parameters formula, or Duhamel's formula). Lemma 4.1.1. Let x E D(A) and let f E C([0, T}, X). We consider a solution uEC([0,T],D(A))nC'([0,T],X) of problem (4.1)—(4.3). Then, we have

u(t) =T(t)x

+

J0 T(t —s)f(s)ds,

(4.4)

t

for all t E [0, T] . Proof. Let t E (0,T]. Set w(s) = T(t — s)u(s), for s E [0, t]. Lets

w(s

+h

E

[0, t] and h E (0, t — s]. We have

) — w(s)

(

— 7t — s — h) {

u(s + — u(s) — T(h) — I u(s)

1

T(t —s) {u'(s)—Au(s)} = T(t —s)f(s) as h . 0. Since T(t — •) f(•) E C([O,t],X), we deduce that w E C 1 ([0,t),X) and that (4.5) w'(s) = T(t — s) f (s), for all s E [0,t). Integrating (4.5) between 0 and obtain (4.4).

rr

< t, and letting T T t, we 0

Inhomogeneous equations 51

Corollary 4.1.2. For all x E D(A) and f E C([O,T],X), problem (4.1)-(4.3) has at most one solution. Remark 4.1.3. For all x E X and all f E C([O,T],X), formula (4.4) defines a function u E C([O, T], X). Now we are looking for sufficient conditions for u given by (4.4) to be the solution of (4.1)-(4.3). Remark 4.1.4. It is clear that if u is a solution of (4.1)-(4.3), then x E D(A). However, this condition is not sufficient. Indeed, assume that (T(t)) tE R is an isometry group, and let y E X \ D(A). Then (see Remark 3.2.4), T(t)y ¢ D(A), for all t E R. Take f(t) = T(t)y, and x = 0 E D(A). It follows easily that (4.4) gives u(t) = tT(t)y D(A), for t # 0. Lemma 4.1.5. For all x E X and f E L 1 ((0,T),X), formula (4.4) defines a function u E C([0,T],X). In addition, we have IIUIIC([O,T],X)
Proof. The result is clear if f E C([0,T],X), and follows by density in the general case. 0 Proposition 4.1.6. Let x E D(A) and let f E C([0,T],X). Assume that at

least one of the following conditions is satisfied: (i) f E L'((0,T),D(A)); (ii) f E W 1,1 ((0,T),X).

Then u given by (4.4) is the solution of (4.1)-(4.3). Proof. We proceed in four steps. Set v(t)

= J t T(t - s)f (s) ds O

= fo

t

T(s)f (t - s) ds,

for t E [0,T]. Step 1. We have v E C 1 ([0,T),X). Indeed, if f E L 1 ((0,T),D(A)), for t E [0, t) and h E [0, t- s], write

f

1 l+h v(t + h) - v(t) - f t T(h) - I h T(t-s) h f(s)ds + h T (t+h-s)f(s)ds,

52 Inhomogeneous equations and abstract semilinear problems

and let h J 0. Note that

T(h)-I —, Af h f

in L 1 ((0,T),X), as h . 0, and apply Lemma 4.1.5. It follows that

d+ v dt (t)

=J

t

T(t - s)A f (s) ds + f(t),

for all t E [0, T). If f E W 1,1 ((0, T), X), for t E [0, T) and h E [0, T - t], we write

v(t + h) - v(t) =

O

T(s) f (t + h

-h

-

f (t - s)

ds+ T(h)

f

h

T (t - s)f

(S)

ds,

and we let h J, 0. Note that (Corollary 1.4.39)

.f(t+h—.) —,f(t—.) h

f'(t—.), as h J 0

in L 1 ((0, t), X) and apply Lemma 4.1.5. It follows that

d+ v dt (t)

= fo

t T(s) f'(t - s) ds + T(t)f(0),

for all t E [0,T). In both cases, we have d+v/dt E C([0,T),X); and so v E

C 1 ([O,T),X)• Step 2. Similarly, we show that (d v/dt) (T) makes sense and is equal to i imv'(t); and so v E C'([O,T],X). -

Step 3. Let t E [0, T) and let h E [0, T - t]. We have

T (h) -I h

v(t) =

11t

I

t

T (t + h - s) f (s) ds

- v(t+h)-v(t) 1 -- [ h h J t

-

f

t T (t - s) f (s) ds

7(t + h- s) f (s) ds.

Letting h 1 0, we deduce v(t) E D(A), and Av(t) = v'(t) - f(t). This is still true for t = T, since the graph of A is closed. It follows that v E C([O,T], D(A)), and that v satisfies (4.2).

Step 4. We have u(t) = T(t)x+v(t) E C([0,T], D(A))f C'([O, T], X), and (4.1) follows. Furthermore,

u'(t) = AT(t)x + Av(t) + f(t) = Au(t) + f(t), for all t E [0, T]. Hence, we have (4.2), and (4.3) is immediate.

❑

Inhomogeneous equations 53

Corollary 4.1.7. Let x E X, f E C([O,T],X) and let u be given by (4.4). Then using the notation of §2.3 and §3.3, u is the unique solution of the problem u E C([O,T],X) nC'([0,T],X); u'(t) = Au(t) + f (t), Vt E [0, T]; u(0) = x. Proof. We apply Lemma 3.3.1 and Proposition 4.1.6.

❑

Corollary 4.1.8. Let x E X, f E C([0,T],X) and let u be given by (4.4). Assume that at least one of the following conditions is satisfied: (i) u E C([0,T],D(A)); (ii) u E C 1 ([0,T],X). Then u is the solution of (4.1)—(4.3). Proof. Assume that (i) holds. By Corollary 4.1.7, we have u' E C([0, T], X); and so u E C i ([0, T], X ), hence the result. Now assume that (ii) holds. By Corollary 4.1.7, we have Au E C([0,T],X); and so (Corollary 2.3.2) u E C([0,T],D(A)); hence the result. ❑ Throughout §4.1, we have supposed that f E C([0,T],X). But in order to give a sense to (4.4), it suffices that f E L 1 ((0,T),X) (Lemma 4.1.5). In this case, we have the following result. Proposition 4.1.9. Let x E X, f E L 1 ((0,T),X) and let u E L 1 ((0,T),X). Assume further that u E L 1 ((0,T),D(A)) or that u E W 1 1 ((0,T),X). Then u verifies (4.4) if and only if u "

u E L 1 ((0,T),D(A)) nW'" i ((O,T),X); u'(t) = Au(t) + f (t), for almost every t E [0,T]; u(0) = x. Proof. First note that if u E W 1 1 ((0,T),X) then u E C([0,T],X) (Corollary 1.4.36) and so the condition u(0) = x makes sense. Let us first show that the assumptions of the theorem are sufficient to have (4.4). To see this, we argue as in Lemma 4.1.1. We consider t E (0, T] and we set w(s) = T(t — s)u(s), for almost every s E (0, t). For all h E (0, t), and for almost every s E (0, t — h), we have '

w(s + h) — w(s) h

_ T(t — s — h)

u(s + h) — u(s) — T(h) — I

(s)}

54 Inhomogeneous equations and abstract semilinear problems

It follows that w is absolutely continuous from [0, T] to Y. Moreover, the righthand member converges for almost every s E (0, t) to T(t – s)u'(s) – Au(s) = T(t – s) f (s), as h . 0 (Theorem 1.4.35). Therefore w is right differentiable almost everywhere on (0, t) and

s

d (s) = T(t – s) f (s)• d Similarly, we show that w is left differentiable almost everywhere on (0, t) and that

v

(s) = T(t – s)f(s).

Consequently (Theorem 1.4.35), w E W 1 1 ((0,T),Y) and w'(s) = T(t – s)f(s) almost everywhere. We deduce (4.4). Conversely, assume that u satisfies (4.4). Let (fn ) n >o be a sequence of C([0,T], X) such that fn —if in L 1 ((O,T), X) as n –+ oo, and let u n be the corresponding solutions of (4.4). Invoking Corollary 4.1.7, we have '

I

u(t) = Au(t) + fn(t), for all t E [0,T]; and so

u(t) = x

+ J t (Aun(s) + fn(s)) ds 0

for all t E [0,T]. Letting n

–ti

oo, we obtain (Lemma 4.1.5)

u(t) = x +

JO t (Au(s) + f(s)) for all t E [0,T]. It follows that u E W ' ((0,T),Y) and that ds,

1 1

u'(t) = Au(t) + f(t), for almost every t E [0,T]. If u E W 1 1 ((0,T),X), we have Au E L 1 ((O,T),X); and so u E L 1 ((0,T),D(A)) (Corollary 2.3.2). If u E L 1 ((O,T),D(A)), we have ❑ u' E L' ((O, T), X ); and so u E W 1 1 ((0, T ), X ). This completes the proof. '

"

4.2. Gronwall's lemma In this section, we give a result which is essential in the study of semilinear problems; not only for showing uniqueness of solutions but also for finding bounds on the solutions.

Semilinear problems 55

Lemma 4.2.1. (Gronwall's lemma) Let T > 0, A E L l (0, T), A >_ 0 a.e. and C1, C2 > 0. Let cp E L' (0, T), cp > 0 a.e., be such that A E L'(0, T) and cp(t) < C1 + C2

J

t

0

A(s)cp(s) ds,

for almost everyt E (0,T). Then we have cp(t)

< Cl exp (C2

rt

A(s)ds Jo

)

for almost every t E (0,T).

Proof. We set

ft

(t) = C l + C2

J A(s)cp(s) ds. 0

i) is differentiable almost everywhere (since it is absolutely continuous), and we have z/^'(t) < C2 A(t)cp(t) < C 2 A(t)O(t), for almost every t E (0, T). Consequently,

(

dt{ ip(t) exp (C2 l

ll t A(s) ds I 1 < 0,

f

I ))J

and so

J

t

< Cl exp (C2 A(s) ds) O

hence the result, since cp < 0.

❑

Remark 4.2.2. In particular, if Cl = 0, we have cp = 0 a.e.

4.3. Semilinear problems Definition 4.3.1. A function F : X —> X is Lipschitz continuous on bounded subsets of X provided that for all M > 0, there exists a constant L(M) such

that IIF(y)-F(x)II <_L(M)IIy

-xII, ex,yEBM,

where BM is the ball of center 0 and of radius M. Throughout s4.3, F: X —4 X is a Lipschitz continuous function on bounded subsets of X. We denote by L(M) the Lipschitz constant of F in BM for M > 0. In particular, L(M) is a non-decreasing function. of M.

56 Inhomogeneous equations and abstract semilinear problems

Given x E X, we look for T> 0 and a solution u of the following problem: u E C([0,T],D(A)) nC'([O,T],X);

(4.6)

u'(t) = Au(t) + F(u(t)), Vt E [0, T];

(4.7)

u(0) = x.

(4.8)

We also consider a weak form of the preceding problem. Indeed, by Lemma 4.1.1, any solution u of (4.6)—(4.8) is also solution of the following problem: u(t) = T(t)x +

JO t

T(t — s)F(u(s)) ds, Vt E [0, T].

(4.9)

Finally, note that, for all u E C([O,T],X), (4.9) is equivalent (following the notation of Corollary 4.1.7) to the problem u E C([0,T],X) nC'([O,T],X);

u'(t) = Au(t) + F(u(t)), Vt E [0, T]; u(0) = x.

4.3.1.

A result of local existence

We begin with a uniqueness result. Lemma 4.3.2. Let T > 0, x E X, and let u, v E C([0, T], X) be two solutions to problem (4.9). Then u = v. Proof. We set M = sup max{IIu(t)II, llv(t)ll}. We have tE [O,T]

t II F(u(s)) — F(v(s)) II ds < L(M) II u(t) — v(t)II <— I0

fo t I u(s) — v(s) II ds,

and we conclude using Remark 4.2.2. Set

1

TM = 2L(2M

+ IIF( 0 )II) + 2 > 0,

for M > 0. We can state a first result of local existence.

IIxII

< M. Proposition 4.3.3. Let M > 0 and let x E X be such that there exists a unique solution u E C([O,TM], X) of (4.9) with T = TM.

Then

Semilinear problems 57

Proof. Lemma 4.3.2 proves uniqueness. Let x E X and let M > K = 2M + IIF(0)II and

IIxII. We let

= { u E C([O,TM],X); [In(t)I[ < K,`dt E [0,TM]},

E

and we equip E with the distance generated by the norm of C([O,TM], X), Let d(u, v) = max IIu(t) — v(t) tE[0 ,TM I

II,

for u, v E E. Since C([0, TM], X) is a Banach space, (E, d) is a complete metric space. For all u E E, we define (D u E C([O,TM],X) by = T(t)x +

f

T(t — s)F(u(s)) ds,

for all t E [0, TM]. Note that for s E [0, TM], we have F(u(s)) = F(0)+(F(u(s)) — F(0)); and so

IIF(0)II IIF(u(s))II <— IIF(0)II + KL(K) < M +TM It follows that II(t)It

0)II < IIxII +JO t IIF(u(s))Ilds <M+t M TM(
Consequently, we have F : E --4 E. Furthermore, for all u, v E E, we have II(t) — (t)II < L(K)

fo IIv(s) — u(s)II ds < TML(K)d(u, v) < 1 d(u, v).

Therefore, 1 is a contraction in E with Lipschitz constant 1/2, and so 1) has a fixed point (Theorem 1.1.1) u E E, which satisfies the requirements of Proposition 4.3.3. ❑ Theorem 4.3.4. There exists a function T : X — (0, oo] with the following

properties: for all x E X, there exists u E C([0, T(x)), X) such that for all 0< T < T(x), u is the unique solution of (4.9) in C([O,T], X). In addition, 2

L(IIF( 0 )II + 2 11u(t)II) ? T(x) — t —2,

for all t E [0, T(x)). In particular, we have the following alternatives: (i) T(x) = oo;

(4.10)

58 In homogeneous equations and abstract semilinear problems

(ii) T(x) < oo and lim Iu(t) ll = oo. ttT(x)

Remark 4.3.5. If property (i) holds, we say that the solution u is global. On the other hand, if (ii) holds, we say that u blows up in finite time. In other words, the alternatives (i)-(ii) mean that the global existence of the solution u is equivalent to the existence of an a priori estimate of IIu(t)II on [0,T(x)). In applications, we establish such a priori estimates by standard methods in the theory of partial differential equations (multipliers, comparison principles, and maximum principles), as well as by various techniques involving differential or integral inequalities more specific to evolution equations (first integrals, Liapunov functions, and variants of Gronwall's lemma).

Proof of Theorem 4.3.4. It is clear that (4.10) implies that if T(x) < oo, then IIu(t)II-->ooastIT(x). Let x E X. We set T(x) = sup{T > 0; 3u E C([0,T],X) solution of (4.9)}.

By Proposition 4.3.3, we know that T(z) > 0. On the other hand, the uniqueness (Lemma 4.3.2) allows us to build a maximal solution u E C({0, T(x)), X) of (4.9). It remains to show (4.10). Inequality (4.10) being immediate if T(x) = oo, we may assume that T(x) < oo. We argue by contradiction, assuming that there exists t E [0,T(x)) such that (4.10) does not hold. We then have T(x) - t < TM,

with M = 1Iu(t)j^. Let v E C([O,TM],X) be the solution, given by Proposition 4.3.3, of

+

v(s) = T(s)u(t)

10

T(s - a)F(v(c)) do, ,

for all SE [0, TM] . We then define w E C([0, t + TM], X) by w(s) =

u(s),

ifs E [0, t];

v(s - t),

if s E [t, t + TM].

We verify easily that w is a solution of (4.9) with T = t + TM, which contradicts the definition of T(x), since t + TM > T(x). Remark 4.3.6. It may very well happen that, for the same equation, T(x) < oo for some initial data, and T(x) = oo for others. For example, choose X = R, A = 0, and F(u) = u 3 - u. This choice corresponds to the ordinary differential equation u' = u 3 - u. If (xi <_ 1 we have T(x) = oo, and if xi > 1 we have T(x) < oo. In the last case, (4.10) gives 12iu(t)1 2 > (T(x) - t) ' - 4. This -

Semilinear problems 59

estimate describes the blow-up phenomenon sharply, since the solutions actually blow up as (T(x) - t) I'. -

4.3.2. Continuous dependence on initial data Proposition 4.3.7. Following the notation of Theorem 4.3.4, we have the following properties: (i) T : X -* (0, oo] is lower semicontinuous; (ii) if x n --* x and if T < T(x), then u n -3 u in C([0, T], X), where u n and u are the solutions of (4.9) corresponding to the initial data x n and x. Proof. Let x E X, and let u be the solution of (4.9) given by Theorem 4.3.4. Let 0 o C X is a sequence such that x n —*x as n -^ oo, then for n sufficiently large T(x) > T and u n --+ u in C([0,T],X). To see this, set

M = 2 sup IIu(t)1I, tE[O,TJ

and Tn = sup{t E [0,T(x n )) ; IIun(s)II < 2M,Vs E [0,t]}. For n large enough, we have (IxII < M; and so Tn > TM > 0. For all t < T, t<Ti,,wehave

Ilu(t) — u n(t)II

(Ix — xn ll + L(2M)

fo t IIu(S) — un(S)IIds;

and it follows from Lemma 4.2.1 that

II u(t) — u(t)ii < IIx — x n lieTL( 2 M),

(4.11)

for t < T, t <_ Tn . In particular, we deduce from (4.11) that for n large enough we have

II un(t)II C Al, for t _< min{T, rrn }; and so Tn > min{T, rrn }, i.e. Tn > T. We then have T(x) > T. Applying (4.11) again, we see that u n --> u in C([0,T],X). This completes the proof. ❑ Remark 4.3.8. Actually, T may be discontinuous. For example, choose X = 1R 2 , A = 0, and F(u, v) = (vu 2 , -2). For x = (1, 2) we have T(x) = 1 and the corresponding solution is ((1 - t) -2 , 2(1 - t)). For x E = ((1 + E) -1 , 2) we have T(x e ) = oo and the corresponding solution is ((e + (1 - t) 2 )',2(1 - t)).

60 Inhomogeneous equations and abstract semilinear problems

4.3.3.

Regularity

In some cases, it is possible to give a more precise result on the regularity of solutions of (4.9). In particular, we have the following. Proposition 4.3.9. Assume that X is reflexive. Let T > 0, x E X, and let u E C([0,T],X) be a solution of problem (4.9). Then, if x E D(A), u is the solution of problem (4.6)—(4.8). Proof. Let h> 0 and let t E [0,T — h]. It is easy to see that u(t + h) — u(t) = T(h)x — x+

J

0

t

T(s){F(u(t + h — s) — F(u(t — s))} ds

+

f

h T(t + s)F(u(s)) ds.

Hence,

IIu(t + h) — u(t)II c IIT(h)x — x1j+h sup IIF(u(s))II sE[O,T]

+ L(M)

J

0

t llu(s

+ h) — u(s) I ds

Frthermore, we have h

T(h)x — x = J T(s)Axds; O

and so IIT(h)x — xII < hilAxil. Applying Lemma 4.2.1, we obtain

u(t + h) — u(t)II < Ch,

for 0 < t X is also Lipschitz continuous. We conclude by applying Corollary 1.4.41 and Proposition 4.1.6. ❑ Remark 4.3.10. If X is not reflexive, the conclusion of Proposition 4.3.9 may fail, as shown by the following example. Choose X = Co(R) x Co(IR), where Co(IR) is the space of functions of C(R) which dies away to 0 as x —^ foo, equipped with the norm L °"(][8). We define the operator A by D(A) = {( u, v) E X f1 C' (R 2 ); (u', v') E X}; {

A(u, v) = (u', v'), `d(u, v) E D(A).

Isometry groups 61

A is m-dissipative with dense domain, and generates the semigroup (T(t)) t >o given by

I

T(t)(u, v) = (u(t + •), v(t + )), for t > 0, x E R. Next, consider the Lipschitz continuous function F : X --+ X given by

F(u, v) = (v,0), V(u, v) E X. For all (x, y) E X, the corresponding solution (u, v) of (4.9) is given by

t

(u, v)(t) = (x(t +) + t y +(t + •), y(t + •))• Taking (x, y) E D(A) such that y(0) = 0 and y'(0) so (u, v)(t) V D(A), for t 0.

54 0, y+ is not in C'(R), and

4.4. Isometry groups In the case in which A generates an isometry group (see Theorem 3.2.3), and in particular when X is a Hilbert space and A is skew-adjoint, we can also solve (4.7) for t < 0. Indeed, solving the problem u E C([—T,0],X) nC'([—T,0],X);

u (t) = Au(t) + F(u(t)), Vt E [—T, 0]; u(0) = x; is equivalent to solving v E C([0,T],X) nC 1 ([0,T],X);

v'(t) = —Au(t) — F(u(t)), Vt E [0, TI; 1. v(0)=x; setting u(t) = v(—t), for t E [— T, 0]. The second problem is solved by Theorem 4.3.4, since —A is m-dissipative and —F is Lipschitz continuous on bounded sets of X. Notes. One finds generalizations of the results of §4.3 in Segal [1] and Weissler [1]. Also consult Ball [1, 2] for an interesting discussion about the blow-up phenomenon.

I

II

• I

5 The heat equation Throughout this chapter, we assume that S2 is a bounded subset of R N with Lipschitz continuous boundary, and we use the notation of §3.5.1. In particular, X = Co(1) and Y = L 2 (1). In addition, we consider a locally Lipschitz continuous function g E C(R, R), such that

g(0) = 0. We define the function F : X --^ X by

I

F(u)(x) = 9(u(x)), for all u E X and x E Sl. It is easy to check that F is Lipschitz continuous on bounded sets of X. In what follows, we denote g and F by the same expression. 5.1. Preliminaries Given cp E X, we look for T> 0 and u solving the problem u E C([0,T], X) n C((0,T], Ho (S2)) n C'((O, T], L 2 (5l));

Au E C((O,T],L 2 (1l));



u t — Au = F(u), Vt E (0, T];



u(0) = cp.

(5.1) (5.2) (5.3)

The result is the following. Proposition 5.1.1. Let cp E X, T> 0, and let u E C([0,T],X). Then u is solution of (5.1)—(5.3) if and only if u satisfies u(t) = T(t) cp +

JO

t

T(t — s)F(u(s)) ds,

(5.4)

foralltE [0,7']. Proof. Let u be a solution of (5.1)—(5.3), let t E (0,T], and let e E (0,t]. We

set v(s) = u(e + s),

Preliminaries 63

for 0 < s < t - e. It is clear that v is a solution of (5.2) on [0, t - e] and that

v(0) = u(e) E D(B). Hence, we have (Lemma 4.1.1) v(s) = S(s)u(e)

+

10, S(s - o,)F(v(u)) ds,

for all s E [0, t - E]. Applying Lemma 3.5.6, we deduce that u s+e T(s)u(e) +

"

T s-v F u v +e ds,

for all s E [0, t - e]. Since u E C([0,T], X), we have, for all s E [0, t), T(s)u(E) -; T(S)W,

as j0; F(u(. + e)) -' F(u(.)),

uniformly on [0, s] as e j 0. Letting first E 1 0, and then s T t, we deduce (5.4). Conversely, let u E C([0, T], X) be a solution of (5.4). We consider 0 < t ( T. By Proposition 3.5.2, we have T(t) cp E Ho (Sl), and the function s H T(t - s)F(u(s)) belongs to C([0, t), Ho (S2)), with IIT(t - s)F(u(s))IIHI < C(1 + (t - s) -1 " 2 ) E L 1 (0, t);

and so (Proposition 1.4.14 and Corollary 1.4.23) u(t) E Ho (S2). A similar estimate shows that actually u E C((0, T], Ho (l1)), IIu(t)IIH1 < C(1 +t -1 / 2 )

Since g is Lipschitz continuous on bounded subsets of R and since the range of u is bounded, we conclude (Proposition 1.3.5) that F(u(t)) E H(l), and that IIF(u(t))IIH= < C(1 +t -1 1 2 ).

It follows that F(u) is weakly continuous as a map from (0, T] to Ho (St). Take 0
L2

< C(t - s) -1 / 2 (1 + s -1 / 2 ) E L l (0, t);

and so Au(t) E L 2 (1l). Consequently, u(t) E D(B), for all t similarly that u E C((0,T), D(B)). We then set v(s) = u(e + s),

E

(0,T]. We show

64 The heat equation

for O < s t - E. We have v(s) = S(s)u(E) +

fo

S(s - a)F(v(a)) ds, s

for all s E [0, t - e]. We conclude by applying Corollary 4.1.8.

o

Note that here it is not possible to invoke Proposition 4.3.9, since X is not reflexive. Remark 5.1.2. Considering the above estimates in more detail, we obtain, as a consequence of Proposition 3.5.2,

IIVu(t)IILa

+ <- CIcI 1 / 2 (t -1 + t)

;

where C depends only on g and sup IIu(t)II. O
Iiou(t)IIL2 < CIQ1 1 " 2 (1 + t'/ 2 ); IILu(t)IIL 2 < CIcj 1/2 (t -1/2 + t), where C depends only on g, sup Iu(t)II andII^PIIH1• O
Remark 5.1.4. The same method also shows that if we assume further that cp E D(B), then

IIAu(t)IIL2 G CIslIli 2 (1 + t),

where C depends only on g, sup IIu(t)II, and IIcolIHl. O
5.2. Local existence Applying Proposition 5.1.1 and Theorem 4.3.4, we obtain the following result. Theorem 5.2.1. For all cp E X, there exists a unique function u, defined on a maximal interval [0, T()), which is a solution of (5.1)-(5.3) for all T E (0, T(,)). In addition, if T (cp) < oc, then IIu(t) II -4:: as t j T(). Remark 5.2.2. u depends continuously on cp (this follows from Proposition 4.3.7). The following proposition gives improved regularity of u if cp is more regular.

Global existence 65

Proposition 5.2.3. Assume that cp E X fl Ho (S2). Then the solution u corresponding to (5.1)-(5.3) is in C([0,T(cp)),Ho(c)). Suppose further that Ap E L 2 (c); then u E C([0,T(p)),D(B)) fl C'([0,T(cp)), L 2 (c)). Proof. Assume that cp E X f1 Ho (S2), and let t E (0, T(cp)). Applying (5.2),

Proposition 3.16, and (3.31), we obtain

u(t) — VIIH'

f t 1 IIF(u(s))Ilds < II S(t)V — VIIH1 + C /o t-s <—IIS'(t)V—VIIH=+C^-->0 astj0.

Therefore u E C({0, T()), Ho (S2)). In particular, if T < T(p), then u is bounded in Ho (SI) on [0, T], and then so is F(u). Consequently, in the case in which A E L 2 (c), it follows from (5.2) and (3.32) that

IIo(u(t) — V)IILZ

I(S(t) — I)oVIIL2

+Cft

t1 S

JIF(u(s))IIHlds tie

and so u E C([0,T(cp)), D(B)). In particular, u(t) — Ap + F(cp) in L 2 (c) as t 1 0. Furthermore, for t Acp + F(cp) as t j 0. V+t t t 0

J

Consequently, (d + u/dt) (0) = t o u t (t); and so u E C'([0, T(cp)), L 2 (c)).

❑

5.3. Global existence We establish here two kinds of results. First, we show that if g satisfies certain conditions for Ix) large, then all solutions of (5.1)-(5.3) are global. Then, in another spirit, some results prove that if g satisfies certain conditions for IxI small, then the solutions of (5.1)-(5.3) with small initial data are global. We begin with the following result (the maximum principle). Proposition 5.3.1. Let T > 0 and cp E X. Let u E C([0,T], X) n C((0,T), H(i)) n C 1 ((0,T),L 2 (fl)) with Au E C((0,T),L 2 (c)), and f E C([0,T],X) be such that ut1=f, VtE (0,T);

1

u( 0 ) = W.

(5.5)

Assume further that there exists a constant C such that

If(t,x)I <— CIu(t,x)I, in [0, T] x Q. Then, if cc > 0, we have u(t) > 0 for all t E [0, T] .

(5.6)

'

66 The heat equation

Proof. For

t E (0, T), we set v(t) = f((t))2.

Multiply (5.5) by -u (t) and integrate over Q. Integrating by parts and applying (5.6), we obtain (see the proof of Lemma 3.5.9) -

v -(t)

< f If(t)Iu (t)


f Iu(t)1u (t) = Cv

Integrating the last inequality, we obtain, for all 0 < s <

(

t . )

t < T,

v(t) < v(s)et-8). Letting s

1 0, it follows that v(t) e ct

I

hence the result.

f

(VP

)

2

= 0 , Vt E



0

Remark 5.3.2. Applying Proposition 5.3.1 with v = -u, we see that if cp <0, then we have u(t) < 0 for all t E [0, T]. We give a first result of global existence. Proposition 5.3.3. Assume that there exist K, C < co such that xg(x) <

CIx1 2 ,

( 5.7)

for Ixl <_ K. Then, for all E X, the solution of (5.1)-(5.3) given by Theorem 5.2.1 is global. Proof. We proceed in two steps. Step 1. Assume first that cp > 0. It is easy to verify that, for all T < T(p), h(t) = F(u(t)) satisfies (5.6). Then we have u(t) > 0 for all t E [0,T(cp)). Furthermore, by (5.7) we have g(x) < Cx,

for x >_ K. Since g is bounded on [0, K] and by possibly modifying C, it follows that g(x) < C + Cx,

I

for x > 0; and so IIF(u(t)) for all

t E [0,T(cp)).

+11 < C + CIIu(t)11,

Global existence 67

Applying (5.4), Lemma 3.5.9, and Gronwall's lemma, we deduce that ct 5.8) u(t)II < (C+ IIwII)e (

for all t E [0,T(cp)); and so (Theorem 5.2.1) T(cp) = oo. Step 2. In the general case, we set 0 = IcpI and we denote by v the corresponding solution of (5.1)—(5.3). Let T < min{T(),T(cp)} = T(cp). We easily verify that w = v — u fulfills the assumptions of Proposition 5.3.1; hence

u(t) < v(t), for all t E [0, T] . We then use z = v + u, where v is the maximal solution of the equation v = T(t) +

fo

t

T(t — s){—g(—v)} ds.

Since —g(—x) verifies the same assumptions as g(x), v is a global solution (Step 1). Applying Proposition 5.3.1, we obtain that u(t) > —v_(t), for all t E [0, T]; and so

IIu(t)II <— max{ IIv(t) II, 112(t)II for all t E [0, T]. It follows that T(cp) > T(i) = oo. This completes the proof. ❑

Remark 5.3.4. If C = 0 in (5.7), it follows from (5.8) that all solutions of (5.1)—(5.3) satisfy

IIu(t)II s K + II^II, for t >_ 0. Actually, this result can be sharpened by the following proposition, whose proof is quite simple. Proposition 5.3.5. If C = 0 in (5.7), then solutions of (5.1)—(5.3) satisfy

IIu(t)II 5 max {K, IIVII}, fort>0.

68 The heat equation

Proof. As in Proposition 5.3.3, we may restrict ourselves to the case cp > 0. Set k = max{K, Ilcpll} and multiply (5.2) by (u — k)+ E Ho (S2). Integrating by parts and setting f (t) = f ((u(t) — k)+) 2 , it follows that f'(t) < C fn g(u(t))(u(t) — k) + < 0.

We conclude that f (t) < f (0) = 0, and so u < k; hence the result.

❑

If the constant C of (5.7) is sufficiently small, we still have a result of this kind. To see this, we consider A given by (2.2). Proposition 5.3.6. If C < .\ in (5.7), then, for all cp E X, the solution u

of (5.1)—(5.3) satisfies

sup Iu(t)II < 00.

O
Proof. We have xg(x) < C1 + Cx 2 . As in Proposition 5.3.3, we may assume that cp > 0. Let c > 0 and let u be the corresponding solution. We multiply (5.2) by u. Integrating by parts and setting f (t) = f u(t) 2 , it follows that

f'(t) < —2a f(t) + 2f u(t) g (u(t)) <— —2(A — C)f (t) + C1IQI. By Lemma 8.4.6 (see below), we have f(t) f( 0 )+C1 A I ^ IC ; and thus sup IIu(t)IIL2 = K < oo. Let p E (N/2, oo). By (3.34) and (3.37), 0
)

E L 1 (O ,+oo).

We then note that there exists a constant C2 (see the proof of Proposition 5.3.3) such that

II9(u(t))+IILP <_ C2 + Cu(t)j,,

for all t > 0; and so

Ilu(t)II ^ IIkII + IIT(.)IIG(Ln,LO°)(C2 +C O S p IIu(S)IILP) C3 +C4 Sup IIu(S)IIL", O<_s<_t

for all t >_ 0. Finally, invoking Holder's inequality, for all e > 0, there exists C(e) such that

IIVIILP < EIIvII +C'(E)IIVIIL2,

Global existence 69

1

for all v E X. Choosing e such that eC4 < 1/2, it follows that C3 + KC(e)C4 + 1 sup IIu(s)II, 2 0<8
IIu(t)II for all t > 0. Therefore

sup IIu(s)II < 2C3 + 2KC(E)C4 i

0<s
for all t > 0; hence the result.

c

Now we show that if g fulfills certain conditions in a neighbourhood of 0, then solutions with small initial data are global. To see this, consider ) given by (2.2). Proposition 5.3.7. Suppose that there exists a > 0 and p < .\ such that xg(x) < µlx1 2 , for jxj < a.

Then, there exists A < oo such that, if JIVII < aA, the corresponding solution u of (5.1)-(5.3) is global and satisfies

IIu(t)II <_ All(pj!e -(a- µ ) t, for t>0. Proof. As in Proposition 5.3.3, it suffices to deal with the case cp > 0. Set T = sup{t E [0,T()); IIu(s)II < a for s E [0,t}} > 0.

Multiply (5.2) by u. Integrating by parts, and letting f (t) = f u(t) 2 , for t E [0, T], it follows that f'(t) <-2f(t) +2

J

and so

u(t)g(u(t)) < -2(.\ - C) f (t);

IlIt2e U u,

IIu(t)IIL2

-

-

for all t E [0, T]. We write e(



)t u (t)

= e U-u)tT(t)^ + f o

t

eU-u)(t-9)T(t - s) (e

3

F(u(s)))ds

70 The heat equation Note that (see the proof of Proposition 5.3.6)

IIe(a—µ)`T(t)IIG(LP, Loo) E L 1 (O,+oo). Consequently, arguing as in the proof of Proposition 5.3.6, we obtain

e (A— µ ) tlIu(t)II <—e tIHwHI+CIISOIIL2+

1 — 2

upteUu(s)Ij , os

for all t E [0, T]. Hence, there exists A such that

sup e a µ sIIu(s)II : AII^jI, (

—

)

O<s
for all t E [0,T]. It follows that if 1IcpII < aA, then T = oo, which completes the proof. 0 Remark 5.3.8. We see that if p 1 0, we may take 5 j a. If g has a higher order near 0, we have a more precise result. Proposition 5.3.9. Assume that there exist p > 0, e > 0, and a > 0 such that

xg(x) < plxI2+E, for xj < a.

Then there exist Q, -y > 0 such that, if 11 pll < 3, then the corresponding solution u of (5.1)-(5.3) is global and 1)u(t) II : 'YII^PIIe - ^ t , for t > 0.

Proof. As in Proposition 5.3.3, we may assume that y > 0. Set

for x > 0, and -^ = min 0 < 0. For all a E (0, ^), there exist 0 < x a < Ya such

that 0(xa) + a = 0(y a ) + a = 0.

Furthermore, we have a < X a
Global existence 71

-EX

Fig. 5.1 Suppose that IIcpjj 0,

1(t) = sup IIu(s)II• o<s
We have

IIF(u(t))+II s µllu(t)II 1 +E, for t E [0, T]. Applying (5.4), (3.37), and Lemma 3.5.9, it follows that

E a9 {e a i eat llu(t)II < MII^II +µMfo e sllu(s)II} +Eds < MII^vII + eMf(t) 1 +E Thus, 0(1(t)) +

> 0,

for all t E [0, T). If we assume further that MII^PII < , then this implies that f (t) E [0, xMII II) U (yMII u oc)• Since f (0) E [0, xMJIWII) and f (t) is continuous in t, we have ,

f (t) E [0, xxMliroli ),

for all t E [0, T). We conclude as in Proposition 5.3.7, and we obtain the result with ,3 = min{a, C/M} and ry = (1 + E)M/E. ❑

72 The heat equation

5.4. Blow-up in finite time

We begin with a result whose proof is quite simple. Consider the function 0 E D(B), such that Lv)+A')=0,

(5.9)

'>0 on S2,

(5.10)

I

(5.11)

Q = 1.

zp is easily obtained by solving the minimization problem (2.2), using the compactness of the embedding Ho (St) L 2 (c). We may choose a positive minimizing sequence, and obtain (5.10). is the first eigenfunction of —'L in Ho (Sl) (see, for example, Brezis [2], Theorem VIII.31, p. 192). Proposition 5.4.1. Suppose that there exist a, /8, e > 0 such that g(x) > ax 1 +E — Ox, for x > 0. Let cp E X, cp > 0 on St, be such that

f^

(-P)

Then T(cp) < oo. Proof. Denote by u the solution of (5.1)—(5.3) given by Theorem 5.2.1. By

Proposition 5.3.1, we have u(t) > 0 on S2, for all t E [0,T(p)). Set f (t) = i u(t)V) > 0,

n

for t E [0, T (cp)). Applying (5.9) and Lemma 2.34, we obtain, for all t E [0, T(W)), f-(t) = fut(t) = I ('^'U(t) + g(u(t)))

=

J

u(t)A +

J g(u(t))V) =

> — (A + li)f (t) + a J u(t)

— Af(t)

1+E

+ J g(u(t)) )

b

On the other hand, by (5.11) and Holder's inequality, ./ (t) <

(fSZ u(t)1+E )

U2 m )

l+f

< (f u(t)1+E 4^ )

and so

f'(t) ? f(t)(—(A+0)+of(t)E),

I}e i

(5.12)

Blow-up in finite time 73

for all t E (0,T(^p)). Let T = sup{t E (0,T(cp)), f' > 0 on (0,t)} > 0. If T < T(cp), we have f'(T) = 0 and f (T) > f(0), which contradicts (5.12). Thus, we have T = T(c') and f' > 0 on [0,T(cp)). Now let b > 0 be such that (a - S)f ( 0 ) 6 = A + ,Q.

I

We deduce from (5.12) that, for all t E (0, T(c )), f , (t)

a f(t) i+E + f(t)(—(A + Q) + (a — 6)f (t)) > of (t) -' + f (t) (—(a + p) + (a — 8)f ( 0 ) E ) ? (5f (t) 1

>

,

i.e.

_

(

f( t) -E ) > (bt) '

I

'

I I

From this, we easily deduce that 0 < f (t) -E <

e

f(0) (0)-E

e

- bt,

for all t E (0,T(cp)); and so EbT(cp) < f(0) E; hence the result.

❑

-

Remark 5.4.2. It is important to note that the above argument only shows that ebT(cp) < f(0) f and not that EbT(cp) = f(0) - E. For further discussion concerning this question, see Ball [1, 2]. -

Remark 5.4.3. If we take E X such that ( > 0, then for k > 0 large enough V = k( satisfying the assumptions of Proposition 5.4.1, and so T(a) < oo. Now we show a second blow-up result, using a different method. We need the functional E defined by

E(u) = 2 f IDu1 2 - fc(u) , for u E X fl Ho (1), where

o

I

G(x) = f g(s) ds,

for

x

x

E

I

][^.

Proposition 5.4.4. Assume that there exists K > 0 and a> 0 such that

xg(x) > (2 + e)G(x),

74 The heat equation

I

K xg(x) and v = o m K G(x). Let cp E X n Ho (.l) be for lxi > K. Set µ =
such that (2 + s)E(cp) < IQI(p — 2ev).

Then T(cp) < oo. The proof makes use of the following two lemmas. Lemma 5.4.5. Let T > 0 and u E C((0,T),X)nC((0,T),D(B))f1C'((O,T), L 2 (S2)). Then

Il '

'

(Du + g(u))ut + E(u(t)) = E(cp),

for all 0<s
1

f(zu

+ g(u))ut = f(_Vu. ^

Du t + g(u)ut) =

and hence the result.

❑

Lemma 5.4.6. Let cp E X and let u be the corresponding solution to (5.1)-

(5.3). Then dt in

u(t) 2 dx + 2j i Vu(t)1 2 dx = 2 f u(t)g(u(t)) dx;

f J ut + E(u(t)) = E(u(s)); t

(5.13)

(5.14)

z

for all0<s
C

Proof of Proposition 5.4.4. Set f (t) = fo u(t) 2 . By (5.13), for t E (0,T()), we have f'(t) _ —2

in

IVU(t) 1 2 + 2

J

juj
u(t)g(u(t)) +2

f f

1 ju>K)

u(t)9(u(t))

G(u(t))• >_ —2 f ^Vu(t)1 2 + 21 u(t)g(u(t)) + 2(2 + e) Juuj>K) {Iu,<x} o

(5.15)



Blow-up in finite time 75

On the other hand, observe that, by Proposition 5.2.3, we may let s = 0 in (5.14); and so 2(2 + e)

fl

G(u(t)) = -2(2 + e) juj^!K}

+(2 + e)

f1

G(u(t)) juj
Vu(t)1 2 - 2(2 +E) {E() - f'f ut} .

(5.16)

Observe further that 2)

u(t)g(u(t)) - 2(2 + e)

JJJ 1luI
4u

j
G(u(t)) > 2j52(µ - (2 + e)v). (5.17)

Putting together (5.15), (5.16), and (5.17), we obtain the following inequality, for 0
J

f'(t) > 2(2+e) f t ut+E J IVu(t)I 2 +2i52I(µ-(2+e)v)-2(2+e)E(cp). (5.18) o st sz In particular,

f'(t)

t

>

2(2

+

e) o f u.

(5.19)

2

Now set h(t)=f t f (s) ds. Applying the Cauchy-Schwarz inequality, we obtain t h'(t) - h'( 0 ) = f (t) - f ( 0 ) = 2 f i uue

o n

t

< Uo ^^2 ^ I

1/2

1/2 12

t

t

< 2h(t) ( 2u ^ =) Cf fu o J 2)

1/2

From (5.19), it follows that

h(t)h"(t) > (1+ 2) (h'(t) - h'(0)) 2 .

(5.20)

For the sake of contradiction, assume that T(^p) = oo. From (5.20), we have (1 + E/2)(h'(t) - h'(0)) 2 >_ (1 + E/4)h'(t) 2 for t >_ to large enough. It follows from (5.20) that (h(t)- e/4 ) 1,

0 ,

(5.21)

for t >_ t o . But h(t) > 0 and h(t) -, / 4 -> 0 as t --> oo. Thus there exists t l > t o such that (h - £/ 4 )'(t l ) < 0. Hence, by (5.21), 0 < h(t) -e / 4 < h(t1) -e / 4 + (t —

76 The heat equation

for t > tj; and so

h(t l)-E/4

t < tl - (h-E/4)'(tl)' for t > t1, which is absurd.

❑

Remark 5.4.7. The condition on g in Proposition 5.4.4 means that g is superlinear. Indeed, in the case in which there exists xo > 0 such that G(xo) > 0, it means that x - ( 2 +E)G(x) is non-decreasing on [xo, oo) (if G(xo) > 0 for a certain xo < 0, then x - ( 2 +E)G(x) is non-increasing on (-oo, xo]). In this case, G(x) > axe - bx 2 , for x >_ so. Thus, if we take ( E X f1 Ho (Sl) such that C > 0, then E(k) -* -oo as k --- oo. In particular, for k > 0 large enough the assumptions of Proposition 5.4.4 are fulfilled and so T(k) < oo. 5.5. Application to a model case We choose g(x) = alxIax, with a > 0 and a 0. We consider cpX, and we denote by u the corresponding solution of (5.1)-(5.3). Then we have the following results. If a < 0, then T() = oo, and u is bounded in X (Proposition 5.3.5). If a > 0, then T(cp) = oc if lI^PII is small enough (Propositions 5.3.7 or 5.3.9). On the other hand, for some we have T(cp) < oo (Remarks 5.4.3 and 5.4.7), and in that case IIu(t)JI > b(T(cp) - t) - * (Theorem 4.3.4). Notes. We have studied the heat equation in the space Co(1). It is also possible to study it in the spaces C'a(il) (see Friedman [1], and Ladyzhenskaya, Solonikov and Ural'ceva [1]) and in the spaces LP(fl) (see Weissler [2]). In LR(1l), we observe certain singular phenomena, such as non-uniqueness (see Baras [1], Brezis, Peletier and Terman [1], and Haraux and Weissler [1]). In some cases, the regularizing effect allows one to solve the Cauchy problem with singular initial data, such as measures (see Brezis and Friedman [1]). We can also consider more general non-linearities, depending on the derivatives of u, and more general elliptic operators than the Laplacian. See, for example, Friedman [1], Henry [1], and Ladyzhenskaya, Solonikov and Ural'ceva [1]. Some non-linearities with singularity at 0 have also been studied; see Aguirre and Escobedo [1]. Concerning systems, consult, for example, Dias and Haraux [1], Fife [1], and Smoller [1]. Various versions of the maximal principle for the heat equation can be found in Protter and Weinberger [1]. For the linear and non-linear regularizing effects, consult Friedman [1], Haraux and Kirane [1], Henry [1], Kirane and Tronel [1], and Ladyzhenskaya, Solonikov and Ural'ceva [1]. For more blow-up results, consult Fujita [1], Levine [1], and Payne and Sattinger [1]. The nature of blow-up is currently rather well known. See Baras

Application to a model case 77

I

and Cohen [1], Baras and Goldstein [1], Friedman and Giga [1], Friedman and McLeod [1], Giga and Kohn [1, 2], Mueller and Weissler [1], and Weissler [2, 3]. For the behaviour at infinity of solutions, consult Chapters 8 and 9, as well as, for example, Cazenave and Lions [2], Escobedo and Kavian [1, 2], Haraux [1], Henry [1], Kavian [2], Lions [1, 2], Weissler [4], and Esteban [1, 2].

I 1 0 I 1 1 1 ^

'

y

n

d 6 The Klein—Gordon equation ii

1

6.1. Preliminaries In this section, we give some technical tools that are essential in this chapter. 6.1.1.

An abstract result

'

Let X be a Hilbert space, let A be a skew-adjoint operator in X, and let (T(t))tER be the isometry semigroup generated by A. We have the following result.

I

Proposition 6.1.1. Let T > 0, x

u (t) = T (t)x +

f

for all t E [0, T] . Then the function t

H

id

o

2 d IIu(t)II 2

1

E

X, and f

E

C([0,T],X). Let u E

C([0,T],X) be given by

for all t

E

T (t - s)f (s) ds,

Il u(t)11' belongs to C' ([O, T]) and

= ( f(t),U(t)),

[0, T] .

Proof. Suppose that x E D(A) and f E C 1 ([0,T],X). By Proposition 4.1.6, we haven E C([0,T],D(A)) nC 1 ([0,T],X) and

u'(t) = Au(t) + f (t), for all t E [0, T]. Thus,

2 dt Ilu(t)11 2

'

= ( u(t),

u'(t)) = (u(t), Au(t) + f(t)) = (u(t), f (t)).

Hence

IIu(t)11 2 = IIx11 2

+ f (u(s), f(s)) ds.

(6.1)

0

In the general case, we approximate x and f by sequences (x)>o C D(A) and (fn)n >o C C'QO,T],X), and then we pass to the limit to obtain (6.1). The result follows since u and f are continuous functions. o

Preliminaries 79

6.1.2.

Functionals on Ho (Sl)

In this section, f is any open subset of RN. We consider a function g E C(R, R) such that there exists 0 < a < oo and C < oo so that g(0 ) = 0;

(6.2)

I9(x) — g(y) I ^ C(I xI

a

+ Iyl a )Ix — yI, b'x, y E R.

(6.3)

In particular, we have Ig(x) I < ClxI«+ 1 and so, for all p > a + 1, g defines an operator F : L o,(cl) —+ L^ ,(1l) by F(u)(x) = g(u(x)), for almost every x E Q. When there is no risk of confusion, we still denote by g the operator F. Applying (6.2), (6.3) and Holder's inequality, we easily obtain the following result. Proposition 6.1.2. Let a + 1 < p < oc. Then F is Lipschitz continuous from bounded subsets of LP(S2) to LT (1l). More precisely, we have 19(u) — g(v)II

^ C(u1j

+ IIvIIip)IIu — vIILP,

(6.4)

for all u, v E LP(52). For g as above, we define G E C(]R, R) by G(x) =

fo x g(s) ds.

(6.5)

Then G verifies condition (6.3), with a replaced by a + 1. Then, it follows from Proposition 6.1.2 that G allows us to define a functional V, Lipschitz continuous on bounded subsets of La+ 2 (Sl), by V(u) _ —

J G(u(x)) dx, `du E La+ (I) 2

(6.6)

More precisely, we have the following. Proposition 6.1.3. V is a functional of class Cl on L 2 (f ). Its derivative (which is a continuous mapping La+ 2 (52) -> (L 2 (1))' = L (1)) is given by V'(u) = for all u E L"+2(1)

— 9(u),

(6.7)

80

The Klein-Gordon equation

Proof. We have, for all x, y E R, x+y

(9(a) — 9(x)) do ;

G(x + y) — G(x) — y9(x) = f x

,

and so, applying (6.3), IG(x + y) - G(x) - y9(x)I

<- C , (Ixl a + Iyl a )Iyl 2 •

Applying Holder's inequality, we deduce that, for all u, v E L"+2 (1),

v(u + v) — V(u) + (v, g(u)) L ^ }2 L I = I v(u + v) — v(v) + f vg(u) dxI < C'(IIuliLa+2 + IIvllL.+2)IIvllL^+2

,

J

hence the result.

We now assume that, instead of (6.3), g satisfies the following weaker condition: I9(x) - g(y) I

C(1 + Ixl + IyI )Ix - yI, Q

a

(6.8)

for all x, y E R. In that case, we write (6.9)

9=91+92,

where 92 verifies (6.3) and gl verifies (6.3) with a = 0. For example, consider 9(x), 9i(x) = g(1),

I. g(-l),

if Ixi < 1; if x > 1; if x < 1.

On the other hand, if (N - 2)p < 2N, we have

HH(Q)'.., Lr(l)

(6.10)

Ln'(S2) , H-1O)

(6.11)

with dense embedding, and so

The result is the following.

< 4/(N - 2). Proposition 6.1.4. Let g satisfy (6.2) and (6.8), with 0 < a Then g is Lipschitz continuous from bounded subsets of Ho(1l) to H-'(Il).

Preliminaries 81

1

Proof.

By Proposition 6.1.2, gl is Lipschitz continuous from bounded subsets of L 2 (1l) to L 2 (fl), and so from bounded subsets of H(l) to H -1 (Sl). Invoking Proposition 6.1.2 again, 9 2 is Lipschitz continuous from bounded subsets of L 2 (52) to L+ (S2). Applying (6.10) and (6.11), with p = a+2, we deduce that 92 is Lipschitz continuous from bounded subsets of Ho (S2) to H -1 (52); hence the result. ❑ Proposition 6.1.5. Suppose that g satisfies the hypotheses of Proposition 6.1.4 with (N — 2)a< 2. Then g is Lipschitz continuous from bounded subsets of Ho (Q) to L 2 (1)_

Proof. The result is immediate for g l . Set p = 2(a + 2). Applying (6.3) and Holder's inequality, it follows that 1192(U) — 92(V)I1L2 < for all u, v E Ho (1l). by (6.10).

C(!I uII2

,

+ IvIIia)l rt — VIIL-+2,

I I

However, we have (N — 2)p < 2N; hence the result, ❑

We now consider G and V defined by (6.5) and (6.6). We have the following result. Proposition 6.1.6. Suppose that g satisfies (6.2) and (6.8), with a >_ 0 such that (N — 2)a < 4. Then V is a functional of class Cl on Ho (Sl). Its derivative (which is a continuous mapping from Ho (52) —> (Ho (S2))' = H -1 (1)) is given by

V'(u) = —9(u),

(6.12)

I

for all u E Ho (S2). Proof. We apply Proposition 6.1.3 to gi and 92, and we use embeddings (6.10) and (6.11).

❑

Corollary 6.1.7. Suppose that g satisfies the hypotheses of Proposition 6.1.6, with (N — 2)a < 2. Let T > 0 and u E C([0,T],Ho(1l)) fl C'([0,T],L 2 (S2)). Then the mapping t H V(u(t)) is in C'([0,T]), and we have

d V(u(t)) for all t E [0, T] .

_-J

sz

9(u(t))u t (t) dx,

I

(6.13) '

82 The Klein-Gordon equation

1

Proof. Suppose first that u E C' ([0, T], Ho (S2)). Then, for all t E [0, T], d V (u(t)) = (V'(u(t)), u (t))H-1,Ho

I I

=

- (g(u(t)),u'(t))x-1,Ho

= - I g(u(t))ut(t)dx.

n

It follows that V(u(t)) = V(u(0)) -

IJ t

st

gu(s))u t (s) ds.

(6.14)

By density, we deduce that (6.14) is still true when u E C([O,T],Ho(Sl)) n C 1 ([O,T],L 2 (Sl)); hence the result. ❑ 6.2.

Local existence

Throughout this chapter, we follow the notation of §2.6.3, §2.6.4, and §3.5.2. In particular, 1 is any open subset of R N , m> -A, X = Hl) x L 2 (cl) and Y = L 2 (0) x H -1 (0). We consider a function g E C(R,R) which satisfies (6.2) and (6.8) with (N - 2)a < 2. Finally, we consider G and V defined by (6.5) and (6.6). We define the functional E on X and the mapping F: X -> X by

E(u, v) =

211(u, v) II X + V (u)

= 21 f {Ivul 2 + mlul 2 + lv1 2 — 2G(u)}dam, F(u, v) _ (0, g(u)),

for all (u, v) E Ho (St) x L 2 (Il).

It is clear, from Proposition 6.1.5, that g defines a Lipschitz continuous mapping from Ho(S1) to L 2 (Sl), and so F is Lipschitz continuous on bounded subsets of X. Given (gyp, zli) E X, we are looking for T> 0, and u a solution of uE C([ 0, T] , Ho($)) nC 1 ([ 0 ,T],L 2 (f))flC 2 ([ 0 ,T],H -1 (Q));

(6.15)

u tt - Lu + mu = g(u),

(6.16)

U( 0 ) = ^P,

ut( 0 ) = V •

for all t E [0, T];

(6.17)

Applying Corollary 4.1.7 and Proposition 4.3.9, and arguing as in the proof of Proposition 3.5.11, we obtain the following result.

Let T > 0 and (p, 1p) E X. Let u E C([0, T], Ho (Sl))n C ([0,T],L 2 (cl)). Then u is solution of (6.15)-(6.17) if and only if U = (u,u t ) is 1solution of Lemma 6.2.1.

Local existence 83

U(t) = T(t)(,

0) +

JO

t

T(t - s)F(U(s)) ds,

(6.18)

for all t E [0,T]. In addition, if A E L 2 (S2) and E Ho(Sl), then we have u E C 1 ([O,T],H0(1)) nC 2 ([O,T],L 2 (SZ)) and Au E C([O,T],L 2 (fl)). Applying Theorem 4.3.4, we deduce a local existence result. Theorem 6.2.2. For all (cp, 0/i) E X, there exists a unique function u, defined on a maximal interval [0, T(, v>)), which is a solution to (6.15)-(6.17) for all T
(cp, V)) E X and let u be the corresponding solution (6.19)

E(u(t),ut(t)) = E(co, l0 ), for all t E [0,T(p,

L)).

Proof. We apply Proposition 6.1.1 and Corollary 6.1.7. It follows that dtE(u(t), ut(t)) = (F(u(t), ut(t)), (u(t), ut(t)))x — for all t E [0,T(cp,0)); hence the result.

f

g(u(t))ut(t) dx = 0, C

Remark 6.2.4. Proposition 6.2.3 justifies the study of the Klein-Gordon equation in the space X. Indeed, the energy E is related to the X-norm and, as we will see in the next sections, the conservation of the energy (6.19) allows us, under certain hypotheses on g and (p, vi), to obtain estimates for the solution in X (and so global existence), or results about blow-up in finite time. Remark 6.2.5. By using §4.4, we can solve problem (6.15)-(6.17) for T < 0 as well as for T> 0. Actually, note that u is a solution of (6.16) on [-T, 0] with u(0) = cp and u t (0) = Ali if and only if v(t) = u(-t) is solution of (6.16) on [0,T] with v(0) = cp and vt (0) = - V.

84 The Klein-Gordon equation

6.3. Global existence As for the heat equation (§5.3), we will state two kinds of result according to the hypotheses on g: global existence of all solutions (i.e. independent of initial data), or global existence of solutions with small initial data. Proposition 6.3.1. Suppose that there exists C < oo such that G(x) _< CIx1 2 for ai: x E R. Then, for all (v, ) E X, we have T(cp, 0) = oo. Proof. Set f(t) = II(u(t),ut(t))IIX, for t E [0,T(cp,)). By (6.19), we have f (t) <- f (0) _2

f

G(cp) +2

f

G(u(t)) <- f (0) -2

J G(cp) + 2C J u(t)

2,

(6.20)

for all t E [0,T(cp,i )). On the other hand, we have t u(s)ut(s) u(t)IIi2 = IH L2 + f t ddt is, Iu(s)I 2 = II^IIi2 +21 o (6.21) 0

IIIIL2 +2/ t f(s). 0

Applying (6.20), (6.21), and Gronwall's lemma, we obtain the result. Remark 6.3.2. If 2C < A+m (A being given by (2.2)), then for all (co, ii) E X the corresponding solution u of (6.15)-(6.17) satisfies sup II(u(t),ut(t))IIx < no. t>o

Indeed, in this case we easily verify that C f u(t) 2 < (1 - e) f (t), with E > 0, and it follows from (6.20) that e f (t) < C'; hence the result. Proposition 6.3.3. Suppose that there exist p < A + m (A being given by (2.2)) and 3 > 0 such that 2G(x) < PIxI 2 for IxJ <_ 3. Then there exists .5,K >0 such that if I I (cp, Ali) II x < 6, we have T (cp, Vi) = no and the corresponding solution u of (6.15)-(6.17) satisfies sup II (u(t), ut(t))II x < I)II x•

Proof. The hypotheses on g imply that there exists a constant C < no and k > 2 with (N - 2)k < 2N, such that 2IG(x)I < C(IxI 2 + Ixlk),

for all x E I1. By possibly taking larger C, we have 2G(x) < plxi 2 + Clxik,

Global existence 85

for all x E R. Sobolev's inequalities show that there exists a constant, which we will still denote by C, such that

2J G(u)

0 such that pfu2 < (1—

v)lIuIIi,

for all u E Ho (a), and consequently 2 f G(u)

< (1 - v)IIuIIHI +CIIuIIHi,

(6.22)

for all u E Ho(S2). Let (cw) E X, with II(^P,V))Ilx <1, and let u be the corresponding solution to (6.15)—(6.17). Using the notation of the proof of Proposition 6.3.1, we deduce from (6.20) and (6.22) that, for all t vf(t)
f

G(cp)+Cf(t) 2 .

(6.23)

Observe that

f G(

C(II (^v,VG)IIX + II(V, V,)IIX) < C(f(0) 2 + f(0) k );

LI I fl I I El I I El

and so, since f(0) < 1, there exists ME [1, oc) such that f(0) _2

f

G(cp) < vM f (0).

(6.24)

Set k/2 = 1 + E (e > 0) and

_

1

I

9( x \_ C x 1+E — x

v all a E (0, X) there exists 0 < x a, < Ya for x > 0, and set —X = min 6 < 0. For such that

O(xa) + a = 6(y a ) + a = 0. In addition, we have a < X a
I

1

1

86

The Klein—Gordon equation

-E2

Fig. 6.1 By (6.23) and (6.24), we have

6(1(t)) + Mf(0) > 0, for all t E [0, T(cp,

v')). Consequently, if we suppose that M f (0) <, we have f (t) E 0 [

,

xMf ( 0 )) U (yM f (0), oo),

and since f is continuous and f (0) < xM f( o ), we deduce that f (t) xMf(0) <

l+e

Mf ( 0 ),

for all t E [0,T(^p, 4 )). The result follows, with 5 = min{1, X/M} and K =

(1

+

e)/EM.

❑

Remark 6.3.4. If A = 0 (for example, if 52 = RN), we may apply Proposition 6.3.3 only if G(x) < 0 for 1st small. Then, we can replace the hypotheses G(x) < µ^x^ 2 by G(x) < / I x l 2 +E, for lxi small (E > 0), the proof being the same. Remark 6.3.5. When 11 is bounded, we may assume that the conditions on G involved in the statements of Proposition 6.3.1 and Remark 6.3.2 hold only for lxi large. The proof is the same; see Proposition 6.4.4.

Blow-up in finite time 87

6.4. Blow-up in finite time The main result of this section is the following. Proposition 6.4.1. Suppose that there exists e > 0 such that

xg(x) ? (2 + e)G(x), for all x E R. Then, if ( 0 and let u E C([0,T],Ho(c)) fl C 1 ([0,T],L 2 (fl)) fl C 2 ([0, T], H -1 (12)). Then the function t H f u(t) 2 belongs to C 2 ({0, T]) and we have

d2

J u(t) 2 dx = 2 J

u(t) 2 dx + 2(u(t), utt(t))H.,H- 1,

for alit E [0, T] . Proof. Set f (t)

_f

u(t) 2 , Vt E [0, Tj, n and assume first that u E C 2 ([0, T], Ho (S2)). Then f"(t) = 2

if, ut(t) 2

+2

f

ut(t)utt(t) =2

J

ut(t) 2 + 2 (u(t), utt(t))Ho,H -1,

for all t E [0, T]. It follows that f (t) = f ' ( 0 ) + 2f t

If

ut(s) 2 +(u(s),Utt(s))Ha,H- 1 } ds, o

'

(6.25)

for all t E [0,T]. By density, we then show that (6.25) still holds for u E hence the result. o

C([0,T], Ho (52)) fl C'([O, Tj, L 2 (5l)) fl C2 ([0,T], H - 1

(c));

Lemma 6.4.3. Let T> 0 and let u E C([O, T], Ho (S2)) n C l ([0, T], L 2 (1l)) fl C 2 ([0,T],H -1 (52)) be the solution of (6.16). Set

f(t)

_f

u(t) 2 ,

for all t E [0, T]. Then f"(t) =2

foralltE [0,T].

fn

ut(t) 2 —2

fn

f

IVu(t)1 2 — 2m n u(t) 2 +2

fn

u(t)g(u(t)),

88 The Klein-Gordon equation

Proof. We apply Lemma 6.4.2 and (6.16). It follows that f"(t) = 21 u(t)2 + 2(u(t), Au(t) - mu(t) + g(u(t)))Ho,H-^, t

for all t E [0, T]. It suffices to see that, for all w E Ho (S2), we have

(w, OW)Ho H-1 = —J IDwI2. This is a consequence of Lemma 2.6.2 if tw E L 2 (S2), and follows by density ❑ otherwise. Proof of Proposition 6.4.1. Let (cp, ') E X be such that E(
f(t) = f u(t)2,

for all t E [0,T(cp,i')). By Lemma 6.4.3, we have

f ut (t) >2 J u(t)

f (t) =2

J u(t) - 2m J u(t)

2

-2 f IVu(t)1 2 - 2m

2

-2

J Vu(t)!

2

2

2

f u(t)g(u(t)) + 2(2 + E) J G(u(t)), +2

for all t E [0, T(, ii')). Applying Proposition 6.2.3, we deduce that f " (t) > E { f IVu(t)1 2 + m f u(t) 2 I+(4+E) z

ll sz

fn

u t (t) 2 -2(2+e)E(cp,z/^), (6.26)

for all t E [0, T(, si)). For the sake of contradiction, assume that T(cp,ip) = oo. In particular, we deduce from (6.26) that f"(t) > -2(2 + E)E( W , 7p) > 0, dt > 0. It follows that f (t) --> oo as t -- oo. On the other hand, applying (6.26) and the Cauchy-Schwarz inequality, it follows that

f (t)f"(t) > (4+E) fn and then

u(t)2

fn u(t)2 > (

4 +E)Y

o

ut(t)u(t)) 2 > ( 1 +

4) f ' (t) 2 ,

f(t) i <0, `dt>0.

We conclude as for Proposition 5.4.4 (inequality (5.21) and below).

❑

Application to a model case 89

In the case in which ci is bounded, we can weaken the hypotheses of Propo-

sition 6.4.1. Proposition 6.4.4. Suppose that 52 is bounded and that there exist K < oo and e > 0 such that

,

xg(x) ? (2 + e)G(x),

for IxI > K. Set µ = min xg(x) and v = max G(x). Then, if (cp, 0) E X - lxj<x lxl
-

(2+e)v),

we have T(cp, 0) < oo.

I

Proof. We argue as in Proposition 6.4.1, using

fn u(t)g(u(t))

= f1juj:5K}

u(t)g(u(t)) + 4uj>K} u(t)g(u(t))

> mjci + (2 + E)

f1 juj>Kj

G(u(t))

J G(u(t)) - (2 + E) J > (m - (2 + e)v) IcI + (2 + e) G(u(t)), J

>_ m^c + (2 + e)


G(u(t))

I

for all t E [0, T(cp, )).

Remark 6.4.5. For some comments on the hypotheses of Proposition 6.4.1 and 6.19, see Remark 5.4.7. In particular, if there exists xo >_ K such that G(xo) > 0, and if we write (<, i) E X with C > 0 a.e. on S2, we have E(k(, k97) -4 -oo as k -> oo. In particular, for k sufficiently large, (kc, ki1) satisfies the conditions of Propositions 6.16 or 6.19, and so T(kc, krl) < oo.

6.5. Application to a model, case We choose g(x) =` aIxI' x, with a # 0, a > 0, and (N - 2)a < 2. We consider (gip, Ali) E X and we denote by u the corresponding solution of (6.15)-(6.17). Then we have the following results. If a < 0, then T(,) = oo and (u, u t ) is bounded in Ho (S2) x L 2 (fl) (Remark 6.3.2).

d

0

90 The Klein-Gordon equation

') IIx is small enough (Proposition 6.3.3). If a > 0, then T (cp, i1) = oo if 1l ( In addition, for some (p,) E X, we have T (cp, i/I) < oo (Remark 6.4.5), and in that case II(u(t),u t (t))11x > 6(T(^,') -t) a (Theorem 4.3.4). Notes. For more about local and global existence, in the framework of Chapter 6, consult Browder [1] and Heinz and Von Wahl [2], and for more about blow-up phenomenon, see Levine [2, 3], J. B. Keller [1], and Glassey [1, 3]. In the general case (ci RN) the behaviour at infinity of solutions is well known only in the dissipative case. See §9.4 and, for example, Haraux [1, 2]. In the conservative case, we only have some partial results, often limited to dimension 1. See Brezis, Coron, and Nirenberg [1], Cabannes and Haraux [1, 2], Cazenave and Haraux [2, 31, Cazenave, Haraux, Vazquez, and Weissler [1], Friedlander [1], C. Keller [1], Payne and Sattinger [1], and Rabinowitz [1, 2]. On conservation laws, see Serre [1]. For S2 = RN, there exist estimates of the same kind as in §7.3; see Brenner [1, 2], Ginibre and Velo [6, 9], and Marshall, Strauss, and Wainger [1]. These estimates allow us, for the local existence, to replace in (6.8) the condition (N - 2)a < 2 by the weaker condition (N - 2)a < 4. See Ginibre and Velo [8, 10] and Jorgens [1]. If condition (6.8) is not satisfied, we only know how to build solutions in the case xg(x) < 0, but we do not know whether uniqueness holds. See Strauss [1,2], as well as the very interesting numerical study of Strauss and Vazquez [1]. Again for S2 = RN, we know how to investigate the dispersive properties of the linear equation to show the global existence for small initial conditions, with non-linearities that depend on derivatives of u (Klainerman and Ponce [1]). If the order of the non-linearity at 0 is not sufficiently high, blow-up in finite time may occur for arbitrarily small initial data. See Balabane [1, 2], Sideris [2, 3], Hanouzet and Joly [1], and John [1-3]. For some non-linearities, there exist solutions of the form u(t, x) = eiwtcp(x). These solutions are called stationary states. See, for example, Berestycki, Gallouet, and Kavian [1], Berestycki and Lions [1], Berestycki, Lions, and Peletier [1], and Jones and Kiipper [1]. The behaviour at infinity of solutions is rather well known in the repulsive case, in which the solutions behave asymptotically as the solutions of the linear equation. See Brenner [1, 2], Ginibre and Velo [8, 10], Morawetz and Strauss [1], Reed and Simon [1], and Sideris [1]. In the attractive case, we know mainly how to study the stability of certain stationary states. See Berestycki and Cazenave [1], Blanchard, Stubbe, and Vazquez [1], Cazenave [3], Cazenave and Lions [1], Grillakis, Shatah, and Strauss [1], C. Keller [1], Payne and Sattinger [1], and Shatah and Strauss [1].

7 The Schrodinger equation

7.1. Preliminaries Throughout this chapter, we use the notation of §2.6.5 and §3.5.3. In particular, 1 (SI,C) and Y = L 2 (cl) = L 2 (SI,C). The isometry groups generated by A and B are both denoted by (T(t)) tER (see Lemma 3.5.12). Given g : Ho (cl) -* H -1 (c), Lipschitz continuous on bounded subsets and E Ho (S2), we are looking for T> 0, and u a solution of the following problem:

X = H '(S2) = H -

-

UE

C([0,T],Ha(cl) flC'([0,T],H-1(f));

iu t + Au + g(u)

= 0, Vt E [0, T];

u(0) = `. Applying Lemma 4,

(7.1) (7.2) (7.3)

and Corollary 4. 1.8, we obtain the following result.

Lemma 7.1.1. Let T > 0, cp E Ho (S1) and let u E C([0, T], Ho (ii)). Then u is a solution of (7.1)-(7.3), if and only if u is solution of

u(t) = T(t) cp +

ifo

t

T(t(7.4) - s)g(u(s)) ds,

for all t E [0,T]. On the other hand, applying Proposition 4.1.9, we obtain a sharpened version of Lemma 7.1.1, which will also be used in what follows. Lemma 7.1.2. Let T> 0, cp E Ho (SI) and let u E L 011 ((0,T), Ho (I )). Then

u is a solution of

u E L((O,T),

Ho (Q)) n W 1

((0, T), H- 1(Q)); (7.5) (7.6)

iu t + Du + g(u) = 0, a.e. t E [0, T];

u(0) = y , (7.3) if and only if u is solution of (7.4). Remark 7.1.3. If (7.1)-(7.3) can be solved for T> 0, then in general it can also be solved for T < 0 (see §4.4). If g satisfies certain symmetry properties,

92 The Schrodinger equation

it is es pecial ly clear. Indeed, if we suppose that g(u) = g(u), then v given by v(t) = u(—t) is a solution of (7.2) on [0, T] if and only if u is a solution of (7.2) on [—T, 0]. 7.2. A general result

Assume that g : R + —4 ll is (globally) Lipschitz continuous and that g(0) = 0. We extend g to the complex plane by setting

g(z) = Izlg(lzI),

(7.7)

for all z E C, z # 0. We also define the function G by G(z) = j o

IZI

g(s) ds,

(7.8)

for all z E C. Then, g defines a mapping L 2 (S2) —> L 2 (1l), which we still denote by g *, by g(v)(x) = g(v(x)),

(7.9)

for all v E L 2 (1l) and for almost all x E f. In addition, g is Lipschitz on L 2 (S2) (see §6.1.2). Furthermore, if we set V (v)

= — J G(v(x)) dx, Vv E L 2 (1l),

(7.10)

then V E C 1 (L 2 (1Z),l18) and V'(v) = —g(v), t/v E L 2 (1l).

(7.11)

Finally, we define the functional E E C l (Ho (Sl),1[8) by E(v)

= f I Vv

2

dx + V(v),

(7.12)

for all v E Ho (1). We have the following result. Theorem 7.2.1. For all cp E L 2 (S2), there exists a unique function u E

C([0, oc), L 2 (Sl)) which is a solution to (7.4) for all T < oo; furthermore,

IIu(t)IIL2 = II(PIIL2,

(7.13)

* This is a common abuse of notation, g denotes both a mapping C —+ C and a mapping L2 --> L 2 . However, the context makes it clear which is used.

A general result 93

for all t >_ 0. If we assume further that E Ho (S2), then u E C([0, oo), Ho (S2)) n C 1 ([O,00),H - '(1l)) and E(u(t)) = E(9), (7.14)

I

for all t >_ 0. If in addition Acp E L 2 (12), then Au E C([0, oo), L 2 (1Z)) and u E C'([O,00),L Z (Sl))• The proof makes use of the following lemma.

I

Lemma 7.2.2. Let T > 0 and let u E C([0,T],D(B)) fl C'([O,T],L 2 (52)). Then the function t '-* Vu(t)II belongs to C 1 ([O,T]) and we have ;

I dt II u(t)II i2 = 2(-^u(t), ut(t)),

for all t E [0,T].

I

Proof.

The result is clear if u E C'([O, T], D(B)), and is obtained by density in the general case (see, for example, the proof of Proposition 6.1.1). ❑

Proof of Theorem 7.2.1.

We proceed in seven steps.

Step 1. cp E L 2 (Il). The global existence in L 2 (cl) is a consequence of Theorem 4.3.4, since K(M) is bounded.

Step 2. cp E D(B). If cp E D(B), the regularity is a consequence of Proposition 4.3.9. In that case, u is solution of (7.2) for all T < oo, and (7.2) is satisfied in L 2 (1l).

Step 3.

The conservation law (7.13). If (p E D(B), and taking the scalar product (in L 2 (cl)) of (7.2) with u, we obtain

2 dt IIu(t) IIi2 = (u t (t), u(t)) = Iiou(t), u(t)) + (i g (u(t)), u(t)) = 0.

(7.15)

Indeed, for all w E D(B), we have (see Lemma 2.6.2) (i^u(t), u(t)) = Ref iL ww = _Ref iIVwl 2 = 0;

I

and (applying (7.8)) (ig(u(t)), u(t)) = Re

Jsz

ig(w)iJ

= Reig(IwI)Iwf = 0. fn

Then (7.13) follows from (7.15). In the general case y E L 2 (52), we obtain (7.13) by density, applying Proposition 4.3.7.

I



94 The Schrodinger equation

Step 4. The conservation law (7.14) for cp E D(B). Taking the scalar product

(in L 2 (Sl)) of (7.2) with u t , we obtain 0 = Ref iIutI 2 = (iut, ut) _ -(AU, u t ) -

(g(u), ut),

for all t E [0, oo). Applying Lemma 7.2.2 and Corollary 6.1.7, we deduce that (d/dt)E(u(t)) = 0; hence (7.14). Step 5. If E Ho (S2), then u is weakly continuous as a map from [0, oo) to

Ho (S2) and we have E(u(t))

<

E(cp),

(7.16)

for all t E [0, oo). Indeed, consider a sequence (W n ) n>o C D(B) such that cp n --i cp in Ho (1) as n -+ oo, and let u n be the corresponding solutions of (7.2). Let T > 0. Since g is Lipschitz continuous on L 2 (St), it follows from (7.13) and (7.14) that u n, is bounded in L° ° ((0, T), Ho (S2)). By Proposition 4.3.7, we also have (7.17) u in C([0, T}, L 2 (Sl)) as n -, oo. u(t) in L 2 (52) as n --> oo, and In particular, for all t E [0, T], we have u(t) —u(t) ^[U n (t)[[H1 is bounded. Therefore, u(t)

-k

u(t)

in Ho (1) as n -+ oo.

(7.18)

Applying Proposition 1.4.24, we deduce that u E L°°((0,T),Ho(fl)). Since u E C([0, T], L 2 (f)), it follows that u is weakly continuous from [0, T] to Ho(Sl). Applying (7.17), (7.18), and the weak lower semicontinuity of the norm in Ho (S2), we deduce (7.16) from (7.14). We conclude observing that T is arbitrary. Step 6. The conservation law (7.14) for

cp E

Ho (S2). For all s E [0, oo), we set

v(t) = u(s - t), `dt E [0,s].

On checking, it is immediate that v is solution of (7.2) on [0, s]. Then, we may apply (7.16). It follows that

I

E('P) E(u(s)).

(7.19)

By putting together (7.18) and (7.19), we obtain (7.15). '

t

Step 7. If cp E Ho(11), we have u E C([0,00),Ho(Q)) n C'([0,co),H -1 (SI))• Indeed, since g is continuous on L 2 (Sl), we deduce from (7.14) that the function t [u(t)11Hi belongs to C([0, oo)). Since u is weakly continuous from [0, 00) to Ho (S2), we then have u E C([0, oo), Ho (S))); and so by Corollary 4.1.8, u E C l ([0, oo), H -1 (I )). This completes the proof. ❑

The linear Schrodinger equation in R N 95 Remark 7.2.3. Theorem 7.2.1 applies only if the non-linearity g is mild. Indeed, if we consider g(u) = IuI "u (a >_ 0), we may use it only if a = 0. However, Theorem 7.2.1 will be useful to prove a more general result for S2 = R N (§ 7.4). It is then necessary to specify the dispersive properties of the Schrodinger equation in R N These properties are described in the next section. .

7.3. The linear Schrodinger equation in R N We suppose that SZ = R N , and we consider T > 0. We are going to apply the results of §3.5.4 to give estimates for the solutions of the inhomogeneous Schrodinger equation

{

iut+Au +f =0; u(0)=gyp.

To do this, we define the operators 4?, 'I', and O t (for t E [0, T]) by

4? f (t)

= 1 t T (t - s) f (s) ds, dt E [0, T];

= J T(s - t) f (t) dt, Vs E [0, T]; Ot,f (s) = f T(s - v)f(o) do, ds E [0, T]; o IY f(s)

T

s

for all f E L 1 ((0,T),H -1 (RN)). We easily verify (see Lemma 4.1.5) that', 'I', and O t are continuous from L I ((0,T),H -1 (RN)) to C([0,T],H -1 (RN)), and from L 1 ((0,T),H'(R' )) to C([0,T],H 1 (R")) Definition 7.3.1. We say that a pair (q, r) of positive numbers is admissible if the following properties hold: (i)2
Observe that if (q, r) is an admissible pair then we have q E (2, oo] (q E [4, oo] if N = 1). The pair (oo, 2) is always admissible. Remark 7.3.2. If (q, r) is an admissible pair, then we have in particular H 1 (RN) . Lr(RN) with dense embedding and L''(RN) H -1 (RN). It follows that C([0,T],H l (1RN)) — LQ((0,T),L''(RN)) and L9'((O,T),Lr (RN)) y

so

L 1 ((0,T),H -1 (R ')). In particular, the operators 4D, 'Y, and O t are continuous from LQ'((0,T),L" (RN)) to C([0,T],H-1(RN)).

96 The Schrodinger equation

Remark 7.3.3. If (q, r) is an admissible pair, and taking suitable test functions, we easily verify that for all u E L 9 ((0,T),Lr'(R N )), we have f IIiIIL9((o,T),L') =supRe

uapdxdt1;

L 9 ((0,T), Lr (RN)), IkPIIL9'((o,T),Lr') ,

w

^P

J

RN

J O

E

1.

By truncation and regularization, we also have ( f T

IlUIIL4((o,T),Lr) = sup {

E

f

Re

J0 JR^

u-pdxdt1;

L 9' (( 0 ,T),L r' (R N )) nC([O,T],Hl(RN)), lfplILa'((O,T),Lt')

1 }.

The main result of this section is the following. Proposition 7.3.4. Let (y, p) be an admissible pair according to Definition 7.3.1 and let f E L-Y'((0,T),LP (RN)). Then 4?f E C([0,T], L 2 (R")). In addition, for any admissible pair (q, r), we have Pf E L 4 ((0,T),Lr'(R N )) Finally, there exists a constant C, depending only on y and q, such that II'Pf II L11((o,T),L•) < C(7, q) II f II L-' ((o,T),LP' )'

(7.20)

for all f E Lry ((O,T), LP (II^ 1`')). Proof. The proof proceeds in six steps. Step 1. For all admissible pairs (q,r), the operator 4' E C(Lq'((0,T),LT(RN)), L 9 ((0,T),L''(R N ))), with norm depending only on q. By density, we need only consider the case in which f E C([0,T],Lr'(R N )). In this case, it follows from Proposition 3.5.14 that (Pf E C([O,T], L''(R')), and that for t E [0, T] we have

II 4 )f(t)IILr

< fo

t

It — sI -- # — f ) IIf(s)I1L, , ds < fT It — sI Q^ 11f (s)IILr , ds.

The classical estimates on Riesz' potentials (see Stein [1], Thm. 1, p. 119) then

give

iii II L9((o,T),Lr) < CII f

IIL9'((o,T),L-')'

where C depends only on q; hence the result.

The linear Schrodinger equation in R'

97

1

Step 2. Similarly, T and O t are continuous from L 9 '((0,T),L"(R N )) to L 9 ((0,T),Lr(R N )) with norms depending only on q. E G(Lq'((0,T),L'(l[8'v )),C([O,T],L Z (R N ))), for all Step 3. The operator admissible pairs (q, r), with norm depending only on q. By density, we need

only consider the case in which f E C([0,T],L''(R N )), and using a regularizing sequence, we may assume further that f E C([0,T],H l (IR N )). In that case, we have f E C([0,T], H l (R")). Applying Corollary 3.2.6, and denoting by (, ) the scalar product in L 2 (R N ), we obtain, for all t E [0, T], IIf(t)IIL2=( f t T(t = I

—

s)f(s)ds,

0 '

fo T(t

—

v)f(a)de)

f t (T(t — s) f (s),T(t — a) f (Q)) da ds

tt

=f f(f(s),T(s — a)f(o)) dads =

f

(f (s), Ot,f (s)) ds.

Applying Holder's inequality, first in space and then in time, and using Step 2, it follows that

IIr(t)IIL2

<-IIfIIL9'((O,T),Lr')Ilot,IIIL9«O,T),Lr) <-C(q)IIfIIL4'((o,T),L-');

I

hence the result, since t is arbitrary. Step 4. Similarly, 'I' and O t are continuous from L 2 (RN)), with norms depending only on q.

((0, T), U'

(RN))

to C( [0, T],

Step 5. The operator c E (L 1 ((0, T), L 2 (RN)), L 9 ((O,T), L''(RN))), for all admissible pairs (q, r), with norm depending only on q. Let cp E C([0, T], Hl (RN)) nC([O,T],L , (RN)) be such that IIcpjIL9(lo,T),L-) < 1. In particular, (f E C([O,TJ, L 2 (R n')), and (by Step 4) IIW,vIIL—((o,T),L2) < C(q). We have fT 0

Re

J

R

N

r

T

JT(f (t), co(t)) dt _ (T(t — s) f (s), cp(t)) ds dt J ,T

c i dx dt =

ft

_ Jo J = JOJ 0

o

ft

T

T

(f(s),T(s — t)^p(t)) ds dt

T

(f (s), T(s — t)^p(t)) dt ds

s

= f (f (s), `T a(s)) ds.

I

98 The Schrodinger equation

Applying the Cauchy-Schwarz inequality, and then Step 4, it follows that fT

rRe

✓0

fRI

4)fipdxdt

< IIfIIL1((O,T),L2)II'c IIL°°((o,T),L2) < C(q)II f II L1((O,T),L2) II (PII L9'((O,T),L. ) < C(q) II f II L1((o,T),LZ).

Then it suffices to apply Remark 7.3.3. Step 6. Conclusion. Let ('y, p) be an admissible pair. By Steps 1 and 3, 'D is a continuous operator Lry'((O,T),LP (R N )) -> C([0,T],L 2 (R N )) and L 7 '((0,T), L° (R N )) --> Lry((0,T),LP(R N )). Let (q,r) be an admissible pair such that 2 < r < p, and let 0 E [0, 1] be such that 1=0+ 1-6

r

p

2

and

1_0+ 1-0

q

ry

o0

Applying Holder's inequality in space and then in time, we obtain II^f IIL4((0,T),Lr)

<_ II4'f II L-,((O,T),LP) II ^f II L4(O,T),L2) < CII f II L7'((O,T),LP')'

where C depends only on -y and q. Then, 'I' is continuous Lry'((0,T), LP (R N )) -^ L 9 ((0,T), Lr'(R N )). Now let (q, r) be an admissible pair such that p < r, and let p E (0, 1) be such that 1

ry'

P+

1

-µ and 1 =u+ 1-,u q' p' 2 r'

1

By Steps 1 and 5, 4 is a continuous operator L 9 '((0,T),L' (R n')) --> L 4 ((0,T), LT(RN)) and L 1 ((0,T),L 2 (R N )) -* L 4 ((0,T),Lr'(R N )). By interpolation (see Bergh and Lofstrom [1], Thm. 5.1.2 p. 107), 1 is a continuous operator L°((0, T),

L 6 (IR N )) -> LQ((0,T),Lr'(RN)),

for all (a,6) such that

1=0+ 1-0 and 1 q'

o

1 - B + 1-0

6

2

r' '

with 0 E (0,1). Choosing 0 = p, we deduce that 4) E L(L 1 '((0,T), LP (R N )), L 9 ((0, T), Lr(R' ))), which completes the proof. ❑ Corollary 7.3.5. Let (y, p) be an admissible pair. Let f E C([0,T], L 2 (R' )) nW 1,7' (( 0 ,T),L P (R N )). Then F1 E C([0, T], H2(RN)) flCl([0,T],L2(RN))•

Proof. We have in particular f E W l " l ([0,T],H -1 (1[x')), and so (applying Proposition 4.1.6) Ff E C([0, T], H'(R n )) f1 C 1 ([0, T], H -i (R' )) and

d

Ibf

= iL4)f + f,

(7.21)

The linear Schrodinger equation in RN 99

for all t E [0, T]. In addition, we have ^f(t) = f t T(s)f(t — s)ds, 0

and so

dt"Ds(t) = T(t)f(0)

+

fo

t

T(s)f'(t — s) ds = 4) f (t).

This relation is clear if f E C 1 ([O,T],L''(R' )) C C l ([O,T],H -1 (R 1")), and is obtained by density in the general case. Applying Proposition 7.3.4, it follows that 'f E C 1 ([O,T],L 2 (RN)). By (7.21), we have A-Df E C([O,T],LZ(]f N)); and so 4? f E C([O,T), H 2 (IRN)). In the spirit of Proposition 7.3.4, we give the following result. Proposition 7.3.6. Let cp E L 2 (1RN). For all admissible pairs (q,r) we have T(•)cp E LQ(1R, L' (RN)). Moreover, there exists a constant C, depending only on q, such that

II7()coIIL9(R,L*) <— C(q)IIpIIL2, for all cp E L 2 (RN) Proof. The proof is similar to that of Proposition 7.3.4, and so we only sketch it briefly. Set +00 +00 A1 (t) _ f T(t —s)f(s)ds, rf = f T(—t)f(t)dt.

We show that (see Step 1 of the proof of Proposition 7.3.4)

IIA1IIL9(R,L.)
)

Next we deduce (see Step 3 of the proof of Proposition 7.3.4) that

IIrfIIL2 < IIfIIL9'(R,L*' ) We conclude (as in Step 5 of the proof of Proposition 7.3.4) by noting that

I

+00

+00

f (T(t), (t)) = I ( f T(—t)(t)) < C(q)IIIIL2IIIIL9'(^Lr'),

and applying Remark 7.3.3.

M

100 The Schrodinger equation

7.4. The non - linear Schrodinger equation in RI'': local existence We extend the result of Theorem 7.2.1 to more general non-linearities. From now on, g is a function R+ --* R such that g(0) = 0 and such that there exist K < oo and a > 0 with (N — 2)a < 4, such that Ig(y) — g(x)I < K(l

+ Ixi" + IyI a )Iy — x1,

for all x, y E R+ . We extend g to the complex plane by formula (7.8) and we define the function G : C —* R + by formula (7.9). g defines a mapping Lj + 1 — Lj , by formula (7.10). We verify (see §6.1.2) that g is continuous from Hl(RN) to H —l (lfB N ), and that V defined by formula (7.11) is continuous on H 1 (R N ). Finally, we define the functional E on H 1 (R') by formula (7.12). The main result of this section is the following theorem. Theorem 7.4.1. There exists a function T : H 1 (R N ) --, (0, 00] that satisfies the following properties. For all cp E H' (RN), there exists a function u E C([0,T(cp)), H 1 (R N )) which is the unique solution of (7.1)—(7.3) for all T < T(cp). In addition, (i) ifT(cp) < oo, then lju(t)IlH1 —; 00; (ii) 11u(t)IIL2 = IIoIIL2, for all t E [0, T('));

(iii) E(u(t)) = E(cp), for all t

E [0, T(cp));

(iv) if cc E H 2 (TR"), then u E C([0,T(cp)),H 2 (R N

))

n C 1 ([0,T(cp)),L 2 (R N )).

Finally, the function T : H'(R") --> (0, oo] is lower semicontinuous, and u depends continuously on cp in the following sense: if cp,, --+ in H l (RN), and if u,,, and u denote the corresponding solutions of (7.1)—(7.3), then u,,, —> u in for allT
Remark. Property (iv) above is a regularity result. As well as being interesting in its own right, this result will be indirectly useful in §7.6 (blow-up in finite time) and §7.7 (behaviour at infinity) to establish the fundamental identity (7.58). This property has been included in Theorem 7.4.1 since its proof requires the approximation argument which we use for the local existence in Hl(R N ). Doing this, the proof of Theorem 7.4.1 turns out to be much more complicated. Consequently, at the first reading, the reader should omit Lemmas 7.4.7, 7.4.9, and 7.4.10, Corollary 7.4.8, Steps 3 and 5 of the proof of Proposition 7.4.12, and Step 3 of the proof of Corollary 7.4.13 (identified by an asterisk). These results, whose aim is to prove property (iv), are somewhat technical and may obscure the proof of the local existence in H'(R N ) In fact, the proof of the local existence itself is not easy. We sketch it briefly and point out the difficulties.

The non-linear Schmdinger equation in ][fi n': local existence 101

In §7.2, it appears clearly that the methods developed so far do not allow to solve (7.1)—(7.3) if g is superlinear (see Remark 7.2.3). In R n', we can get round this difficulty by using the inequalities proved in §7.3. Recall, however, that these inequalities are specific to R N . The idea of the proof of Theorem 7.4.1 is the following. We approximate g in a convenient sense by a sequence (g n,),, >o of globally Lipschitz non-linearities, which satisfy the same assumptions as g uniformly with respect to m. In a preliminary section (§7.4.1), we apply the estimates of §7.3 to establish various inequalities for g„ i and for g. In general, they derive from Proposition 7.3.4 and Holder's inequality. We apply these inequalities in §7.4.2. First, (Lemma 7.4.11), we obtain immediately a uniqueness result (note that, at this stage, we do not use approximation but only Proposition 7.3.4 and Holder's inequality). Next (Proposition 7.4.12) applying Theorem 7.2.1 to problem (7.1)—(7.3) with g replaced by g.,,,,, we build solutions u,,,, of the approximate problems. On a small time interval, we estimate these solutions Urn , uniformly with respect to m, using mainly conservation of energy. These estimates allow us to pass to the limit as m —> oo, and to obtain a local solution u. Solving the backward problem, we demonstrate that energy is conserved (Corollary 7.4.13). Finally, we complete the proof showing alternative (i) and continuous dependence. To establish property (iv), we may suppose that cp E H 2 (R') and deduce estimates of U rn in H 2 (R N ) which are independent of m. We then pass to the limit in these estimates as in —> oo. 7.4.1.

Some estimates

I

The following notation will be useful in what follows. We define k E C(C, C) by k(z)

Izi < 1; if IzI > 1.

g(z), =! l zg(1),

if

for all m E N, we define g,,, E C(C, C) and h,n E C(C, C) by

9 m (Z) —

if

m9(m),

hr,(z) (We have in particular g l

Iz) < m; if Iz) > m;

1 9(z),

= 9rn(z) -

k(z)•

= k.) Also, let 900=9; h00=g—k.

Form E N U {oo}, we define G m E C(C, R) by Gm(z)

f o

IZI

9,,,.(s) ds.

I j

102 The Schrodinger equation Observe that for all m E N, 9 m is globally Lipschitz continuous. For all u E H l (R N ), we set Vm(u) = — J Gm(Iul);

E () = 2 f m

N

IVul 2 dx

+V,n(u).

We set V^ = V and E.. = E. Finally, for all T> 0, u E L-((0,T),H 1 (R N )) and mENU{oo},weset

Qm(u)(t) = f T(t — s)9m(u(s)) ds; o xmlu)(t) = f T(t — s)h m (u(s)) ds; o

IC(u)(t)

= fo

t T(t — s)k(u(s)) ds;

for alit [0,T], and we let H =71, 9.=G. By possibly modifying the value of K, we readily verify the following inequalities:

(7.22) K( 1 + Iz2I" + Izi1 a I)Iz2 — z1I; «+1. (7.23) I9m(z) — g(z) < KIzl (7.24) Ihm(z2) - hm(zi) <- K(Iz2I" + IziIl)Iz2 - ziI; I9m(z2) — gm(zi)

Ik(z2)

— k(zi) < K^z2

(7.25)

— z1I;

for all m E N U {oo} and all z, zi, Z2 E C. In what follows we set r = a + 2 and a = 4(a + 2)/(Na), so that (a,T) is admissible. Lemma 7.4.2. Let M < oc. There exists C(M) < oo and v > 0 such that II k(v) — k(u))1 L2 < C(M)I )v — uIIL^;

(7.26)

II hm(v) — hm(u)II L,' < C(M)II v — UIILT;

(7.27)

II9m(u) — 9(u)IIL*' < C(M)m ";

(7.28)

—

for all u,v E H I (R N ) such that (IuIIHl < M and IIvIIHl < M and for all m E N Proof. (7.26) is an immediate consequence of (7.25). (7.27) is a consequence of (7.24), Holder's inequality, and of the embedding H l (R N ) y LT (R N ). Finally, we observe that 0,

I9m(z) - 9(z)^ < {

KIzI

if IzI < m;

«+i

if IzI > m.

The non-linear Schrodinger equation in RN: local existence 103

Let r > T be such that (N - 2)r < 2N. Then we have

IR N 19m(U) - g(u)IT < K 412:m IuIT < Km -(r

-T)

flul>m

Iulr

< CKm - (r - T) II uII Hl From this, we deduce (7.28).

❑

Lemma 7.4.3. Let M Goo. There exist C(M) < oo and > 0 such that

IVm(v) - Vm(u)I < C(M) (IIv IVm(u) - V (u)I < C(M)m -1`; for all u,v E Hl(RN) such that N U {oo}.

IIUIIHI M

+ IIv - uliy, 2

uIIL2

) ;

( 7.29) (7.30)

and

IIvIIHI <_ M and all m E

Proof. We observe that, by (7.23), Icm(z2) — Gm(zi)I < K(Iz2

+ Izil + Iz 2 I «+ 1 + (zila +1 )Iz2 — z 1 I;

and so, by Holder's inequality,

Ivm(V)-Vm(u)I <- C( IIuIIL2+IIVIIL2) IIV - UIIL2+C(IIuIIL± 1

+IIvjj L ± 1 )IIv- ulILr.

On the other hand, for all u E H 1 (RN), we have (Theorem 1.3.4)

IIuIIL

,

< IItII'HI IiuII7, with a = N

(2 - T) = a;

(7.31)

and hence (7.29). (7.30) is proved as (7.28), observing that

I Cm(z) l

-

IzI < m ; if IzI > m;

j 0, Ti f

G(z)I < KIzI ,

for all m E N.

❑

Lemma 7.4.4. Let M < oo and let (q, r) be an admissible pair. There exist C(M, q) < oo and v > 0 such that

II K(v) - K(u)II L9((o,T),Lr.) < C(M, q)T II v - UIIL-((o,T),L2); (7.32) IINm(v) - xm(u)II L9((o,T),L.) < C(M, q)T' -2 /°II v - uII Lfl((o,T),L.); (7.33)

IIGm(u) - c(u)II L9((o,T),L-) < C(M, q)T m - ' ;

(7.34)

104 The Schrodinger equation for all T > 0, all m E N U {oo} and all u,v E L 00 ((0,T),H l (R N )) such that IIUIILo((O,T),H1) < M and IIvllL—((O,T),H1) < M. Proof. We apply (7.20), next Holder's inequality on (0, T), and finally inequal❑ ities (7.26), (7.27), or (7.28). Lemma 7.4.5. Let T > 0 and let u E L°°((0,T),H 1 (R N

))

n W l "°°((0,T),

H - ' (RN)). Set K = max {IIuIILoo((O,T),H 1 ), I u II L°°((O,T),H -1 '

)}-

Then u E C([0,T],L 2 (R N )) and IIu(t) — u(s) IIL2 < 2KIt _812 , for all t, s E [0, T]. Proof. We have u E C([0,T],H -1 (R N )) (Corollary 1.4.36), and so U: [0,T] --> H'(1) is weakly continuous. In addition, we have, for all t, s E [0, T], u(t) — u(s) IIL 2 = ( u(t) — u(s) , u(t) — u(s))H-i Hi < 2KII u(t) — u(s) IIH -1 < 2K hence the result.

f

t 9

IIu'(a)IIH

-1

dQ < 2K 2 It — sI,



C

Lemma 7.4.6. Let T > 0, u E LOO((0,T), HI (lI N)) n C([0,T], L 2 (R N )). Let r >2 be such that (N — 2)r < 2N. Then u E C([0,T], L""(lR' )) and

II'IIL-((0,T),L*) <_ IIUIIL-I(O,T)H1)IIUIIL( (O,T),L2)),

(7.35)

where C does not depend on u. Proof. Applying (7.31) to w = u(t + h) — u(t), we obtain the continuity and ❑ then, letting w = u(t), we obtain (7.35). Similarly, we show the following lemma.

Lemma 7.4.7 (*) Let T> 0, u E L°°((0,T),H 2 (lRN)) n C([0,T],L 2 (R N )) Let r >2 be such that (N — 4)r < 2N. Then u E C([O,T], L'(RN)) Corollary 7.4.8 (*). If u E L((0,T),H 2 (R N )) n C([0,T], L 2 (R N )), then g(u) E C([ 0 ,T] , L2(RN))•

The non-linear Schrodinger equation in R N : local existence 105

We have in particular u and the result follows readily.

Proof.

E

C([0, T], L 2 (a+l)(R"')) n C([0,T], L 2 (R N )), ❑

Lemma 7.4.9 (*). For all M, there exists C(M) such that

I

Il9m(u)IIL2 <_ 2IIAullL2 + C(M),

for all m E N and all u E HI(RN) such that IIuUIx1 <M. Proof. It suffices to estimate h.,,,. If N < 2, we have H 1 (RN) L 2 (c, + 1 )(R ), and then I[h m (u)II L 2 < C(M). If N >_ 3, we may suppose that a >_ 2/(N — 2), so that 2(a + 1) > 2N/(N — 2). In that case, we have (Theorem 1.3.4)

Ilhm(u)IIL2 < Cc-4- a+l) < cHDu1IL(«+1)Ilull(1

(ck+1)

< Cll, UlIL2a+1) M( 1— a)(«+ 1 ) '

with a/N = 1/2 — 1/N — 1/(2(c + 1)). In particular, we have a(a + 1) < 1; hence the result. ❑

1

Lemma 7.4.10 (*) For all M, there exists C(M) with the following properties. For all T > 0, all m E N U {oo} and all u E L°O((0,T),H l (R N )) n YT' 1 ' 0 o((0,T),L 2 (R N )) nW l,- ((0,T),LT(R rs')) such that IIUIILOO((o,T),H 1 ) < M, we have k(u) E W 1 '°o((0,T),L 2 (R")) and h m (u) E W 1 ' 0"((0,T),LT'(R )); in addition, Ilk(u)'IIL1((o,T),L2)
IIL°o((o,T),L2);

(7.36)

C(M)T 1_2/° II IILL°((O,T),Lr).

(7.37)

Proof. By Theorem 1.4.40, u is Lipschitz [0, T] -4 L 2 (R N ); and then so is k(u),

by (7.26). Applying Theorem 1.4.40, we thus obtain k(u)' E L°°((0, T), L 2 (Rn')) and Ilk(u)'IILoo((O,T),L2)
hence (7.36). Applying (7.27) we show that h ,(u) E Wyo'((O,T),LT'(Rn')) and that 7

Ilhm(U)'IIL°((o,T),L)

(7.37) follows.

^(M)IIu^IIL°((o,T),LT),

❑

l

1

106 The Schrodinger equation

7.4.2. Proof of Theorem 7.4.1 We begin with a uniqueness result. Lemma 7.4.11. Let T > 0 and E H 1 (RN). Let u,v E L 00 ((0,T), H 1 (RN)) be two solutions of (7.4). Then u = v.

Proof. Observe first that by Lemmas 7.1.2 and 7.4.5, u,v E C([O,T],L 2 (RN)). Let 8 = sup{t E [0,T]; u = v on [0,t]}. If 8 = T, we have u =von [0,T]. If 8 < T, we may suppose 8 = 0, using the transformation t -* t - 8. Set M = max{IIullL- ((o,T),xl), IIVIILOC((o,T),Hl) }, and let (q,r) be an admissible pair. We have

u-v=i(1Cu-)Cv)+i(7iuand applying (7.32)-(7.33), we obtain, for all t E (0, T]

Ilu

—

VIILq((O,t),L.)
—

VIIL-((o,t),LT)).

Choosing successively (q, r) = ( oo, 2) and (q, r) = (Q, r), making the sum and then taking t sufficiently small, we obtain

Ilu — VII L—((o,t),LZ) + In — VIIL. ((O,t),L , ) = 0 ; We therefore have the desired contradiction with the hypothesis 0 = 0.

❑

Proposition 7.4.12. For all M, there exists TM > 0 with the following properties. For all cp E H 1 (RN) such that II pIIH' <_ M, there exists a solution u E L 00 ((0,TM);H i (RN)) f1 W" ((0,TM);H - i(TI ')) of (7.4). This solution satisfies: (i) IIUIIL-((o,TM );Hi) <- 2M; (ii) Ilu(t)IIL 2

=

IICPIIL2, for alit E [0,TM];

(iii) E(u(t)) < E(cp), for alit

E [0,TM];

(iv) if cc E H 2 (1R"'), then u E C([0, TM], H 2 (R n

)) fl C' ([0, TM], L 2 (RN)).

In addition, if u and v are the solutions corresponding to the initial data cc and we have:

In — VIILOO((0,Tna);L2) < KII CP — Y'IIL -, where K is a constant which does not depend on M. (v)

Proof. We proceed in that

IIWPIIHI << M.

six steps. We consider M > 0 and cc E H 1 (lRN) such

The non-linear Schrodinger equation in RN: local existence 107

Step 1. Construction of approximate solutions. For all m E N, the function g,,, is globally Lipschitz continuous. Thus we may apply Theorem 7.2.1. There exists a unique function U rn E C([0, oo), H 1 (RN)) fl C'([0, oo), H -1 (R"')), a solution of the following equation um(t) = T (t)v + igm(um)(t),

(7.38)

for all t > 0. In addition, we have I[U rn (t)L2 = IIc IIL 2 ;

(7.39)

D'm(um(t)) = Em(co);

(7.40)

for all t > 0. Step 2. A priori estimates on the solutions. Let Tm. = sup{t > 01 H U IIC([o,t],H')< _ 2M}. Our intention is to show that there exists TM, depending only on M, such that

T,,,, > TM. To see this, assume that T. < oo. In that case, we have

IIum(Tm)IIH1 = 2M.

(7.41)

But, by (7.39) and (7.40), we have IIum(t)I^HI = II^PI^ 2r i +2 (V.(um(t))

-

Vm(P)) •

(7.42)

On the other hand, applying Lemma 7.1.1, (7.26), (7.27), and Sobolev's embeddings, we obtain, for all t E [0, Tm ], IIumIIL—((0,t),H -1 ) ^ II bUmf[Loo((O,t),Jf_ 1 ) + Itk( Urn) IIL°°((0,t),H -1 ) + tIhrn(Um)IILoo((0,t),H_ 1 ) ^ 1[Um 1IL °° (( 0 ,t),H l ) + I k(um) 1 IL o° ((0 ,t),L2 ) + C' II h m (u m) 1 ILoo((o,t),LT')

2M + 2MC(M) + 2MC"C(M). By Lemma 7.4.5 and (7.29), it follows that there exists D(M), depending only on M, such that 2 IVm(um(t)) - Vr(cp)I < D(M)(t 1i2 +t 1 / 2-1 i°),

for all t E [0, T,,,]. Putting this into (7.42) and applying (7.41), we obtain

4M 2 < M 2 + D(M)(

T,m 2 +

108 The Schrodinger equation

It follows readily that there exists TM > 0 depending only on M, such that Tm

> TM , and so

sup{IIu,,,(t)IIH1; t E [0,TM]} < 2M,

(7.43)

which is the desired estimate. Step 3 (*). The case cp E H 2 (RJ"). In that case, we know (Theorem 7.2.1) that u rn E C([0,00),H 2 (R N )) nC 1 ([0,00),L 2 (R N )). Since g,,,, is Lipschitz, we have in particular g m,(u,,,,) E W 1, O0 ((0,TM),L 2 (R N )) and, for all t E [0,TM]:

u;,, (t) = T (t) (iAcp + ig rn (cp)) + i

f T(t - s)g.,,,,(u 0

rn )'(s) ds.

(7.44)

From Propositions 7.3.4 and 7.3.6, it follows that u,,,,, E W 1, °((O,TM);L T (R N )) Then, we write (7.44) as u' (t) = T(t)(iAcp + ig, n,(p)) + i J

t

T(t - s)k(um )'(s) ds

°

+i

I

(7.45) t

T(t - s)h m (u,)'(s) ds.

By Lemma 7.4.9, App + g(p) is bounded in L 2 (R N ). We estimate the first term on the right-hand side of (7.45) by Proposition 7.3.6 and the integrals as a consequence of Proposition 7.3.4 and Lemma 7.4.10. It follows that, for all T
+ IIumIIL°((°,T),LT) C( p) + K(M)(T + T 1-2k )(IIumIIL00(c°,T),L2) + IIumhIL°((0,T),LT))

IIumIIL—((O,T),L2) ,

,

where K(M) depends only on M and C depends on II'PIIH2• Choosing if necessary TM smaller (but depending only on M), we may assume that K(M)(TM + Z '1-2 < 1/2. We then obtain M )IIUmhhL0o((°,TM);L2) + IfmblL'(c°,7M);Lr) < 2C(cv).

(7.46)

According to the equation satisfied by U rn , we also obtain /urhhL-((°,T,,);L2) ^ II u'M II L-((°,TM);L 2 ) + II gn(um.)I )L-((°,TM);L2).

Next, we apply (7.46) and Lemma 7.4.9. It follows that IIoLrhIL00((o,TM);L2) < D(cp),

where D depends only on II cPII H 2 .

(7.47)

The non-linear Schrodinger equation in R N : local existence 109

Step 4. Convergence of the sequence (u,,,,),,,,> o . Let in and p be two integers We write that, for all t > 0, u rn (t) - u(t) =i ( IL ( u m)(t) - K ( UP)( t ))

+ i (flr( U, )( t ) - llm(uP)(t))

+i( 7-tm(uP)(t)

—

xP(uP)(t))•

Apply Lemma 7.4.4 successively with (q, r) _ (oo, 2) and (q, r) = (a, T). We deduce that there exists C(M), depending only on M, such that

hi u m -up lIL—((o,TM);L 2 )+chum — uphIL-((O,TNM);Lr)
21

°) (hkum — UPI[L—((O,TM);L 2 ) + (jum — UPII L°((O,TM);L-)).

Taking possibly TM smaller (but depending only on M as above) we may suppose that

C(M) (TM +T 2k) <1 Therefore (u.m ) m,, o is a Cauchy sequence in L°O((O,TM); L 2 (RN)) n L°((0,TM); LT (RN)) and there exists u E C([0, TM ], L 2 (RN)) n L°((O,TM ); LT (R N )) such that um --j u in C([0, TM], L 2 (R N )) n L°(0, TM); L T (R N )) as m — oo. (7.48) Furthermore, (7.43) and Corollary 1.4.24 imply that u E L 00 ((0,TM);H 1 (R N )) and estimate (i) follows. Invoking Lemma 7.4.4, we may immediately pass to the limit in equation (7.38) and conclude that u is a solution of (7.4) on [0, TM]. The conservation law (ii) is a consequence of (7.39) and (7.48). Lemmas 7.4.3 and (7.48) prove that

V,l.(un (t)) --->V(u(t)) as m —* oo,

(7.49)

for all t E [0,TM]. Next let t E [0,TM]. Since u,(t) —k u(t) in H l (R N ) as m —> oo, we also have

J N IVu(t)^ dx < liiminf JR N ^Vu

m (t)I dx.

(7.50)

Applying (7.49) and (7.50), we obtain (iii). Step 5 (*). Returning to the case H2(RN). By ( 7.47), u m is bounded in L 00 ((0,TM );H 2 (RN)), and so (Corollary 1.4.24) u E L°°((0,T,^,r );HZ(II2n')). In particular (Corollary 7.4.8), g(u) E C([O,TM ], L 2 (R" )). Applying estimate (7.46) and Corollary 1.4.42, we obtain in addition that U E W100((0, TM); L 2 (R N )) n W 10 (( 0 ,Tm); L T (R N )). Applying Lemma 7.4.10,

r

110 The Schrodinger equation

we have k(u) E WC 3 ((O,T,yI);L 2 (1I n')) and h(u) E WN 0 "((0,Tm);L T (R N )). Finally, Corollary 7.3.5 implies u E C l ([O,TM], L 2 (R N )) fl C([O,TM], H 2 (R N )). Step 6. The continuous dependence with respect to cp. Let cp and be such

that Ik IIHI <M and II^'IIHI <_ M. Let u and v be the corresponding solutions of (7.4). We have u—v=T(.)(cp—V))+i(Ku—)Cv)+i(liu— on [0, TM]. We estimate the first term on the right-hand side with Proposition 7.3.6 and the following two terms by Lemma 7.4.4, by taking successively (q, r) = (oo, 2) and (q, r) = (v, ,r). We deduce that there exist K < oo and C(M), depending only on M, such that

IIu — VII L°°((o,TM);L 2 ) + IIu — VIIL°((O,TM);L*) < KIIp — iljL 2 +c(M) ( TM

+TM 2k ) (IIu — VII L((o,TNM);L2) + IIu — VII L°((O,T.);L')).

Taking TM smaller if necessary (but depending only on M), we may assume that T2) C(M) (TM +TM We deduce (v).

❑

Corollary 7.4.13. Let T> 0, cp E H l (R N ), and let u E L 00 ((0,T),H l (lR t ')) be a solution of (7.4). Then u E C([0,T],H l (R N )) f1 C 1 ([0,T],H' 1 (R N )), and we have:

(i) IIu(t)IIL2 = IIcpll L 2, for all t E [0,T]; (ii) E(u(t)) = E(cp), for alit E [0,T]. In addition, if cc E H 2 (R"), then u E C([0,T], H 2 (R N )) f1 C 1 ([0, T], L 2 (R N )). Proof. We proceed in three steps. Step 1. The conservation laws (i) and (ii). Let u be as in the statement of the corollary. Observe that u : [0, T] —+ H'(R N ) is weakly continuous. Set M = (IUIILc((o,T),H1) and 6 = sup{t E [0, T]; (i) and (ii) hold on [0,t]}. We want to show that 6 = T. To do this, we argue by contradiction, assuming that 6 < T. Then, we may consider only the case 0 = 0, since otherwise we use the transform t —* t —6. In particular, there exists 6 E (0,TM), such that

lIIa(6)lIL — IIcPIILzI + IE(u(6)) — E(cp)) > 0, 2

(7.51)

The non-linear Schrodinger equation in RN: local existence 111

where TM is given by Proposition 7.4.12. Denote by v the solution of (7.4) given by Proposition 7.4.12. Since 6 E (0, TM), we have n = v on [0, b] (Lemma 7.4.11). In particular, we have II u( 6 )IIL 2 = II'IIL2 and E(u(b)) < E(cp) ((ii) and (iii) of Proposition 7.4.12); and so (7.51) is equivalent to E(u(b)) < E(p). (7.52)

Set w = u(t - 6) for all t E [0, 6]. Since u satisfies (7.6), w satisfies iwt+pw-I-g(w)=0,

for almost all t E (0, 6). Therefore (Lemma 7.1.2) w is solution of (7.4) on [0, d], with cp replaced by w(0). But (Iw(0)II H 1 = IIu(6)II H 1 <_ M. It follows that w coincides with the solution given by Proposition 7.4.12 on [0, d]. In particular, we have E(w(b)) < E(w(0)), and so E(cp) < E(u(b)), which contradicts (7.52). Consequently, we have (i) and (ii). Step 2. The continuity in Hl(RN). Since u : [0,T] -a HI(RN) is weakly continuous, (i) implies that u: [0,T] -> L 2 (RN) is continuous. By (7.29), V(u) : [0, T] --+ R is also continuous. We then deduce from (ii) that IIuIIHI : [0, T] -* R is continuous. It is now clear that u E C([0,T],H 1 (RN)). Since g : Hl(RN) ---> H -1 (RN) is continuous, we have u E C([0,T],Hl(RN)) nC l ([0,T],H -1 (RN)) Step 3 (*). The H 2 (RN) regularity. Let m E N and 9 > 0 be such that 9 < TM and m9 = T. u coincides with the solution of (7.4) given by Proposition 7.4.12 on [0,9], and so we have u E C([0,T],H 2 (RN)) n C l ([0,T],L 2 (RN)). Iterate, replacing cp successively by u(j9), 1 <_ j < m - 1, in order to obtain u E C([o,T], H 2 (R ')) n Cl([o,T], L 2 (RN))

o

End of the proof of Theorem 7.4.1. Using Proposition 7.4.12 and Lemma 7.4.11,

and arguing as in the proof of Proposition 4.3.4, we show the existence of a maximal solution which satisfies (i). Properties (ii), (iii), and (iv) are consequences of Corollary 7.4.13. It remains to show the continuous dependence. To see this, we consider E H'(R'), and a sequence ('p,,,,) m>o , such that cp,,,, -+ cP in H l (RN), as in -+ oo. Let u and U rn be the corresponding maximal solutions of (7.4). It suffices to show that, for all T E [0,T(')), we have T(',,,,) > T for m large enough, and that U rn -> u in C([0,T], H 1 (R N )), as m -> oo. Set

M = 4IIuiIL—((o,T),H1); 0(m) = sup{t E [0,T(cp m )),t < T; IIurIILoo((O,t),H1) <M}. Let k E N and 6 > 0 be such that 6 < TM and k6 = T. Applying Proposition 7.4.12, and since 9(m)
(7.53)

112 The Schrodinger equation Let us show that 0(m) = T, for m large enough. Without loss of generality, we may assume that 0(m) -> 0 E [TM, T], as m -4 oo. Let t E (0, 0). U rn (t) is defined for m large enough and, according to (7.53), we have u m (t) --. u(t) in L 2 (R N ), as m -p oo. On the other hand, we have E(u rn (t)) = E(cp m ) E(cp) = E(u(t)). It follows (see Step 2 of the proof of Corollary 7.4.13) that H (RN) as lib —> IW. particular, In art,cu,ai we IlUm(t)IIH iiavc u i <_ M/2 have u t - u i1 in l(RN), t + TM/2}. >_ min{T, 0(m) large enough, and so (Proposition 7.4.12(i)) m for large enough. for = T, m <0 being arbitrary, it follows that 0(m) t Thus, it remains to show that U rn -* u in C([0,T], H l (R N )), as m -* oo. We argue by contradiction, and we suppose that there exists a sequence (t m ) m >o in [0,T] and e > 0, such that IIu n (t m ) - u(tm)IIH 1 > e. We may assume t m -> t e [0,T], which implies that IIu,,,,(t rn ) - u(t)II H l >_ E/2. By (7.53) and since u E C([0, T], L 2 (R N )), we have u m (t m ) -> u(t) in L 2 (R N ), as m - oo. Furthermore, we have E(u,,,,(t rn )) = E(cp m ) --> E(cp) = E(u(t)). It follows (see Step 2 of the proof of Corollary 7.4.13) that u m (t) u(t) in H l (R N ), as m -* oc; hence the contradiction. This completes the proof of Theorem 7.4.1. ❑

m( )

7.5. The non-linear Schri dinger equation in R N : global existence We suppose that ci = R N and that g satisfies the assumptions of Theorem 7.4.1. As for the heat and Klein-Gordon equations, we are going to establish two kinds of results: global existence for all initial data when g satisfies some additional growth assumptions and global existence for small initial data, without any additional hypothesis on g. Proposition 7.5.1. Suppose that there exist 0 < 3 < 4/N and two constants A and B such that

G(z) < AIz1 2 + BIzja+ 2

for all z E C. Then, for all cp E H l (1R"), we have T (co) = oo, and sup I u(t) IIjp < t>_o

00.

Proof. Observe first that, for all w E H 1 (R N ), we have

IV(w)I <— A f N Iu1 2 +B f N I u Ip+2 Applying Theorem 1.3.4, we deduce that f r

V(w)I < A

J

Iwi 2 + C RN \ JI]^N Iwi2 /

e+i - 4

( IRNIDwi

2 )

(7.54)

The non-linear Schrodinger equation in RN: global existence 113

Now let cp E H l (R N ) and let u be the corresponding solution of (7.4). Let f(t) = IIVu(t)I^L z , for t E [0,T(cp)). By the conservation of energy, we have f (t) <E(') + 2IV(u(t))I, for all t E [0, T(cp)). Applying (7.54), and the conservation of the norm in L 2 (RN), we deduce that f(t) < E(p) + C(cp) f (t) 4p , for all t E [0,T(cp)); hence the result, since No < 4. Remark 7.5.2.

I ❑

If 3 = 4/N, the above inequality becomes

f(t) <E

()

+ CII^jI f(t),

for all t E [0, T()). Therefore, if IIcc IL2 is sufficiently small, we again have

I

T( p) = oo. ,

Proposition 7.5.3.

Let g be as in Theorem 7.4.1. There exist 6, K > 0 such

that if IIVIIH1 << 6, then we have T(cp) = oo and sup IIu(t)IIH1 < KII(PIIHI e>o Proof.

Observe that

G(z) < AIz1 2 + BIzI'+z for all z E C. We may as well assume that a > 0. It follows that 2

Iv(w)I <- CIIwIIi2 +DIIwIIt 2 ,

I

(7.55)

for all w E H l (R N ). Now let cp E H l (R N ) and let u be the corresponding solution of (7.4). Let f(t) = II u(t) II H l for t E [0,T()). By the conservation laws (ii) and (iii), we have f(t) <_ f( 0 ) + 2 IV(^P)I +2IV(u(t))I,

I

for all t E [0, T()). Applying (7.55) and conservation of the L 2 norm, and letting e = a/2, it follows that f(t) < f(0)+Cf(0)+Df(0) 1+f +Cf(0)+Df(t) 1 for all t E [0, T()). Therefore, if we suppose that f(0) _< 1 then, letting M=1+2C+ D, we have

f(t) <Mf(0)+Df(t) 1 +E,

I

(7.56)

for all t E [0,T(,p)). Set 9(x) = Dxl+E — x for x > 0, and let —X = min0 < 0. For all a E (0, X) there exist x a and ya, with 0 <x < ya such that

114

The Schrodinger equation

l

+ a = B(ya) + a = 0. In addition, we have a < x , < a(1 + e)/e (see Figure 7.1). B(xa) a

G

-

-Ex

Fig. 7.1 By (7.56), we have 9(1(t)) + for all t

M f (0) > 0,

e [0, T (p) ). Consequently, if we suppose that M f f(t)

(0) <, then we have

E [ 0,x Mf( 0 )) U (YMf(0) 00).

In addition, since f is continuous and f (0) < f(t)

<_ XMf(o) <_ 1

E

x,M f( o ),

we deduce that

E Mf( 0 ),

E [0,T(cp)). The result follows, with S = min{1, X/M} and K = (1 + e)M/e. ❑

for all t

7.6. The non - linear Schrodinger equation in RN: blow-up in finite time The blow-up results for the non-linear Schrodinger equation are based on the following proposition. Proposition 7.6.1. Let cp E Hl(RN) and let u E C([0,T(cp)),H'(R")) be the ), then we have I u(t, •) E corresponding solution of (7.4). If I (•) E L 2 C([0,T(cp)), L 2 (RN)). In addition, the function t '—+ f IxI 2 Iu(t,x)I' dx belongs to C Z ([O,T(cp))), and we have (['

The non-linear Schrodinger equation in

dt f N

RN :

Ix1 2 1u(t, x)I 2 dx = -4(iu r , ru) = 4Im

blow-up in finite time

J

N

rziu r dx,

115

(7.57)

d2 IxI 2 Iu(t,x)j 2 dx = 16E(u(t)) dt 2 f N - 4N f g(Iu(t)^)1u(t)j dx + 8(N + 2)f G(u(t)) dx, (7.58) ^N

for alit E [0,T(cp)). Remark. Identity(7.58) is very important. It is useful not only for establishing blow-up results, but also for studying the global behaviour of solutions (Section 7.7). The proof of (7.57) is rather simple. Formally, it suffices to multiply the equation by r 2 u. However, this is not correct because of the term r 2 . We overcome this difficulty multiplying the equation by e - " 2 r 2 u and passing to the limit as e J 0 (Lemma 7.6.2). The proof of (7.58) is much more complicated. Formally, we would multiply the equation by ru, though this is not possible for two reasons: firstly, because of the term in r; secondly, u is only in H 1 (RN) so the equation makes sense in H -1 (Rn'), but u,. is only in L 2 (RN). Therefore, we cannot multiply the equation by u,. (for this we need u r. in H l (RN)). To show (7.58), we then proceed as follows: we approximate cp by a sequence of functions co of H 2 (RN) , and we denote by un the corresponding solutions. We multiply the equation satisfied by u, 1 by e - 9r8,.u,, and then we let E j 0. We obtain (7.58) for the solution u, (Lemma 7.6.3) and then we let n -> oo. Lemma 7.6.2. Let p E H 1 (R") and let u E C([0,T(cp)),H 1 (R' )) be the corresponding solution of (7.4). If (•) E L 2 (RN), we have • Ju(t, •) E C([0,T(,p)), L 2 (RN)). In addition, the function t f xj 2 lu(t,x)j 2 dx is in C 1 ([0,T(cp))), and identity (7.57) holds. Proof. For e > 0, we define the functions 8 and p E by z

0 E (r) =re;

p(r) = r 2 e -Er2

We set fE (t) = J I B E UI 2 dx, RN

116

The Schrodinger equation

for t E [0,T(cp)). It is clear that f E C'([0,T(cp))), and that, for all t E [0,T(co)), we have E

f(t) = 2 (Beu, OeUt)Hl,H -1 = 2 (peu, ut) = 2 (peu, iAu + i9(u)) = 2(p f u,iLu) = -2(p 8 Vu,iVu) -2(Vp,u,iVu)

(7.59)

= -2(V p,u, iVu) = -2 (p'u, iur)•

We easily verify that p'' (r) < CO.,(r), and so we obtain f(t) <2CIIu(t)IIxlfe(t)1 /2

It follows immediately that f(t) is bounded as E J. 0, uniformly on [0, T], for all T < T(cp). By monotone convergence, we deduce that f IxI 2 Iu(t,x)I 2 dx < oo for all t E [0,T(cp)), and that u(t, •)11 L 2 is bounded on any interval [0,T], with T < T(p). In particular, t i I • Iu(t, •) is weakly continuous from [0, T()) to L2(RN). Integrating (7.59) between 0 and t E [0, T(cp)), we obtain fe(t) =

ff( 0 )

-4

J0 ((1 - Er )e t

2

-E

for all t E [0,T(cp)). Letting E J 0, it follows that

J

Ixu(t)[ 2 dx =

IR

t

Ixcpl2 dx — 4 (ru, iur.), 0

N RN

for all t E [0,T(p)). We deduce

(7.57).

In addition, the function

tHIII'Iu(t,-)IIL2 is continuous, and then I • Iu(t, -) E C([0, T(cp)), L 2 (R N )).

❑

Lemma 7.6.3. Let cp E H 2 (R N ) and let u E CC[0,T(cy)),H 2 (R N )) be the corresponding solution of (7.4). If I • Icy(•) E L 2 (R N ), we have I • ( u(t, •) E C([0,T(ep)),L 2 (R N )). In addition, the function t > f IxI 2 Iu(t,x)I 2 dx is in C 2 ([0,T(cp))), and (7.58) holds. Proof. We proceed in five steps. Step 1. Let 8 E S(l[8 N ) be a radially symmetric real-valued function. Let T> 0 and u E C([0,T],H'(lR N ))nC l ([0,T],L 2 (R r')). Set h(t) = (r9u,iu,.), for t E [0,T]. Then h E C 1 ([0,T]) and h'(t) = (u t , i(20ru,. + (NB + rO')u)),

(7.60)

The non-linear Schrodinger equation in R N : blow-up in finite time 117

for all t E [0,T]. Indeed, suppose first that u E C 1 ([0,T],H 1 (lR N )). In that case, it is clear that h E C 1 ([O,T]) and we have

h'(t) = (rOu t , iu r ) + (rOu,iu tr ) = (u t ,irOu r ) + (Bu,ix • Vu t ). Observe that

(7.61)

f

(Bu, ix • Vu t ) = Im and that

JRN 6x • VT,

t

dx,

Ox V i = V • (xOuut ) — NOuut — Ox • Vu — rO'uut . We then have

(Bu, ix • Vu t ) _ (u t , i(Ox • Vu + (NO + rO')u)) = (u t , i(Oru r + (NO + rO')u)). Applying (7.61), we deduce (7.60), and then

h(t) = h(0) +

fo

t (u t ,

i(20ru r + (NO + 8')u)),

(7.62)

for all t E [0,T]. By density, we obtain (7.62) as well as (7.60) for u E C([ 0 ,TJ, H 1 (R N )) n C' ([ 0 ,T], L 2 (R N )).

Step 2. Let u be as in the statement of the lemma and set h(t) = (rOu, iu r.), for t E [0,T]. Then, by Step 1, we have h E C 1 ([0,T(cp))) and h'(t) = (Au + g(u), 20ru r + (NO + rO')u),

(7.63)

for all t E [0, T}. Step 3. Let w E H 2 (RN) be such that I • Iw(•) E L 2 (RN). Then we have

(Ow + g(w), 20rw r +(NB + rO')w) = —2

JR

N

+N f 9(g(IwI)jwj-2G(w))

+7

rO'(g(Jwj)jw) — 2G(w) — 2 ^wT[ 2 )

N

— Re

J

r((N + 1)0' + rO")w T w. (7.64)

]RN

Note first that, by density, it suffices to consider the case w E D(RN). In that case, we have (g(w), (NO + rO')w) = f (NO + r0')g(Jwj)Iw^;

(7.65)

118 The Schrodinger equation

= Re

(g(w), 20rw r )

J

N

20rg(w)w,. =

fR N 20rG(w) .. r

But 20rG(w) r = 29x•OG(w) = V (x20G(w))-2(N0+r9')G(w). Consequently, (g(w),

28 r w r ) = —2

J (NB + rO')G(w).

(7.66)

On the other hand, we have (Aw, (NB + r8')w) _ —

IR

N

(NB + r9') I I 2 — Re

J

N

(Aw,20rw r ) _ —(20Vu,V(rur))— 2

r((N + 1)0' + r0")w r w; (7.67)

fR

rO'Iwr 1 2 .

N

(7.68)

Note also that the following identity holds: Re(20Vw • V(rwr )) = Re(20Vw • 0(x • V5 )) = ((2 — N)9 — r8')IVwI 2 + V _ (zOIVwi 2 ) From (7.68), we then deduce that (1w, 20r w r.) =

J^'

((N — 2)0 + r8')IVW1 2 —2 / rO'Iw r 1 2 . JJ1R

N N

(7.69)

Applying (7.65), (7.66), (7.67), and (7.69), we obtain (7.64). Step 4. Let u be as in the statement of the lemma and set h(t) = (r0u, iu r.),

for t E [0,T(cp)). Then, by Steps 2 and 3, we have h'(t) = —2

fR

+

J

9IVU1 2 + N

RN

J

9(g(1u1)1u1 — 2G(u))

r9'(9(juj)juj — 2G(u) — 2Iu r 1 2 )

— Re

(7.70)

JRN r((N — 1)9' + r8")u u, r

for all t E [0, T(cp)). Step 5. Take 0 E = e 2 , fore > 0. We have 10,1 1 and 0, —* 1 as E J 0. We easily verify that Ir0EI <_ C, r9E —40 as e J. 0, and r[(N+ 1)0E +r0E]I < C and that Ir[(N+1)0 +r0E]I —+0 as E 1 0. We deduce immediately from (7.70) that

(r9

u,

t iu r ) = (r0 E Cp, icp r ) + 1 fe (s) ds, 0

(7.71)



The non-linear Schrodinger equation in R N : blow-up in finite time 119

where f f is a bounded function on [0, T] for all T E [0, T (gyp)) and is such that fE(t) —> _2f

N

(Vu1 2 + N

J

N

(g(^u^)lul — 2G(u)) as E 1 0.

(7.72)

Observe that, due to conservation of energy, we have

_2f jvul 2 +N f (g(IuD)!ul - 2G(u)) N

N

_ —4E(u(t)) + N

J g(^uJul — (2N + 4) f G(u). N

Letting e 1 0 in (7.71), we obtain the following: (ru, iu r ) = (i , r) — f ( 4 E(u(t)) — N J g(Iul)lul

°

+ (2N + 4)

f

RN

RN

(7.73)

G(u)).

Now, (7.58) follows readily from (7.57) and (7.73).

❑

Proof of Proposition 7.6.1. Applying Lemma 7.6.2, it remains to verify that

the function

tH

J

N

I xl 2 ju(t, x)I 2 dx

is in C 2 ([0,T(cp))), and that we have (7.58). Let T < T(cp) and let (cp m ),,,, >o be a sequence of functions in H 2 (R N ), such that cp 71 —4 cp in H 1 (R N ), as m —> oo. Denote the corresponding solutions of (7.4) by u rn . We know (Theorem 7.4.1) that, for m sufficiently large, we have T(cp m ) > T and u,,,, —, u in C([0,T], H l (R N )), as in —4 oo. We write identity (7.73) for u,n and t E [0,T] and we let in —•+ oo. We deduce that u satisfies (7.73); hence the result. ❑ Theorem 7.6.4. Let g be as in Theorem 7.4.1, and suppose further that sg(s) > (2+ N)) G(s),

(7.74)

for all s >_ 0. Then if cp E H l (R N ) is such that yep(•) E L 2 (R') and E(cp) < 0, we have T(ep) < oo.

Proof. Let cp be as above. Let u be the corresponding solution of (7.4) and set 1(t) = I

RN

IxI2Iu(t,x)l2dx,

118 The Schrodinger equation

(g(w), 20rw,.) = Re

JR 29rg(w)w,. = I

26rG(w) r .

RN

N

But 20rG(w), = 20x•OG(w) =V.(x20G(w))-2(N6+rO')G(w). Consequently, (g(w), 20,w,) = —2

J (NB + rO')G(w).

(7.66)

On the other hand, we have

(Aw, (NB + r9')w) _ —

(NB + r8')IVW1 2

fR N

— Re

J r((N + 1)0' + r0")w w; (7.67) r

R, N

(Aw,20rw r ) _ —(20Vu,V(ru r ))-2

f

RN

rO'Iw r 1 2 .

(7.68)

Note also that the following identity holds: Re(20Vw • V(rw,.)) = Re(20Vw • 0(x • Vw))

= ((2 — N)8 — rO')IVwI 2 + V • (xOIOwI 2 ). From (7.68), we then deduce that (Aw, 20 r W,) =

IR

((N — 2)0 + rO')IVwl 2 — 2 N

IR

rG'jw r.I 2 .

(7.69)

N

Applying (7.65), (7.66), (7.67), and (7.69), we obtain (7.64). Step 4. Let u be as in the statement of the lemma and set h(t) = (r0u, lU r ), for t E [0, T(cp)). Then, by Steps 2 and 3, we have

h'(t) _ —2

JR

+

f

N

OIVuI 2 + N

RN

I

0 — 2 G(u)) RN (g(IuI)Iuj

r9 '(g(jul )ju^ — 2G(u) — 2Iu r.I 2 )

— Re

fR

N

(7.70)

r((N — 1)0' + r8")u r u,

for all t E [0,T(^p)). 2 , fore > 0. We have I B E I < 1 and B E --4 1 as e J 0. We Step 5. Take 0 E = e easily verify that I r0 I <_ C, r0 —40 as e 1 0, and r [(N + 1)0 + r0E'] I
t

(rB E u,iu,) = (r0e ,i^Pr) + f fe(s) ds,

o

(7.71)

The non-linear Schrodinger equation in R N : blow-up in finite time 119

1

where fE is a bounded function on [0, T] for all T E [0, T (cp)) and is such that —2 fe(t) —9-2

fit

N IVu1 2 + N JR N (g(lul)lul — 2G(u)) as e j 0.

(7.72)

Observe that, due to conservation of energy, we have

—2 f N IVu1 2 +N f

N

(g(lul)lul — 2G(u))

_ —4E(u(t)) + N

I

J g(lul)Jul — (2N +4) fG(u). ]RN

Letting E 1 0 in (7.71), we obtain the following: (ru, iu r.) = (rev, i^vr) — ft (E(u(t)) — NJ g(luIuI + (2N + 4)

f

R N

RN(7.73)

G(u)).

Now, (7.58) follows readily from (7.57) and (7.73).

❑

Proof of Proposition 7.6.1. Applying Lemma 7.6.2, it remains to verify that

the function t—

f N Ix1 2 Iu(t,x)I 2 dx

is in C 2 ([0, T())), and that we have (7.58). Let T 0 be a sequence of functions in H 2 (R N ), such that cc —> W in H l (R N ), as m —+ oo. Denote the corresponding solutions of (7.4) by u rn . We know (Theorem 7.4.1) that, for in sufficiently large, we have T(cp T..) > T and u rn —> u in C([0,T],H l (R N )), as m —^ oo. We write identity (7.73) for u,,,, and t E [0, T] and we let m —+ oo. We deduce that u satisfies (7.73); hence the result. ❑ Theorem 7.6.4. Let g be as in Theorem 7.4.1, and suppose further that sg(s) > (2 + N I G(s), for all s >_ 0. Then if cc E H I (R N ) is such that I Icp(•) E we have T (gyp) < oo.

(7.74) L2(RN)

and E(cc) < 0,

Proof. Let cp be as above. Let u be the corresponding solution of (7.4) and set

f(t) =Ixl 2 1u(t,x)I 2 dx, fRN

120 The Schrodinger equation

for t E [0,T(cp)). It follows from (7.58), (7.74), and the conservation of energy that (7.75) f"(t) < 16E(), for all t E [0,T(cp)). From (7.75), we deduce that f(t) <1(0) + t f'(o) + 8t 2 E( V ), for all t E [0,T(cp)); this implies that T(cp) < oo, since f (t) > 0 and E(cp) < 0. ❑ Remark 7.6.5. If there exists x > 0 such that G(x) > 0, (7.74) implies that G(s) > (s/x) 2 + 4 I N G(x) for s > x. In particular, if we take E H l (R N ), then E(kcp) < 0 for k large enough; and so if I • Icp(•) E L Z (R N ), then T(kcp) < oc. 7.7. A remark concerning behaviour at infinity Identities (7.57) and (7.58) allow us to prove directly the pseudo-conformal conservation law, which provides information about the behaviour at infinity in time of the solutions, in some cases (see §7.8 below). The following proposition is related to this conservation law. Proposition 7.7.1. Let cp E H'(R N ) be such that u be the corresponding solution of (7.4). Set f(t)

=f

I(x + 2itV)u(t, x) 12 dx — 8t2

N

JR

I Icp(•) E L 2 (R N ) and let

G(u(t, x)) dx,

(7.76)

N

for alit E [0,T(cp)). Then f E C 1 ([0,T(cp))) and f'(t) = 4t

J (Ng(IuI)IuI — 2(N + 2)G(u)),

(7.77)

.N

for alit E [0, T(cp)). Proof. Developing the right-hand side of (7.76), we obtain f(t)

= IIxuIIi2 + 4t(ru, iu r ) + 4t 2 IIVuIIi2 = IIxuIIi2 + 4t(ru, iu r ) + 8t 2 E(cp).

—

8t 2 V(u)

It follows immediately from Proposition 7.6.1 that f E C 1 ([0, T())) and that identity (7.77) holds. ❑ Remark 7.7.2. If g(s) = As 1 + 4 /N, (7.77) means f'(t) = 0, and then we have

f(t) = f(0) = f IxVI2•

Application to a model case 121

Remark 7.7.3. Let u be as in the statement of Proposition 7.7.1, and set v(t, x) = e ' U u(t, x), -

for x E R N and t E [0,T(cp)). It is clear that

I(x+2itV)u(t)I 2 = 4t 2 IVv(t)I 2 > 4 t 2 IVIu(t)I I 2 ; and, consequently,

f (t) = 8t 2 E(v(t)) > 8t 2 E(Iu(t)I).

(7.78)

7.8. Application to a model case We choose g(s) = als^as, with a > 0, (N — 2)a < 4, and a 54 0. We consider cp E H 1 (IRN) and we denote by u the corresponding maximal solution of (7.4), which exists by Theorem 7.4.1. We then have the following results. If a < 0, then T(cp) = oc and u is bounded in H 1 (R N ) (Proposition 7.5.1). If a > 0 and if a < 4/N, then T(cp) = oc and u is bounded in H 1 (RN) (Proposition 7.5.1). If a > 0 and if a = 4/N, then T(cp) < oo for some initial data cp (Theorem 7.6.4 and Remark 7.6.5); on the other hand, if II PIIL 2 is sufficiently small then T(o) = oo and u is bounded in H'(Rn') (Remark 7.5.2). If a> 0 and if a > 4/N, we have T(cp) Goo for some special initial data cp (Theorem 7.6.4 and Remark 7.6.5); on the other hand, if II PIIH1 is small enough, then T(cp) = co and u is bounded in H 1 (RN) (Proposition 7.5.3). Observe that in the case in which a <0 and if I • Icp(•) E L 2 (RN), (7.77) shows that the solutions converge to 0 as t —+ oo, in certain spaces LP(RN). Indeed, it follows from (7.77) that

f'(t) = -4(4 — Na)t a +

2 fR N

Iu(t) I-+ 2 =4(4 — Na)tV(u(t)),

(7.79)

for all t >_ 0. In particular, if a > n,, we have f'(t) <0, and so, with (7.78), it follows that 8t2E(I u(t)I)

s f Ix^I 2 ,

for all t > 0. If we apply Theorem 1.3.4, we obtain, in particular,

I^u(t)IIL-+z
-) ,

122 The Schredinger equation

for all t >_ 0. Observe that we obtain the same negative exponent of t as for the linear equation (Proposition 3.5.14). If a < 4/N, it follows from (7.79) and (7.78) that

f'(t) <4(4- Na)tE(I u(t) I) < (2_ Na) t f (t), Na-4

for all t > 0. Therefore, t 2 f (t) is non-decreasing and, (by (7.78)), we have

8t

E(Iu(t)I) <_ f(1),

for all t > 1. Consequently, we again obtain

IIu(t)IIL-+2
_

Na

a+2

i.e. the same negative exponenent of t as for the linear equation. Notes. For Sl 54 RN, N > 2, a few results are known. See Brezis and Gallouet [1], Kavian [1], Y. Tsutsumi [2, 3], and Yao [1]. Note also that Cazenave and Haraux's results [1] apply in any open subset St C R N . For §7.3, see Yajima [1], and Strichartz [1]. A regularizing effect of Hs(R")type also exists; see Constantin and Saut [1-3], Sjolin [1], and Vega [1]. The problem of local existence (§7.4) has been studied extensively. Our presentation is based on Kato [3] and Cazenave and Weissler [1, 3]. See also Baillon, Cazenave, and Figueira [1], Cazenave [2], Ginibre and Velo [1, 2, 5], Hayashi [1], Hayashi and Tsutsumi [1], Lin and Strauss [1], and Weinstein [1]. The Cauchy problem has also been studied in H 9 (R n') (s # 1); see Cazenave and Weissler [2, 4], Ginibre and Velo [4], and Y. Tsutsumi [4]. There exists a regularizing effect for the non-linear equation; see Hayashi, Nakamitsu, and Tsutsumi [1, 2], Hayashi and Ozawa [2], and Kato [3]. Various kinds of non-linearities (possibly non-local) have been considered. See, for example, Baillon, Cazenave, and Figueira [2], Baillon and Chadam [1], Cazenave [2], Cazenave and Haraux [1], Cazenave and Weissler [1], Chadam and Glassey [1], Dias and Figueira [1], Ginibre and Velo [3], Klainerman and Ponce [1], Lange [1], and Schochet and Weinstein [1]. For more about blow-up in finite time, consult M. Tsutsumi [1], Glassey [2], and Weinstein [4]. For some non-linearities, there exist solutions of the form u(t, x) = e u)t ^p(x). These solutions are called stationary states. See, for example, Berestycki, Gallouet and Kavian [1], Berestycki and Lions [1], Berestycki, Lions, and Peletier [1], and Jones and Kiipper [1]. The behaviour at infinity of solutions is rather well known in the repulsive case, if the solutions behave asymptotically as the solutions of the linear equation. See Ginibre and Velo [1, 2, 7, 8, 10], Hayashi [2], Hayashi and Ozawa [1],

Application to a model case 123

Lin and Strauss [1], Reed and Simon [1], and Y. Tsutsumi [1]. In the attractive case, we only know how to study the stability of some stationary states. See Berestycki and Cazenave [1], Blanchard, Stubbe, and Vazquez [1], Cazenave [3], Cazenave and Lions [1], Grillakis, Shatah, and Strauss [1], Jones [1], Shatah and Strauss [1], and Weinstein [2, 3]. For more references concerning these questions, consult Cazenave [4].

S !

l

^...

Bounds on global solutions The study of the behaviour at infinity of global solutions is one of the most important problems in the study of non-linear evolution equations. The problem can be formulated as follows. If u(t), 0
u'(t) = Au(t) + F(t, u(t)), for t >_ 0, how does u(t) behave as t —> oo? The results concerning the behaviour at infinity of global solutions are, in general, based on compactness properties of U {u(t)}, and in particular on bounds for U {u(t)} (see Chapter 9). In some t

>o

t>O

cases, the solutions are bounded by construction (see §5.3, §6.3, §7.5). On the other hand, even for linear equations, global solutions may be not bounded. For example, u(t, x) = e t sin(xrx) is a solution of u t = u xX + 2ir 2 u, in cl = ( 0,1); u=0, in 852; and u is bounded in no function space as t --f oo. The available methods allow us to study the behaviour of the solutions in the following two situations: • Semilinear autonomous equations (see §8.1, §8.2, §5.3, §6.3, and §7.5). • Semilinear non-autonomous equations, with dissipation (§8.4) or with repulsive interaction (see §8.3). 8.1. The heat equation We use the notation of Chapter 5. In particular, f is a bounded open subset of RN with Lipschitz continuous boundary, X = Co(1) and (T(t)) t >o denotes the semigroup associated with the heat equation. g is a locally Lipschitz continuous function IR —4 ]R such that g(0) = 0, and we consider G and E to be defined as for Proposition 5.4.4. In the following subsection, we gather some preliminary results that will be useful in establishing the main result.



The heat equation 125

8.1.1. tion

A singular Gronwall's lemma: application to the heat equa-

Lemma 8.1.1. Let T > 0, A >_ 0, 0 < /3 <_ a, and 1 _< y <_ oo. Let cp E C([0,T]), cp > 0 and f E Lry(0,T), f > 0. Suppose that a + 1/b < 1 and that

p(t) < At

+ ti

(t -sp )

f (s)^p(s) ds,

for all t E (0,T). Then cp(t) < CAt - a, for alit E (0,T), where C depends only on T, a, p, y, and IIIf II LI (0,T) Proof. Set zl)(t) = t(t) for all t E [0,T], and let 0(t) = sup O(s) for t E O<s
[0, T]. It is clear that 0 E C( [0, T]) and that we have

O(t) < A+ t- fo

(t

1

s) a s f(s)0(s) ds,

-

for all t E (0,T). We have t -0 E L 7 '(0,T), and so there exists E E (0,1) such that

(1 - E) allt "IIL,'(O,ET)IIfIILI(O ,T) <_ 1/2. -

-

It follows that f (s) ds

t 1

^i(t) < A + t^B(t)

(,-E)t (t -s) S 1

+ to

r (i-€)t

J

1 1 f(s)9(s) ds (t s)$ —

< A + 20(t) + pt^ a i(1-E)t s f(s)0(s)ds

We deduce immediately that T

0(t) < A + 2 B(t) + AT` Q j S f (s)B(s) ds; -

and we obtain the conclusion by applying Lemma 4.2.1, since s, `Y f (s) E ❑ L 1 (p T) -

We will also make use of the following comparison lemma, which generalizes Proposition 5.3.1.

126 Bounds on global solutions Lemma 8.1.2. Let T > 0, p > 2N/(N + 2) (p >_ 1 if N = 1, p >1 if N = 2), cp E X, u E C([0,T],X) n L 2 ((0,T),Ho(cl)) nW 1 " 2 ((0,T),H -1 ) and f E Ll((0,T),L'(Sl)) such that ut = Au + f in H'(1), almost everywhere in (0, T); Jtl u(0) = W. Suppose further that there exists a constant C such that f (t, x) < CIu(t, x)I almost everywhere in S2, for almost every t E (0,T). Then if cp > 0 on Sl, we have u(t) > 0 on 52, for all

t E [0, T] . Proof. Note first that the hypotheses imply that f E L 1 ((0, T),H 1 (S )), and so the equation makes sense in H -1 (52). Now, the proof of Proposition 5.3.1 can be adapted immediately, since -

dt fz

u (t) 2 dx = 2 (ut(t), u (t))H-1,Hl = —2 I IVu (t)1 2 —2

Jo

almost everywhere in Q.

1 f (t)u (t),

Jo

1

Lemma 8.1.3. Let T > 0, a, yy >_ 1 be such that N/(2Q) + 1/y < 1. Let cp E X, u E C([0,T],X) and f E L' ((0,T),L°(S2)) be such that u(t) =T(t)cp+

e
fo tT(t—s){f (s)u(s)}ds,

for all t E [0,T]. Then for all E E (0,T], we have sup Jju(t)IILo <_ K, where K depends only on e,o,ry, 1IccII L l and 11 f + IjL1((o,T),L^)

Proof. We proceed in two steps. Note first that the hypotheses imply fu E L((0,T), L°(1)), so the above integral makes sense, since by (3.34) T(t) can be extended to an operator in £(LP, Lp) for all p E [1, oo). On the other hand, invoking Lemma 8.1.2, we may restrict ourselves to the case in which cp > 0, and so u > 0 (see the proof of Proposition 5.3.3). Step 1. Let 1 < p < r < oo be such that Nil —r1<1 and -+-<1.

The heat equation 127

Let us show that, under the hypotheses of the lemma, we have

sup IIu(t)IIL- < C,

e
where C is a constant that depends only on II f°), e, v, y, r, p,T, and + 1IL7((o,T),L IIVIIL' . Indeed, by Holder's inequality, (3.34) and Lemma 3.5.9, we have

I[u(t)IILr
-

)

I

t

JO (t —s)

IIf(S) + IIL°IIu(S)IILrdS,

for all t E (0, T]. We conclude by applying Lemma 8.1.1.

❑

Step 2. Let m be an integer such that m < a < m+1. Let E > 0 and 6 = e/(m + 1). Applying Step 1 with p = 1 and r = a/(a — 1), we conclude

that u(6) is bounded in L° ( i t (cl), with respect to the above parameters. We iterate this argument, translating the time of 6 at each Step and taking successively p = Q/(o — j) and r = a/(Q — j — 1), for 1 < j < m —1. We obtain that u(rn6) is bounded in L°/ ( ° "n ) (Sl) We conclude by applying Step 1 again with p = o- /(o- — m) and r = oo, to find that u is bounded in L°°(1) on [e,T], with respect to the above parameters. ❑ Corollary 8.1.4. Let or >_ 1 be such that or > N/2, cc E X, u E C([0, oo), X ) and let f E C([0, oo), L° (S2)) be such that ft

u(t) = T(t) cp +

J

0

T(t — s){ f (s)u(s)} ds,

for all t >_ 0. Suppose that sup Ilu(t)IILI < oo and that sup 1 f (t) + I[L° < oo. t>O

t>O

Then, for all e > 0, we have sup IIu(t)IIL— <00. t>e

For all s > 0, we apply Lemma 8.1.3 with T = 1 + e and cp = u(s). We obtain in particular that u(s + e) is estimated in L°° (1)-norm, only in terms of c, s, sup Z[u(t)IIL1 and sup II.f (t) + II L°; hence the result, since s >_ 0 is arbitrary. ❑ Proof.

t>O

Corollary 8.1.5.

t>O

Suppose that g satisfies xg(x) < C(x 2 + (xl"),

Vx E R,

(8.1)

where p > 2 and (N — 2)p < 2N. Then, for all M, there exist t(M) > 0 and K(M) < oo with the following properties: if cc E X n Ho (1l) is such that IIccI[x7 < M and if u denotes the corresponding maximal solution of (5.4)

I

128

Bounds on global solutions

(see Theorem 5.2.1), then T(cp) > t(M) and JIu(t)IILOO < K(M) for all

t E [t(M)/2,t(M)]. Proof. Applying Proposition 5.3.1 and arguing as in the proof of Proposition 5.3.3, we may restrict ourselves to the case in which cp > 0, and so u > 0. From (8.1), it follows that

(8.2)

IIg(u(t))+II L <_ A ( 1 + I ju(t)II LP)P ' , -

for all t E [0,T(cp)). If we set /.3 = N(p - 2)/(2p) E [0,1), we deduce from (3.34), (8.2), and Lemma 3.5.9 that

u(t)IILP <— II ^vIILP +A J t

- s) O (tR

(1 + Iju(s)IILP)P

-1

ds,

for all t E [0,T(cp)). But, using Sobolev's inequalities, we have j II Setting f (t) = 1 + sup IIu(s)IlLP, we then obtain

LP

< CM.

o<s
f (t) < (1 + CM) +

A

tl f (t)P-1,

(8.3)

for all t E [0,T(cp)). Set / (1 _ 3)(1 + CM)2- P 1T '

T(M) = I 2PA

We easily deduce from (8.3) that f (t) < 2(1+CM) for 0< t < min{T(cp), t(M)}. Applying (8.1) we conclude immediately that g(u(t))

I C u(t)

<+ _B, I IL

for 0'< t < min{T(cp), rr(M)}, where B depends on u only through the value of M. We then write g(u) = hu, with h = g(u)/u and we apply Lemma 8.1.3, to obtain that, for all e E (0, min{T (cp), T(M)}, sup


e
where K depends on u only through the value of M. It follows that T() > Tr(M) ❑ and we complete the proof taking e = T(M)/2.

The heat equation 129

8.1.2. Uniform estimates

The main result of this section is the following. Theorem 8.1.6. Suppose that g satisfies (8.1), and that there exist M < oc and E > 0 such that xg(x) > ( 2 + e)G(x),

for IxI > M. Let cp E X and let u be the corresponding maximal solution of (5.4) (see Theorem 5.2.1). Then ifT(o

Proof. Applying (5.14) and (5.18), we obtain, for all 1 < s < t < oo, the following inequalities:

ft J

0

d Jr

ut dxdt = E(u(1)) — E(u(t));

u(t) 2 dx > 2(2 + e)

+e

ff t



(8.4)

ut dx dt

fn I Vu(t)1 dx 2

+ k — 2(2 + E)E(u(s));

(8.5)

where k = 2l521(µ — (2 + e)v), and u, v are defined by Proposition 5.4.4. But if k — 2(2 + e)E(u(s)) > 0, we conclude as in the proof of Proposition 5.4.4 that T(p) < oo (inequality (5.19) and what follows), which is absurd. We then have 2(2 + E)E(u(t)) > k for all t > 1, and we deduce immediately from (8.4) that ffudxdt
Set r= {t> 1;

(8.6)

f ue(t) dx
For t E F, we apply (8.5) with s = 1. It follows that E

f IVu(t)1

2

dx < 2(2 + e)E(u(1)) - k + 2 2(2 + e)E(u(1)) - k

f uu

t dx

+ Ilu(t) II L

(

8.7

)

2.

From Poincare's inequality, it follows that

Ilu(t)IILZ <—

2j IVu(t)1 dx 2

+C(E).

We then deduce from (8.7) that there exists M < oo such that, for all t E F, we have IIu(t)MH1 < M. By Corollary 8.1.5, we then have Ilu(t + s)II L- < K(M),

(8.8)

130 Bounds on global solutions

for all t E r and s E [r(M)/2,T(M)]. Set E t = is > t,s v F} for t >_ 1. It follows from (8.6) that there exists 0 < oo such that (E0I < T(M)/4. We then set T = 0+Tr(M). Since u E C([0,T],X), we have in particular Sup uu(t)IIC°° <00.

O
On the other hand, if t >_ T then there exists s E [t — r(M), t — 1 T(M)] such that s E F. Hence we have t E [s + 2T(M), s + T(M)] and so, by (8.8), IIu(t)IIL" < K(M). Consequently, sup Iiu(t)llL— < K(M). ❑ t>T

Remark 8.1.7. If g does not satisfy (8.1), we do not know whether the conclusions of Theorem 8.1.6 still hold. The following proposition sharpens the results obtained in §5.3. Proposition 8.1.8. Suppose that there exist p < A/2 (A given by (2.2)) and M < oo such that G(x) < PX 2 , for lxi >_ M. Then, for all cp E X, the corresponding maximal solution u of (5.4) is global and sup iiu(t)IIL— <00. t>o Proof. By Lemma 5.3.3, we know that u is global. In addition, (8.4) holds, so

that

2

f

J

f IVu(t) 1 2 dx< E(u(1))+C+p ^ u(t) 2 dx,

for t > 1. Applying (2.2), we then deduce that sup 1ju(t)ii H l < oo. On the other —

t>1

hand, g(u)/u is bounded in L(SZ), and we complete the proof by applying ❑ Corollary 8.1.4.

8.2. The Klein-Gordon equation In this section, we use the notation from Chapter 6. In particular, ci is any open subset of RN, m> —A (where A is defined by (2.2)), X = Ho(S2), x L 2 (SZ), g is a function of C(R, R) such that g(0) = 0, and which satisfies (6.8) with (N — 2)a < 2. G and V are defined by (6.5) and (6.6). E is defined at the beginning of §6.2. We now state the result. Theorem 8.2.1. Suppose that N > 3 and that there exists e > 0 such that

xg(x) > (2 + e)G(x),

The Klein—Gordon equation

131

for all x E R. Let (ep, z)) E X and let u be the corresponding maximal solution of (6.15)—(6.17) (see Theorem 6.2.2). Then, if T(cp,) = oo, we have sups>o II (u(t), u t (t))I1 x < oo. Proof.

We proceed in five steps. Let u be as above and set

f (t) =

f

u (t) 2

d x,

for t > 0. Step 1.

Some inequalities. By (6.26), we have

I

f"(t)>e{ i IVu(t) 1 2 dx +m fu(t)2dx

n

+ (4 + e)

fn u (t) t

2

J

dx — 2(2 + -)E(W, 0),

for all t > 0. Therefore f" (t) > E!I u(t) II H , + (4 + E)

in u (t) t

2

dx — 2(2 + e)E(cp

)

(8.9)

for all t > 0. In particular, there exist rl, p > 0 such that f"(t) ? ref (t) — 2(2 + e)E(co, t/5),

(8.10)

f"(t) ? µII (u(t), ut(t))11 i21i — 2(2 + e)E(^P,

),

(8.11)

for all t >_ 0. On the other hand, we deduce from the Cauchy—Schwarz inequality that there exists 6 > 0 such that µII (u(t), ut(t))II X ? 26 In u(t)ut(t) dx = 6 f'(t)I •

i

Then it follows from (8.11) that

f"(t) > Sl f '(t) I - 2(2 + e)E(^P, 0),

(8.12)

for all t > 0. Step 2.

We claim that (8.13)

E() > 0 ; at (ijf (t) - 2 ( 2 + e)E(^P,

,

if(t) < max rl f (0), 2(2 + e)E(cp, 0),

Vt > 0, Vt > 0.

(8.14)

(8.15)

I

132 Bounds on global solutions

Indeed, if (8.13) does not hold, we know that T(cp, Ali) < oo (Proposition 6.4.1). On the other hand, if (8.14) does not hold, then there exists t > 0 such that, setting g(t) = 77f (t) — 2(2 + e)E(cp, zL ), we have g(t) > 0 and g'(t) > 0.

(8.16)

Applying (8.10), we also obtain 9'(t) ? gg(t), Vt > 0.

We deduce from (8.16) and (8.17) that g(t) there exists to > 0 such that f"(t) > (4 + e)

fo

-4

(8.17)

oc, as t -+ oc. Therefore, by (8.9),

to. u t (t) 2 dx, Vt >t0.

b) Goo, which is absurd. We As for Proposition 6.4.1, this implies that then have (8.14), and (8.15) follows immediately. Step 3. We claim that 61 f'(t)I <— max{bI f'(0)I, 2(2 + E)E( cc, 0)}, Vt > 0.

(8.18)

Indeed, set h(t) = b f'(t) — 2(2 + e)E(cp, ), for all t >_ 0. We deduce from (8.12) that h'(t) >_ 6h(t). We then have h(t) >_ e s(t- s h(s), for t >_ s >_ 0. If there exists t >_ 0 such that h(t) > 0, then h(t) --* oo, as t -4 oo; and so f (t) —' oo, as t --> oo. This contradicts (8.15), and so )

b f'(t) < 2(2 + e)E(ip,

0),

Vt > 0.

(8.19)

Now set k(t) = —bf'(t) — 2(2 + e)E(cp,V)), for t > 0. From (8.12), we have —k'(t) >_ bk(t). Therefore, k(t) <_ e -bt k(0) for t > 0; and so k(t) < max{k(0), 0}. Consequently, we have

—s f'(t) < max{-6 f'(0), 2(2 + e)E(,p, 0) }, Vt > 0.

(8.20)

Putting together (8.19) and (8.20), we obtain (8.18). Step 4. We have

/'e+i sup] II (u(s), u t (s)) Il x ds < oo. t>O t

(8.21)

To verify (8.21), it suffices to integrate (8.11) between t and t + 1, and next to apply (8.18).

The Klein-Gordon equation 133

Step 5. Conclusion. Set w(t) = IKu(t),u t (t))IIX, for t > 0. We have (see Proposition 6.2.3 and Corollary 6.1.7)

w'(t) =2! g(u(t))u t (t)dx < 2II9(u(t))IIL2IIut(t)IIL2 Il

(

< CII9(u(t))IIL2w(t)

8.22

)

1/2 ,

for all t > 0. Observe then that N > 3, and so N/(N - 2) <3. Consequently,

CII IzI + IZI'° IIL2 _< C (ii + IIHIIL ) < CIIzII H (1 + IIzlIHI) ,

II9(z)IIL2 <

(

8.23

)

I

for all z E Ho (S2). It follows from (8.22) and (8.23) that w'(t) < Cw(t)(1 + w(t)), Vt > 0.

(8.24)

We deduce from (8.24) and (8.21) that, for all 0 < t < s < t + 1, we have

w(s) < Cw(t) exp

t +1 (ft

w(a) do) < Kw(t),

(8.25)

where K depends neither on t nor on s. In particular, for all t >_ 1 and all T E [0, 1], we have w(t) < Kw(t-r).

Integrating this last inequality in T on [0, 1] and applying (8.21) again, it follows that w(t)
t



Jt -1

w(s) ds < K'.

where K' does not depend on t. Since sup w(t) < oo, the proof is complete. ❑ o
following weaker condition (see Proposition 6.4.4): xg(x) > (2 + large.

E)G(x),

for Ixi

Remark 8.2.3. We have supposed that .N >_ 3. Modifying only the end of the proof (Step 5) we can show that the conclusions of Theorem 8.2.1 remain valid if N = 2 and a < 4 (a appears in (6.8)), and if N = 1 for any value of a >_ 0 (see Cazenave [1] and Sili [1]).

134 Bounds on global solutions 8.3. The non-autonomous heat equation In this section, we use the notation of Chapter 5. In particular, S2 is a bounded open subset of RN with Lipschitz continuous boundary, X = Co(1l), and (T(t)) t >o denotes the semigroup associated with the heat equation. g is a locally Lipschitz function R -4 1[8 such that g(0) = 0. On the other hand, we will use the space H(l) defined in §2.6.4 and §2.6.5. We also consider a > 1 such that v > N/2. We have in particular Ls -* H -1 . Given T > 0, cp E X, and h: [0, T] -> L° (S2), we are going to study the solutions of the following problem:

E C([0,T],X) n L 1 ([O,T],Ho(1)) n W' ,1 ((0,T),H -1 (Q);

(8.26)

u t (t) _ Lu(t) + g(u(t)) + h(t), almost everywhere in (0, T);

(8.27)

u(0) _ ,.

(8.28)

U

In the following subsection, we gather some preliminary results concerning problem (8.26)-(8.28). 8.3.1. The Cauchy problem for the non-autonomous heat equation Lemma 8.3.1. Let T> 0, 1 < p < oo, and

f E LP((O,T),L 0 (S2)), and let w

be given by w(t) = T(t) cp + J t T(t - s) f (s) ds.

(8.29)

0

Then, w E C([0,T], L 3 ) n LP((0,T), Ho (9)) n L1 ((0,T),X). Ifp = oo, we have in addition that w E C([0, TI, X) nC([O, TI, H'(9)). Proof. Observe first that, by (3.34), the integral appearing in (8.29) does make sense. On the other hand, in view of (3.34) and (3.31), the result is clear if f c C([0,T], L °(52)). The general case follows by density since, by (3.34), (3.31), and Young's inequality, we have:

II

!lW LP(( 0 ,T),X

+ 1IwIILP((0,T),H1 < CII f 11 LP((o,T),L° •

Corollary 8.3.2. Let T > 0, 1 < p < oo, and f E Lp((0,T),L °(S2)), and w E C([0, T], L °(0)). Then w is a solution of (8.29) if and only if w is a solution of the following problem:

1!

1 w E C([0 ,T],L ° ( l)) n L

P ([ 0 ,T] ,

((0 T),H -i ( 1 ); Ho( 9 )) n W

(8.30)

wt (t) = Lw(t) + f (t), almost everywhere in (0,T);

(8.31)

w(0) = 0.

(8.32)

Proof. Denote by (S(t)) t >o the semigroup generated in H -1 (52) by the operator C considered in Proposition 2.6.14. It is clear that (S(t))t>o coincides with

u The non-autonomous heat equation 135

1

(T(t)) t >o on L°(52). In particular, note that f E LP((O,T),H -i (St)) and then apply Lemma 8.3.1 and Proposition 4.1.9. ❑

Corollary 8.3.3. Let T > 0, cp E X, and h E L°°((0,T),L°(S2)), and let C([0,T], X). Then, u is a solution of (8.26)-(8.28) if and only if u is a solution of u E

u(t) = T(t)cp +

T(t - s)g(u(s)) ds + T (t - s)h(s) ds. fofo

(8.33)

In addition, u E C((0,T],Ho(Q)) fl L Z ((0,T),Ho(Q)) nW 1 2 ((0,T),H''(S2)). "

I

Proof. We apply Lemma 8.3.1 and Corollary 8.3.2 to f (t) = g(u(t)) + h(t) and w(t) = u(t) - T(t)
-

Now we can state a result of local existence for problem (8.26)-(8.28). Proposition 8.3.4. Let E X, h E LOO ([0, oo), L°(1l)). Then, there exists a unique maximal solution u E C([0,T(cp)),X) of (8.33). We haveT(p) > 0, and ifT( oo as t T T(cp). Proof. Applying Lemma 8.3.1, we easily adapt the proof of Theorem 4.3.4. ❑ Remark 8.3.5. We see that the condition g(0) = 0 is not necessary to solve problem (5.1)-(5.3) or problem (8.26)-(8.28). Indeed, if g(0) # 0, we can replace g by g - g(0) and h by h + g(0)1c. 8.3.2.

A priori estimates

Proposition 8.3.6. Under the hypotheses of Proposition 8.3.4, and if there exist M, C > 0 such that xg(x) < Cx 2 , for lxi > M,

(8.34)

then T(cp) = oo. Proof. Let w and z be the maximal solutions of the following problems: w(t) = T(t) cp+ + J T(t - s)g(w(s)) ds 0 ft

z(t) = T(t)cp

-

+ J0

T(t - s)h(s) ds,

+ J T(t - s)(-g(-z(s))) ds + o

t

fo T( t - s)h

-

(s) ds.

I

136 Bounds on global solutions

Applying Lemma 8.1.2, we easily verify that -z(t) < u(t) < w(t), for all t >_ 0 such that u, w, and z are defined. In addition, applying Lemma 8.1.2 again, we readily obtain w(t) > 0 and z(t) >_ 0. We may restrict ourselves to the case in which cp > 0 and h >_ 0, and so u >_ 0. We deduce from (8.34) that there exists C' > 0 such that (see Step 1 of the proof of Proposition 5.3.3)

Iig(u(t)) + IIL—
,

,

for t E [0,T(cp)). Applying (3.34), (3.37), and Lemma 3.5.9, we obtain

+ MCe—At ^ e A8 Ilu(s)IILo ds u(t)II L — < II'IIL^ + MCA t +

II

fo

L

(8.35)

T(t - s)h(s) ds(IL-,

for t E [0, T(co)). Next, observe that, by (3.34) and (3.37), IIT(t—s)h(s)(IL—
-

and so T(t — s)h(s) dsII L — < KIIhltL—((o,t),L°f e 2 (t 9) (t — s) 2 ds —

" J

-

—

0

K'II hII L((O,t),L° Therefore, it follows from (8.35) that

II u(t)II L— <_ IHIL— +

MC' +MCe—at f t 0

eA3jju(S)IIL—ds

+ K' II hII L-((o,t),L-). We conclude by applying Lemma 4.2.1.

(8.36)

D

The main result of this section is the following. Theorem 8.3.7. Suppose that g satisfies (8.34) with C < A, and let h E L°°(ll+, L°(2)) n L°°([8 + , L 2 (Sl)). Then, for all cp E X, the maximal solution u of (8.33) is global and satisfies sup Iu(t)IILo <00. t>o

Proof. We know (Proposition 8.3.6) that T(cp) = oo. To establish the bound, as in the proof of Proposition 8.3.6 we may assume that cp > 0 and h > 0,

The dissipative non-autonomous Klein-Gordon equation 137

and so it >_ 0. Multiplying the equation by u, integrating by parts, and setting f (t) = fo u(t) 2 dx, it follows that

f'(t) < -A f(t) +

I

n

u(t)g(u(t)) dx + i u(t)h(t) dx. n

But xg(x) < Cl + Cx 2 and

I

u(t)h(t)dx < Ilu(t)IIL211h(t)IIL2 <_

2 G f(t)+c2llh(t)Ili2

a-c

< 2 f(t)+C3. Consequently,

f'(t)

c -A 2 C f(t)+C3.

It follows (see Lemma 8.4.6 below) that sup Hu(t)IIL2 < 00. We conclude as in t >o the proof of Proposition 5.3.6, using (see the proof of Proposition 8.3.6 above) ft

T(t — s)h(s) ds < K II h11 L—((O,T),L°), '

for t E [0, T].

❑

Remark 8.3.8. Applying the estimates of Lemma 8.3.1, we easily show that, for all E > 0, we have sup Ilu(t)IIH1 < 00. t >E

8.4. The dissipative non-autonomous Klein-Gordon equation In this section, we follow the notation of Chapter 6. In particular, 1 is any open subset of RA', m> -A (where A is defined by (2.2)), X = Ho (1l) x L 2 (S2), (S(t) ) tER is the isometry group associated with the Klein-Gordon equation in Y = L 2 (l) x H -1 (f ), and g is a function of C(lR,l1) such that g(0) = 0, and which satisfies (6.8) with (N - 2)ci < 2. G and V are defined by (6.5) and (6.6). E is defined at the beginning of §6.2. We denote by F the function defined by F((u,v)) = ( 0,g(u)), for (u,v) E X. F is Lipschitz continuous on bounded subsets of X. We also consider 'y > 0, and we define the operator r E G(X) given by I, ((u, v)) = (0, ryv), for (u, v) E X. For T > 0, (cp, 0) E X, and h: [0, T] ---> L 2 (fl), we are going to study the solutions of the following problem: it E C([ 0 ,T],Ho(l)) ❑ C'([O,T],L 2 ( 1 ))nW 2 ' 1 (( 0 ,T),H -1 ( 1 )); utt -

u + mu + yu t =

u( 0 ) =
ut( 0 )

= 0.

g(u) + h in H -1 (S2), a.e. in (0, T);

(8.37) (8.38) (8.39)

138 Bounds on global solutions

We have a result which is similar to Lemma 6.2.1. Lemma 8.4.1. Let T>0, (cp, Vi) E X, and h E L 1 ((0, T), L 2 (Sl)), and let u E C([0, TI, Ho(Sl))nC'([O,T],L 2 (Sl)). Define H E L'((0,T),X), H(t) = (0, h(t)), for almost every t E [0,T]. Then u is a solution of (8.37)-(8.39) if and only if U = (u, u t ) is a solution of U(t) = S(t)(, b) + f

o

t

S(t - s){F(U(s)) - F(U(s) + H(s)} ds,

(8.40)

for all t E [0, T]. Proof. We apply Proposition 4.1.9 in the space Y.

❑

Proposition 8.4.2. Let h E Li ,(R+ , L 2 (Sl)). For all (cp, V,) E X, there exists a unique maximal solution u E C([0,T(cp,0)],H01 (1l)) n C 1 ([0,T(p,)],L 2 (SZ)) of (8.37)-(8.39). We have T (cp, Ji) > 0, and if T (cp, zJ) < oo, then jj(u(t),ut(t))IIx -; oc as t I T(^P,' )•

In addition,

E(u(t), u(t)) + ry for

ffr f ut dxds = E(, ) + J

o

z

t

bu t dxds,

^

(8.41)

all t E [0, T(,p, )).

Proof. We apply the method of Theorem 4.3.4 to solve (8.40), and we apply (

Lemma 8.4.1 to show (8.41). We note (see the proof of Proposition 4.3.7) that u depends continuously on h and, by density, we need only consider the case in which h E C(l[8 + , L 2 (cl)). In that case, we apply Proposition 6.1.1 and (6.13), and we obtain

dt

E(u(t),

J dx + f hut dx;

ut(t)) _ -ryu

and hence (8.41).

❑

Corollary 8.4.3. Suppose that there exists C such that G(x) < Cx 2 , Vx E l[8,

(8.42)

and let h E L' (R + , L 2 (1l)). Then, for all (cp, 0) E X, we have T(,) = oo.

The dissipative non-autonomous Klein-Gordon equation 139

Set f (t) _ II (u(t),u t (t))II' , for t E [0,T(co, u')). It follows from (8.41) and (8.42) that

Proof.

f (t) < 2E(cp, ')

+2

J0

t

^Ih(s) IIL2 f (s) 1 / 2 ds + 2C

J

o

f (s) ds,

t

for t E [0, T(cp, l)); hence the result, applying Lemma 4.2.1.

El

Remark 8.4.4. If 1 is bounded, we may assume that (8.42) holds only for (xI large (see Proposition 6.4.4). The main result of this section is the following. Theorem 8.4.5. Suppose that y > 0, that g satisfies (8.42) with 2C < . + m, and that there exist K > 0 and c < .A + m - KC (A given by (2.2)) such that xg(x) - KG(x) <2, Vx E R.

(8.43)

Let h E L°° (R+ , L 2 (S2)). Then, for all (cc,) E X, we have T (cp, 7b) = oo, and the corresponding maximal solution u of (8.37)-(8.39) satisfies sup 1(u(t), u t (t)) x < t>o oo. Proof. We know that T(co, zl^) = oo (Corollary 8.4.3). Take E > 0 and set f (t) = E(u(t), u t (t)) + e

I

n

uut dx, b't > 0.

It is easy to verify that f is absolutely continuous and that we have f'(t) =

J {-(ry - e)ut - eIVu^

2

- emu 2 - Equu t + eug(u) + ehu + hu t } dx,

R

almost everywhere. Let 6 E (0, 2E). We have

f'+bf= l {—

(

^

2 —s ut — ^E— 6 ) IVU1 2

l

-m E - 2 u 2 - e(7 - 6)uu t + eug(u) - 6G(u) + ehu +hu t dx. (8.44)

Observe that, by (2.2),

f -(E-2 IVuI 2 dx-m(e-)f u 2 dx<-(a+m)) 2

^

2

^



2 ^

u 2 dx.



'

140 Bounds on global solutions

On the other hand, if 6 <_ eK, by applying (8.43) and (8.42), we obtain e

f

ug(u) dx — b

J

f

f

G(u) dx < ec u 2 dx + (eK — 6) G(u) dx

sz

< ((eK — 6)C + ec)

I

f^ u^ dx.

Thus, it follows from (8.44) that f'+6f < f {—(y—e-6)U —E((A+m)(1

1

2)

— (K — S) C — c) u 2 — e(y — 6)uu t + ehu +h u t } dx. (8.45) e JJJ Suppose further that 6
xy < 2x 2 + ay 2 ,

with a = 2e(ry — E)/'y, it follows that

1

1

(7 ty 6)2 —gy(1 — 6) f uut dx < E 2 t

<_ e2y

j u 2 dx + 4 in u dx

J u2 dx + ry4 J ut dx.

(8.47)

Applying (8.46) with a = 1/(2e), we find that u2 dx;

(8.48)

j but dx < 1ly fJ"h2 dx + —4 ff2 ut dx.

(8.49)

E

^

hu dx < 4 f h2 dx +e 2 n

and next, applying (8.46) with a = 2/y, t

Combining (8.45), (8.47), (8.48), and (8.49), we obtain f'+6f < fit {— (2 —e—

1

(

l

— K—S \ EI

1 ) ut —E \(.\+m) 1\\\ 1 — c — e(ry+ 1)

l

u2 +

2 / +1h

\ 4 Y/ /

2 dx.

(8.50)

Note that, for e sufficiently small, we can take 6 = e 2 . (8.50) then reads e 2 f

I

f'+el f <—(2—e— 2 ) (

+m

Jsz utdx—e((.^+m—KC—c) c++1))

f

2d+(1 +1) fh2d

t

The dissipative non-autonomous Klein-Gordon equation 141

Recall that we assume that A + m - KC - c> 0, and so, if e is small enough, we have f' + e 2 f <

(

41 +

1

)

f

h2dx.

(8.51)

In addition, applying (8.42) and (8.46) with a = 2e, we obtain

J

f(t)>_4 s^

utdx+2

IVu1 dx+(2 -C -e 2) j u 2 dx. Jsi o 2

Since 2C < A + m, applying (2.2), we see that if e is small enough, then there exists b > 0 such that 1(t) ? EIi((u(t),ut(t))I1 2 (8.52)

The result is now a direct consequence of (8.51), (8.52), and of the following lemma. ❑ Lemma 8.4.6. Let T > 0, ,a > 0, and H _> 0. Let f E C([0, T]) be an absolutely continuous function such that f' + lif < H, almost everywhere on (0,T). Then, we have f (t) < - + e - ^` t f ( 0 ),

for alltE[0,T]. Proof. Set w(t) = eu t (f (t) -H/µ) for t E [0, T]. We have w'(t) <_ 0 almost everywhere; and so w(t) < w(0) for all t E [0, T]; hence the result. ❑ Remark 8.4.7. If ci is bounded, we may suppose that (8.42) and (8.43) hold

only for lxi large (see Proposition 6.4.4). Notes. About §8.1, see Cazenave and Lions [1], Giga [3], and Ni, Sacks, and Tavantzis [1]; and for §8.2, see Cazenave [1] and Sill [1]. Concerning nonautonomous problems (§8.3 and §8.4), see, for example, Haraux [1, 2].

0 The invariance principle and some applications

9.1. Abstract dynamical systems Throughout this section, (Z, d) is a complete metric space. Definition 9.1.1. A dynamical system on Z is a family {S t } t >o of mappings on Z such that:

(i) St E C(Z, Z),Vt > 0; (ii) So = I;

(iii) St +s = St o S9 , Vs, t > 0; (iv) the function t

H

St z is in C([0, oo), Z) for all z E Z.

Remark 9.1.2. In what follows, we write St S 8 instead of St o S. Remark 9.1.3. It is clear that if F is a closed subset of Z such that SF C F for all t > 0, then {(St)I F } t >o is a dynamical system on (F,d). Definition 9.1.4. For all z E Z, the continuous curve t '--> St z is called the trajectory from z. Definition 9.1.5. Let z E Z. The set w(z)={yEZ;St n —>oo, St z—*yarn--+c},

is called the w-limit set of z. Proposition 9.1.6. We have w(z)

= fl U

3>0 t>s

{St z}.

Proof. The proof is straightforward, by Definition 9.1.5.

o

Liapunov functions and the invariance principle 143

Proposition 9.1.7. For all z E Z and all t > 0, we have w(St z) = w(z),

(9.1)

St (w(z)) C w(z).

(9.2)

In addition, if U {St z} is relatively compact in Z, then t>o

(9.3)

St(w(z)) = w(z) 54 0.

Proof. (9.1) is an immediate consequence of Proposition 9.1.6. Let y E w(z). There exists t o -- oo such that St „ z —^ y. For all t >_ 0, and setting -r = t i-, + t, we have S1-,z --+ S t y, and so St y E w(z); hence (9.2). Now suppose that U {S t z} t>o

is relatively compact in Z. Then there exists a sequence t,,, ---> oo and y E Z 0. It remains to show such that St „z --> y. Therefore y E w(z) and w(z) such that that w(z) C St w(z). To see this, consider y E w(z) and t,,, — St „z —> y. Set rr, = t o — t. There exists a subsequence r,,, k ---> oc such that ST,.k z —^ w E w (z). Thus, St w = St lim ST, z = lim St , k z = y;

❑

hence (9.3). Theorem 9.1.8. Suppose that U {St z} is relatively compact in Z. Then: t>o

(i) St (w(z)) = w(z) 0, for all t > 0; (ii) w(z) is a compact connected subset of Z; (iii) d(S t z, w(z)) --> 0 as t --> oo. Proof. (i) is a consequence of (9.3). On the other hand, for all s > 0, U {St z} t>s

is a relatively compact connected set. By Proposition 9.1.6, w(z) is then the decreasing intersection of connected and compacts subsets. Hence we have (ii). To show (iii), assume by contradiction that there exists a sequence t o —* 00 and e > 0 such that d(St „z,w(z)) >_ e. There exists y E Z and a subsequence t,lk --> oo such that St „ k z —* y E w(z). Therefore d(St „ k , w(z)) —> 0 as k —> oo, which is absurd. ❑ 9.2. Liapunov functions and the invariance principle Definition 9.2.1. A function 4P E C(Z, R) is called a Liapunov function for {St } t >o if we have 4)(Stz) < (D(z),

for all z E Z and all t > 0.

144 The invariance principle and some applications

Remark 9.2.2. If 4) is a Liapunov function for {St}t>o then, for all z E Z, the function t H 4)(St z) is non-increasing. Theorem 9.2.3. (LaSalle Invariance Principle) Let 4) be a Liapunov function for {St }t>o, and let z E Z be such that U {St z} is relatively compact in t>o

Z. Then: (i) .£ = lim t - 0 4)(Stz) exists;

(ii) 4)(y)=£, for ally Ew(z). Proof. 4)(Stz) is non-increasing (Remark 9.2.2) and bounded since U {Stz} t>_o

is relatively compact. Hence we have (i). If y E w(z), there exists a sequence ❑ t n, --> oc such that S t „z —^ y. Therefore, 4)(St „z) —> 4)(y); hence (ii). Definition 9.2.4. An element z E Z is called an equilibrium point of {St}t>o if Stz =z for alit >0. Remark 9.2.5. In practical applications, Theorem 9.2.3 is used mainly to establish that some trajectories of {S t } t >o converge to equilibrium points. Definition 9.2.6. A Liapunov function 4) for {St } t >o is said a strict Liapunov function if the following condition is fulfilled. If z E Z is such that 4)(St z) = 4)(z) for all t > 0, then z is an equilibrium point of {St}t>o. Theorem 9.2.7. Let 4) be a strict Liapunov function for {St } t >o, and let z E Z be such that U {Stz} is relatively compact in Z. Let £ be the set of

t>o

equilibrium points of {St } t >o. Then: (i) £ is a non-empty closed subset of Z; (ii) d(St z, £) --+ 0 as t —

(i.e. w(z) C E).

0. Let y E w(z). Applying Theorem 9.1.8(i) again, and then Theorem 9.2.3(ii), we obtain Proof. By continuity of St, £ is closed. By Theorem 9.1.8(i), w(z)

4)(Sty) = 4) (y), Vt > 0; therefore y is an equilibrium point. From this, we deduce (i) and then (ii) by ❑ applying Theorem 9.1.8(iii). Remark 9.2.8. Theorem 9.2.7 means that the set of equilibrium points attracts all the trajectories of {St}t>o.

A dynamical system associated with a semilinear evolution equation 145

Corollary 9.2.9. Suppose that the hypotheses of Theorem 9.2.7 are fulfilled. Let P = lim 4D(St z) and Ee = {x E E, 4?(x) = Q}. Then Ee is a non-empty closed subset of Z and d(St z, EQ) —> 0 as t —+ cc (and so w(z) C Ee). If, furthermore, Ee is discrete, then there exists y E Ej such that St z --+ y as t —> oo.

Proof. Since E is closed and 1 is continuous, E is closed. The remaining part of the corollary is a consequence of Theorems 9.2.3, 9.2.7, and 9.1.8 (ii).

❑

9.3. A dynamical system associated with a semilinear evolution equation

We consider in this section a Banach space X, an m-dissipative operator A with dense domain, and a function F : X -- X that is Lipschitz continuous on bounded subsets. We use the notation of Chapter 4, and in particular we denote by (T(t)) t > o the contraction semigroup generated by A. We recall that, for all x E X, there exists a unique maximal solution u E C([0,T*(x)),X) of u(t) = T(t)x +

JO

t

T(t — s)F(u(s)) ds, `dt E [0,T*(x)).

(9.4)

For x E X and t E [0,T*(x)), we set St x = u(t).

We consider a subset P C X such that there exists M < oc with T(y) = oo, Vy E P;

(9.5)

^IStyll <_ M, Vy E P,dt > 0;

(9.6)

We set Z = U U {St y}, and we denote by d the distance induced on Z by the norm of X.

,EP t>O

Lemma 9.3.1. We have the following properties: (i) T*(z) = co, Vz E Z;

(ii) IISt zU <M, Vz E Z, Vt > 0; (iii) StZ E Z, Vz E Z, Vt > 0. Proof. Let y E P. Then u(t) = St y is the solution of (see §4.3) u E C([0, cc), X) n C' ([O, oo), Y);

(9.7)

u'(t) = Bu(t) + F(u(t)), Vt > 0 (9.8) ;

u( 0 ) = y.

(9.9)

146 The invariance principle and some applications

Therefore, for all s > 0, v(t) = u(t + s) is the solution of (9.7), (9.8), and v(0) = u(s). Thus, St(S s y)) = St (u(s)) = u(t+s) for all s,t >_ 0. Consequently, we have T*(S s y) = oc for ally E P and all s> 0, and jjStS s yjj < M for ally E P and all s, t > 0. Now take z E Z. There exists a sequence (t n ) n,> o c [0, oo) and a sequence (yn,),,, >o C P such that Sy - n - z as n -; oo. Let T < T*(z). By Proposition 4.3.7, we have StSt „y n —' Stz,

(9.10)

as n -> oo,

uniformly on [0, T]. In particular, we have l(StzIl _< M, for t E [0, T]. Since T < T*(z) is arbitrary, we deduce (i), and next (ii). (iii) is then a consequence of Theorem 9.3.2.

(9.10).

❑

{St } t >o is a dynamical system on (Z,d).

Proof. We have So = I. In addition, for all z E Z, if as n -i oc then, by Proposition 4.3.7, we have

(z,,)n >o

C Z and zn -4 z

St zn ---> St z, as n -- oo,

for all t > 0. Hence St E C(Z, Z) for all t > 0. Furthermore, since for all y E Z, u(t) = Sty is the solution of (9.7)-(9.9), we deduce easily that 5t59 = St +s for all s, t > 0. Finally, we have Stz E C([0, oo), Z) for all z E Z; hence the result. ❑ 9.4. Applications to the non-linear heat equation

We follow the notation of Chapter 5. In particular, S2 is a bounded open subset of RN, with Lipschitz continuous boundary, X = Co (l), and (T(t)) t >o denotes the semigroup associated with the heat equation. g is a locally Lipschitz function R -* R such that g(0) = 0, and we consider G and E to be defined as in Proposition 5.4.4. Let cp E X be such that T(cp) = oo and let u be the corresponding maximal solution of (5.1)-(5.3) (see Theorem 5.2.1). If we have sup J[u(t)lILt >o

< 00,

(9.11)

then we may apply the results of §9.3, choosing Y = {cp} to associate to cp a co mplete m etric (Z, d), where d is the distance induced by the norm in X, Z = IJ {u(t)}, and a dynamical system {St } t > o on (Z, d). On the other hand, t >1

we know that there exist sufficient conditions to have (9.11); see, for example, §5.3 and §8.1.

Applications to the non-linear heat equation 147

Lemma 9.4.1. Let yp and u be as above. Then we have the following proper-

ties:

1

(i) U {u(t)} is relatively compact in X; t> o

(ii) for all e > 0, we have sup II u(t) II H 1 < co; t>E

t

(iii) for all e > 0, U {u(t)} is relatively compact in Ho(Sl); t>E

(iv) E is a strict Liapunov function for {St}t>o• Proof. The proof proceeds in three steps. Step 1. Let e > 0 and s > 0. Applying Remark 5.1.2, replacing cp by u(s), we obtain in particular that IIu(s + E) H1 <_ C(e 1 / 2 + e -1 / 2 ), where C does not depend on s. Hence we have (ii), since s is arbitrary. Step 2. To establish (i) and (iii), we need only show that if t o —4 oc, then there exists a subsequence t flk and w E X n Ho (1l) such that u(t fk ) --> w in X n Ho (S2) as k —4 oo. Set Tn = t o — 1, cp n = u(r) and u n (.) = u(rrn + ). It is clear that u(t) = u n (1). By Step 1, cp n is bounded in X n Ho (1 ), and so there exists E L°O n Ho (Sl) and a subsequence (nk) such that cp nk 0 in L 2 (1), as k —> oo. Since Il^pnk — IILOO is bounded and I ^Pnk — 0IIL 2 —> 0, it follows from Holder's inequality that II^Pnk — I Lp --> 0, for all 1 < p < oo. In particular, conk —^ 0 in L N (1), as k --f oo. From (3.37) and (9.11), we deduce that, for all k, f E N, we have

IlUnk

— un, II L — <

tll^nk

t

— Wne II LN +

i ^ 2 llWnk — one IILN

fo

+ Ci

t II9(unk (S))

J

0

t

Ilu nk (s)

— 9(un, (s)) II L _ ds

— ufl

( s ) IIL°° ds,

for all t E (0,1]. Consequently (Lemma 8.1.1), u(t flk ) = u nk (1) is a Cauchy sequence in X. Let w be its limit. Now applying (3.32) and (9.11), we obtain, for all k, f E N, the following inequality: Ikknk — U., 11 ' < (1 + t-1/2)Ilçonk — cne IIL 2 +li

J

0

t (1 + (t —

s)-1/2)IIunk (s)

— carne (s )IIL2

ds,

for all t E (0,1]; from this, it follows (Lemma 8.1.1) that u(t nk ) = u nk (1) is a Cauchy sequence in Ho (1); and so that u(t nk ) —* w in X n Hp (S2) as k —f 00. We have shown (i) and (iii).

1 1 1 1 1

148 The invariance principle and some applications

Step 3. E is a strict Liapunov function on Z. Indeed, E is continuous on

X n Ho (S2) and so, by (i) and (iii), E is continuous on (Z, d). Let z E Z and let v(t) = S t z. It follows from (5.14) that, for all 0 < s < t, we have

Js

f

z

vt dx da + E(v(t)) = E(v(s)).

(9.12)

We then have E(v(t)) <_ E(v(s)), and consequently E is a Liapunov function. On the other hand, E(v(t)) = E(z) for all t > 0, and we deduce from (9.12) that

J

o

J vt dx du = 0. s^

Consequently, v t = 0 for almost all t > 0, and it follows from this that v is constant in L 2 (51), and then is also constant in X. Thus, z is an equilibrium ❑ point and E is a strict Liapunov function. This completes the proof. Theorem 9.4.2. Let g be as above. Set £ = {u E D(A); Du+g(u) = 0}, and

£o, = {u E £; E(u) = a}, for a E R. Let cp E X and let u be the corresponding maximal solution of (5.1)-(5.3) (see Theorem 5.2.1). Suppose that T(cp) = oo and that u satisfies (9.11). Then, we have the following properties: (i) E(u(t)) converges to a finite limit a, as t --^ oo; (1 i)

£a

0;

(iii) dist(u(t), £a ) —> 0, as t — oo, where dist denotes the distance in Xf1Ho (St Proof. We apply Lemma 9.4.1 and Corollary 9.2.9. It suffices to note that the set of equilibrium points of the dynamical system associated with u is included in S. ❑ Remark 9.4.3. If N = 1, we can give a sharper result (see Matano [1]). There exists w E £a such that u(t) — w, as t —# oc. If N >_ 2, this remains valid if we suppose that g is analytic (see Simon [1]). In the general case, it remains true for most of the solutions (see Lions [1, 2]) but, except in some special cases (see Louzar [1] and Remarks 9.4.4 and 9.4.5 below), we do not know whether it remains true for any solution, apart from the recent results of Hale and Raugel [1] and Haraux and Polacik [1]. Remark 9.4.4. If we suppose that xg(x) < Cx 2 , with C < A (A given by (2.2)), then we verify immediately by applying (2.2) that S = {0}. In that case, all bounded solutions of (5.1)—(5.3) converge to 0 in X n Ho (52) as t —> oo. Remark 9.4.5. If g is strictly concave on (0, oo), £ fl {u >_ 0} = {0, co}, where co is the unique positive solution of —L = g(cp), cp E Ho (Sl). In that case, w(u0) is either 0 or ca, for all 'ao > 0 (cf. Haraux [5]).

Application to a dissipative Klein—Gordon equation 149

9.5. Application to a dissipative Klein Gordon equation —

In this section, we use the notation of Chapter 6. In particular, S1 is any open subset of R N , m > —A (where A is defined by (2.2)), X = Ho(S2) x L 2 (f ), (S(t))t>o is the isometry group associated with the Klein—Gordon equation in Y = L 2 (52) x H — '(1), g is a function of C(R,R) such that g(0) = 0, and which satisfies (6.8) with (N — 2)a < 2. G and V are defined by (6.5) and (6.6). E is defined at the beginning of §6.2. We denote by F the function defined by F((u,v)) = (0,g(u)), for (u,v) E X. F is Lipschitz continuous on bounded subsets of X. We also consider 'y > 0, and we define the operator r E G(X) given by F((u, v)) = (0, yyv), for (u, v) E X. For T > 0 and (cp, Vi) E X, we are going to study the solutions of the following problem: U E C([0, T], Ho (SZ)) n C 1 ([0, T], L 2 (S2)) n C 2 ([0, T], H — ' (S2)); utt

—

'Lu + mu + 7u t = g(u), b't > 0;

u(0) = cp,

(9.13) (9.14)

u t (0) = z/^•

(9.15)

We know (Lemma 8.4.1) that u E C([0, T], Ho (S2)) f1 C 1 ([0, T], L 2 (S2)) is a solution of (9.13)—(9.15) if and only if U = (u, u t ) is a solution of U(t) = S(t)() +

J

0

t

S(t — s){F(U(s) — F(U(s))}ds,

(9.16)

for all t E [0, T]. We also know (Proposition 8.4.2) that it is possible to solve locally (9.16) and that the solutions satisfy E(u(t), u t (t)) +'Yf t f ui = E( ^

,

(9.17)

for all t E [0, T]. In particular, we have E(u(t), u t (t)) _< E(,); and so, if there exists C such that 2C < A + m and

G(x) < Cx 2 , Vx E IR, then we have T(co,0) = oo for all (cp,0) E X and sup II (u(t), ut(t))II x < 00, t>o

where u is the corresponding solution of (9.13)—(9.15) (see the proof of Proposition 6.3.1, and Remark 6.3.2). On the other hand, if S2 is bounded, it suffices that g is such that G(x) < Cx 2 ,

for x large.

150 The invariance principle and some applications

For the end of this section, it is useful to formulate (9.16) in a different way. To do this, define the operator A. on X by ) = {(u, v) E X; Au E L (1), v E Ho (1l)}; 5 D(A A-,(u, v) = (v, Au - mu - yv), for all (u, v) E D(A). 2

7

Lemma 9.5.1. The operator A. y is m-dissipative on X, with dense domain. In addition, if we denote by (T7 (t)) t >o the contraction semigroup generated by A y on X, and if we suppose that ry > 0, then there exist M and a > 0 such that

I1T7(t)IIc(x) < Me -

' t ,

(9.18)

for all t > 0. Proof. We show that the operator A. y is m-dissipative on X, with dense domain, as in Proposition 2.6.9. It then remains to establish (9.18). To do this, we argue as in Theorem 8.4.5. We consider e > 0, (cp, J) E D(A,). We set T^,(t)(co,b) _ (u(t), v(t)) and

.f (t) = 2 J v 2 + 1 Vu1 2 + M f u 2 + e f uv. We verify that, for E small enough, we have f (t) > 611(u(t), v(t)) 0. We deduce (9.18), with a = e 2 .

and f'+e 2 f < El

We verify that U E C([0,T],X) is a solution of (9.16) if and only if U is a solution of U(t) = E y (t)(^p,V') +

J

0

t

Ty (t - s)F(U(s)) ds,

(9.19)

for all t E [0, T]. Let (cp, 0) E X, and let u be the corresponding maximal solution of (9.13)—(9.15). Suppose that T (p, ) = oo and that sup I (u(t), u t (t)) lI x < oo, t>o

and set Z = U {(u(t),ut(t)}. The results of §9.3 allow us to associate with u t>_o

a dynamical system {St}t>o on (Z, d), where d is the distance induced by the norm in X. We have the following result. Lemma 9.5.2. Suppose that S2 is bounded and that -y > 0. Let (p, /i) E X be as above. Then, we have following properties:

(i) Z is compact; (

n E is a strict Liapunov function for S }

Application to a dissipative Klein—Gordon equation 151

1

Proof. We proceed in four steps. Set U(t) = (u(t), u(t)) and H(t) = F(U(t)), for t > 0. Step 1.

By (9.19), we have U(t) = T,(t)(cp,7P) + W(t), where

f

W(t) =

(s)H(t — s) ds.

By (Lemma 9.5.1) T,,(t)(^, ) — 0 in X, as t oo, and so there exists a compact subset K l of X such that U {2 (t)(cp, ^l )} C Kl. Then we need only verify that t >o

there exists a compact subset K 2 of X such that U {W(t)} C K2 . t>o

Step 2.

Since S2 is bounded, we see by applying Theorem 1.3.2 and Remark 1.3.3, as well as the estimates of §6.1.2 (see in particular the proof of Proposition 6.1.5), that the range by the mapping u H g(u) of a bounded subset of Ho (Sl) is a relatively compact subset of L 2 (Sl). Since u is bounded in Ho(1l), there exists a compact K of X such that U {H(t)} C K. t>o

Step 3.

Let E > 0, and let T be such that (see Lemma 9.5.1)

IIHIIL—(o,00;x) FT IITryIIG(x) <6. For t >_ T, we then have T

'

§W(t) — f Ty (s)H(t — s) ds1Ix <6. 0

Consequently, U {W(t)} C K' + B(0, e),

(9.20)

t>_T

where we have set

K'= U

T

1

0 ^ t>T

T.^(s)H(t — s) ds J

Observe that the mapping (s, x) '-- Ty (s)x is continuous from [0, oo) x X to X. Consequently, U = U {Ty (t)K} is compact in X. Therefore, F = T • conv(U) O
is relatively compact in X. Since K' c F, K' is relatively compact in X. By (9.20), we can cover U {W(t)} by a finite union of balls of radius 6. On the t>T

other hand, WE C([O, oo), X); hence U {W(t)} is compact and it can also o
152 The invariance principle and some applications

be covered by a finite union of balls of radius a. Finally, we can cover U {W(t)} t>o

by a finite union of balls of radius E. Since e is arbitrary, U {W(t)} is relatively t>o

compact and, by applying Step 1, we obtain (i). Step 4. E is continuous on X, and thus also on (Z, d). (9.17) shows that E is a Liapunov function. Finally, if E is constant on a trajectory (v, v t ) of {St } t >o, we deduce from (9.17) that v t = 0 for all t >_ 0; therefore v does not depend on t; z = (v, 0) is then an equilibrium point, and E is strict Liapunov function. This completes the proof. ❑ Theorem 9.5.3. Suppose that Sl is bounded and that ry > 0. Set £ = {u E Ho (S2); ,u — mu + g(u) = 0}, and E. = {u E £; E(u, 0) = a}, for a E R. Let (cp, 0) E X, and let u be the corresponding maximal solution of (9.13)—(9.15). Suppose that T(cp,z//) = oc and that sup Il(u(t),u t (t))llX < oo. Then, we have the following properties:

t>o

(i) E(u(t), u t (t)) converges to a finite limit ,Q, as t —f oo; (ii) £A ^ 0 ;

(iii) IIut(t)IIL2 —4 0, as t --4 oo; (iv) dist(u(t), £p) — 0, as t —> oo, where dist denotes the distance in Ho (S2). Proof. We apply Lemma 9.4.1 and Corollary 9.2.9. It suffices to observe that the set of equilibrium points of the dynamical system associated with u is in❑ cluded in E. Remark 9.5.4. If we suppose that xg(x) < Cx 2 , with C < A + m (A given by (2.2)), then we verify immediately by applying (2.2) that S = {0}. In that case, all bounded solutions of (9.13)—(9.15) converge to 0 in X as t --4 oo. If g does not satisfy this condition, sufficient conditions to ensure that w(cp, v) is reduced to one point can be found in the literature (cf. Haraux [4], Hale and Raugel [1]). Notes. See Ball [3], Dafermos [1-3], Dafermos and Slemrod [1], Hale [1], Haraux [1, 2], Henry [1], LaSalle [1], and Sell [1]. The w-limit sets also appear in the theory of maximal attractors. Consult Babin and Vishik [1-3], Ghidaglia and Temam [1, 2], Hale [2], Haraux [2], and Ladyzhenskaya [2]. The invariance principle is also very useful in the study of the behaviour at infinity of positive solutions of reaction—diffusion systems. See, for example, Masuda [1], Haraux and Kirane [1], and Haraux and Youkana [1]. There has recently been substantial progress on asymptotic behaviour of gradient-like systems as a consequence

Application to a dissipative Klein—Gordon equation 153

of the work of Hale and Raugel [1] (cf., e.g., Haraux and Polacik [1] where the condition of Hale and Raugel is used in an essential way). On the other hand, negative results are beginning to appear in the literature (see Polacik and Rybakowsky [1]) when the non-linearity depends on x.

10 Stability of stationary solutions In this chapter, we describe an extension of the Liapunov linearization method to establish the (local or global) asymptotic stability of equilibria. The perturbation argument developed here is applicable to various semilinear evolution problems on infinite-dimensional Banach spaces. We also discuss the connection between stability and positivity in the case of the heat equation. 10.1. Definitions and simple examples Let (X, d) be a complete metric space and {S(t)

}

t >o

a dynamical system on X.

Definition 10.1.1. A trajectory v(t) = S(t)a of the dynamical system {S(t) t >o is called (positively) stable in the sense of Liapunov if }

de > 0, 35 > 0 such that x E

•

X and d(x, a) < S = Vt > 0, d(S(t)x, v(t))

(10.1)

Definition 10.1.2. A trajectory v(t) = S(t)a of the dynamical system {S(t) t >o is called (positively) asymptotically stable in the sense of Liapunov if it is stable in the sense of Liapunov and }

•

1S > 0 such that x E X, d(x, a) < Si = lim d(S(t)x, v(t)) = 0.

(10.2)

In particular, an equilibrium a of {S(t) t >o is called stable (resp. asymptotically stable) in the sense of Liapunov if the constant trajectory v(t) - a satisfies (10.1) (resp. (10.1) and (10.2)). }

In the easiest cases for X = RN, stability of the equilibrium a of an equation

= f (u(t)), t> 0,

(10.3)

with f E C' (X, X) can be seen from the linearized equation

z' = ( Df)(a)(z(t)), t > 0.

(10.4)

More precisely, the exponential asymptotic stability of a for (10.3) is related to exponential asymptotic stability of 0 for the linearized equation (10.4). We recall here the Liapunov stability theorem (for a proof, cf., e.g., Haraux [5]).

Definitions and simple examples 155

Theorem 10.1.3. (Liapunov) Let X be a finite-dimensional normed space, and f E C 1 (X,X) a vector field on X. Let a E X be such that f(a) = 0 and assume that all eigenvalues si, 1 < j < k of D f (a) have negative real parts.

(10.5)

Then a is an asymptotically Liapunov stable equilibrium solution of equation (10.3) in the following sense: for each b < v = min1<^ 0 and M(5) > 1 such that, if l]x — all 0, II u(t) — all <_ M(b)]Ix

t• — a]1e -5

I

In the opposite direction, we have the following result (cf. Haraux [5]). Proposition 10.1.4. Let X be a finite-dimensional normed space, and f E C' (X, X) a vector field on X. Let a E X be such that f (a) = 0 and assume that

all eigenvalues Si, 1 <j < k of Df (a) have positive real parts.

(10.6)

Then a is a completely unstable equilibrium solution of (10.3) in the following sense: there is a neighbourhood w of a in X such that, for each b E X, b a, the unique solution of (10.3) with initial condition b leaves w for ever if t >_ T large enough. Remark. Stability of an equilibrium cannot always be seen on the lineariza tion.As a simple example, we may consider u' = f (u) = ±u 3 , in which cases D f (0) = 0. When we choose the (—) sign, 0 is asymptotically (but not exponentially) stable, while when we choose the (+) sign it becomes completely unstable. In this last case, 0 is the only global solution.

-

To illustrate the general ideas of this section, we give two simple examples. Example 10.1.5. Let f E C 1 (IR) and consider the first order scalar ODE u'(t) = f (u(t))•

It is known (cf., e.g., Haraux [5]) that each bounded global solution u(t) of this equation on IR+ tends to a limit c with f (c) = 0. The stability of such an equilibrium c is delicate only when f'(c) = 0. Indeed, • If f'(c) < 0, c is exponentially stable.

Ii

156 Stability of stationary solutions

• If f'(c) > 0, c is completely unstable in the sense of Proposition 10.1.4. As an illustration, the simple first order ODE

u'+u 3 —u=0 has exactly three equilibria {-1, 0, 1}. The equilibria 1 and (-1) are exponentially stable and they attract, respectively, the positive solutions and the negative solutions of the equation. On the other hand, the equilibrium 0 is completely unstable in a very strong sense: it attracts no solution except itself. Example 10.1.6 Let f E C 1 (R), c> 0 and consider the second order ODE

u"(t) + au'(t) = f (u(t)). It is known (cf., e.g., Haraux [5], Hale and Raugel [1]) that each bounded global solution u(t) of this equation on 1R+ tends to a limit c such that f (c) = 0 (and u'(t) tends to 0). The stability here is defined in the sense of the phase space H x H for the corresponding first order system in (u, u'). The situation is more complicated than in the previous example: • If f'(c) < 0, then (c, 0) is exponentially stable in the phase space R x R. • If f'(c) > 0, then (c, 0) is unstable in the phase space R x R but attracts some other trajectories than the equilibrium itself. We have here a typical example of a hyperbolic point. As an illustration, the simple second order ODE

u"+u'+u 3 —u=0 has exactly three equilibria {(-1, 0); (0, 0); (1, 0)}. The equilibria (1,0) and (-1,0) are exponentially stable in the phase space H x R. On the other hand, the equilibrium (0,0) is a hyperbolic point. 10.2. A simple general result

Let X be a real Banach space, let T(t) = e^ t S(t) with c E H, and let (S(t)) t > 0 be a contraction semigroup on X (it is easy to check that the family of operators (T(t)) t > o has the semigroup property, cf. Definition 3.4.1), and F : X --> X locally Lipschitz continuous on bounded subsets. For any x E X, we consider the unique maximal solution u E C([0,T(x)), X) of the equation u(t) = T(t)x + f T(t — s)F(u(s))ds, Vt E [0, (x)).

a

(10.7)

A simple general result 157

By a stationary solution of (10.7) we mean a constant vector a a = T(t)a

+ fo

t

E

X such that

T(t - s)F(a)ds, Vt > 0.

(10.8)

The following result is an easy consequence of the general theory of strongly continuous linear semigroups, and can easily be verified. Let L = cl + A, where A is the generator of S(t) (L can be considered the generator of T(t) in the sense of Definition 3.4.2). Then we have the following. Lemma 10.2.1. A vector a

if we have

E

X is a stationary solution of (10.7) if and only

a E D(L) and La + F(a) = 0.

(10.9)

We are now in a position to state the main result of this section. Theorem 10.2.2.

Assume that, for some constants 8> 0, M > 1, we have Vt > 0,

Let a

E

IIT(t)II < Me

be.

(10.10)

X be a stationary solution of (10.7) such that

2Ro > 0, 3v >0: IIF(u) — F(a)II < vllu — all for

IIu — all < Ro ,

(10.11)

with

1 v

<

Then, for all x

(10.12)

b1M. E

X such that

Ijx - all < R i = Ro /M,

(10.13)

the solution u of (10.7) is global and satisfies Vt > 0,

II u(t) - all < MII x - all e - r t ,

(

10.14)

with y=b-vM>0. Proof. On replacing u by u - a and F by F - F(a), we may assume that a = 0 and F(a) = 0 with IIF(u)ll < vjjujj whenever IIujj C Ro . In particular, setting

T = sup{t > 0, IIu(t)II < R o } < oc, we find that Vt E [0, T), jju(t)II < MIIxjje -6 t +vM

f

0

e — b ( t — s) Ilu(s)11 ds.

158 Stability of stationary solutions

Letting cp(t) = e bt ^Iu(t)11, we obtain t

cp(t) G C1 + C2

J p(s) ds, for all t E (0, T) O

with C l = MIIxIl and C2 = vM. By applying Gronwall's lemma, we deduce that Vt E [0, T), e at lju(t)lI

< MIIxIIe" Mt (10.15)

Since 8> vM, we conclude that if MI(xlI <_ R o , then T = +oo and (10.15) holds ❑

true on [0, oo). This completes the proof of (10.14).

Remark 10.2.3. It is not sufficient for our purposes to state Theorem 10.2.2 with c < 0 (in which case, T(t) itself is a contraction semigroup). Indeed, in the examples given below in §10.3, the generator of the linearized equation will not be dissipative in general, especially when working in C0(S2). 10.3. Exponentially stable systems governed by POE In this paragraph, we show how the stability theorem 10.2.2 can be applied to partial differential equations. (a) We first consider the semilinear heat equation

u t -Au+f(u)=0 inR xS2, u=0 onR + x9Q

(10.16)

where Il is a bounded domain in R N and f is a function of class C l : ]R -4 J such that f(0) = 0 and f'(0) > -Ai,

(10.17)

where A l = A 1 (fl) is the smallest eigenvalue of (-0) in Ho (Sl). We have the following simple result. Proposition 10.3.1. Under the above hypotheses, the stationary solution u 0 of (10.16) is exponentially stable in X = Co(1l) in the following sense: for each y E (0, A l + f'(0)), there exists R = R(-y) such that for all x E X with lixil < R, the solution u of (10.16) such that u(0) = x exists and is global, and satisfies Vt > 0, §u(t)I < MiIxIIe - ry t ,

with M independent of -y and x.

(10.18)

Exponentially stable systems governed by PDE 159

Proof. We have shown in Corollary 3.5.10 that the contraction semigroup To(t) generated in C o (12) by the equation u -Au=0 inR + x52, JSlu=0 onIII+x81 t

satisfies (10.10) with S = Al and some M > 1. It is therefore sufficient to apply Theorem 10.2.2 with T(t) = e - f' (°) To(t), since for f E C 1 (R), F(u) _ f (u) - f'(0)u satisfies (10.11) with a = 0 and v arbitrarily small. ❑ (b) Another situation: this time we assume some conditions which are in a sense opposite to (10.17): f is strictly convex on [0, oo) and f (0) = 0, fd(0) < -A1(S2)

(10.19)

where A 1 (l) is the smallest eigenvalue of (-0) in Ho (52). Here the solution 0 is unstable and we have the following. Theorem 10.3.2. (i) There exists one and only one positive solution cp of the problem: cp E X n Ho (52), -O p + f(p) = 0.

(10.20)

(ii) For each uo E X, no > 0 and not identically 0, the solution u of (10.16) such that u(0) = uo tends to cp as t -> oo. Moreover, we have Vt > 0,

jIu(t, .) - cp(t, .)IILo < C(uo)exp(-yt),

(10.21)

where -y> 0 is independent of uo. Proof. The proof is divided into several steps. Step 1. We already know by Theorem 9.4.2 and the positivity preserving property that the solution u(t,.) asymptotes towards the set of non-negative solutions of (10.20) as t -> oo. We now show that if no # 0, u(t,.) cannot tend to 0 as t - oc. Indeed, assuming that lim j^u(t, .) II L= = 0, then, for each a > 0,

there is T(e) such that

coo

Vt > T(e), f (u(t, x)) < { fd(0) + e}u(t, x) on 52.

Choosing e > 0 so small that -f(0) - e - A 1 (1l) > 0, multiplying the equation by a positive eigenfunction cp l corresponding to the first eigenvalue A1(Q) of (-Li) in Ho (1), and then integrating over 52, we find, d ju(t,x)^o,(x)dx>O for all t > T(e). dt

i

I I I I

u

160 Stability of stationary solutions

1

I I

Since the function: t ---> fs , u(t, x)cpl (x) dx is non-decreasing on [T(E), oo) and tends to 0 as t --> oo, it must vanish identically on [T(E), oo). Because eO 1 is positive on S2, this implies that u(t,.) = 0 for all t >_ T(E). Then a classical connectedness argument shows that u0 = 0. Therefore if n o # 0, the w-limit set of u0 under S(t) contains at least a non-negative solution cp # 0 of (10.20). Step 2. By the strong maximum principle, we must have f() <0 and then

cp > 0 in Q. We now prove the following. Lemma 10.3.3. Let f be as above and cp > 0 in 1 be a solution of the equation E C(1l) n Ho (S2), —A^p + f (gyp) = 0. On the other hand, let 0 > 0 satisfy E C(S2) n Ho (S2), —AV) + f (0) ? 0.

'

Then either = 0 or b >_ cp. Proof. We first establish Vw E Ho (1),

J {IVwI

2

+ k(ep)w 2 }dx >_ 0, with k(cp) := f(o)/co. (10.22)

In fact, denoting by D(fl) the set of real-valued C°° functions with compact support in 52, we have the following sequence of identities

f f

{Ow VW E D(Q), f{VwI 2 + k()w 2 }dx = z =

2

{^Vw^ 2

+ (A/)w 2 }dx

—

DAP V(w2/ )}dx.

Since V (w 2 /cp) = 2(wVw/cp) — (w 2 / p 2 )O p, we obtain the formula Vw E D(S2),

J {^^w^

2

+ [f (p)/ p^w 2 }dx =

J IVw — (w/cp)V pI

2

dx. (10.23)

This establishes (10.22) when w E V(1). Then, by passing to the limit in (10.22) in the sense of Ho (52) along a sequence of functions w,,, E V(l) tending to w, we find that '

bw E Ho (1),

f

{IVwI 2 + [f (cp)/cp]w 2 }dx >_ 0.

We may, in particular, use (10.22) with w = (p — )+ E C(SZ) n Ho (Sl). On the other hand, we have, by the properties of cp and V):

-o(v^ - 0) + f (^) - f (VG) <_ 0.

Exponentially stable systems governed by PDE 161 ]

Multiplying by w = (cp — 0)+ and integrating over S2, we find that I { V w l 2 + [(f() — f())/( - 0)]w ' }dx < 0.

(10.24)

By combining (10.22) and (10.24), we finally obtain f^{[(f (^P) — f())/( — V)] — [f ( ,P)/ ]}w 2 dx < 0.

(10.25)

As a consequence of strict convexity of f, (10.25) now implies that wO = 5(cp — V,)+ = 0, everywhere in f.

(10.26)

Since cp > 0 in S2 and E C(Il), the conclusion follows at once from (10.26). ❑ Lemma 10.3.3 implies in particular the uniqueness part of (i). By combining this with Step 1 we conclude that w(u o ) _ { p}, which means that u(t,.) tends tocpast —>oo.

Step 3. We now establish (10.21). We begin with the identity

dtfin

- I 2 dx =

-2f {IV(u -x)12 +

f( ) _ V(^) lu -

2

}dx. (10.27)

From the convergence of u(t,.) to cp in X, it follows in particular that, fixing some non-empty open set w contained in a compact subset of 1, we have for t > T (depending on the solution u), Vt > T, u(t, x) ? (1/2)cp(x)

in w.

(10.28)

Now, from (10.27) and (10.28), we easily deduce the inequality

dJ t

Ju

2 dx < —2

with c(x)

J {V(u

2 + c(x)iu — cp^ 2 }dx,

(10.29)

2 — .f (P) (P/2)

if x E w

0

if x ^{ W.

The result will now become a consequence of the following lemma. Lemma 10.3.4. There exists 6 > 0 so that, c(x) being given as above, we have the inequality Vw E Ho (S2), f {iVwi 2 + c(x)w 2 }dx >_ 6

J w 2 dx. f1

(10.30)

162 Stability of stationary solutions

Proof. We introduce 6 = inf {

J {IVwI 2 + c(x)w

l ^z

2 }dx,

f^

52 w 2 d x = 1 } . w E H0'(), JJ

(10.31)

Since c E L(1l), a standard argument shows that the infimum in (10.31) is achieved for w = ( >_ 0, where (E Ho (S2) satisfies f ri ( 2 dx = 1 and is a solution of the elliptic problem ( E C(1l) n H' (l), —A( + c(x)( = 6C.

(10.32)

Multiplying by cp and integrating over S2, we immediately obtain 6

J Ccp dx = J (—A(+ c(x)C)cp dx = jo (—Acc(+ c(x)Ccp)dx = f[c(x) — k(^G)(x)]C(x)^P(x) dx,

where k(cp) = f(cp)/cp. By the strict convexity of f, it now follows that c(x) — k(cp)(x) >_ 0 in Il and c(x) — k(cp)(x) > 0 in w. In addition, we have C > 0 everywhere in Il by (10.32) and the strong maximum principle: in particular, we find 6 > 0. The result (10.30) follows at once by homogeneity. ❑ Proof of Theorem 10.3.2 continued. We deduce from (10.29) and (10.30) the

simple inequality d d (Mu(t) — GII)

—2611 u(t, •) —



(10.33)

From (10.33), we first deduce that

Vt > T,

IIu(t, •) — W112 <— II u(T, •) — (112•

In fact, (10.27) and the convexity of f also implies that II u(t, .) — W11 2 is nonincreasing; hence

Vt > T, JIu(t, •) — W112 < exp(6T) Iluo — w112 exp(—St) Kuo — cp^^. exp(—bt),

(10.34)

for some K > 0. Then, since u and z remain bounded in C 1 , from (10.34), we deduce that Vt > 0,

IIu(t, .) — cp^^. < C(uo) exp(— yt),

(10.35)

by replacing 6 by a slightly smaller positive constant, denoted by y. Hence ❑ Theorem 10.3.2 is completely proven.

Exponentially stable systems governed by PDE 163

Remark 10.3.5. The main result of Theorem 10.3.2 can be viewed as a property of global exponential stability of the positive stationary solution cp(x) in the metric space Z \ {0} = {u E Co (); u >_ 0, u # 0). Here, three remarks are in order. 1. The constant C(uo) in (10.35) does not remain bounded with IIuoIIL —• In fact, let A > 0 arbitrary and select t = T such that exp( -yT)II pII L°° > A. By letting uo --> 0 in Co(1), we deduce from (10.5) with t = T the estimate lim inf{C(uo), uo —+ 0 in C o (S1)} > A. Since A is arbitrary, we conclude: lim{C(u o ), uo —> 0 in C0 (S2)} = oo. 2. Assuming that f : JR — 1[8 is odd, locally Lipschitz continuous with f (s) >_ 0 for s —> oo, and satisfies (10.19), it also follows from the proof of Theorem 10.3.2 that the positive stationary solution cp(x) is exponentially stable in the larger space C 0 (1). Indeed, the linearized equation around u = cp(x) is z t — Oz+f'(cp)z=0 inlR xS2,

(10.36)

xaSl

z=0 onR +

I 1 I

which, using the convexity off on R+, turns out to be exponentially damped in Co (cl) by the method of Lemma 10.3.4. 3. Theorem 10.3.2 and the two remarks above are applicable, as a typical case, to the non-linearity f (u) = clul au — Au (10.37) for some positive constants c, a, and A. (c) Similarly, we can consider the semilinear wave equation u tt — Au + f (u) +.\ut = 0 in R x Q; u = 0 on I[8 + x

aci

(10.38)

where S2 is a bounded domain in R N f is a function of class C 1 : JR --> JR such that f (0) = 0 and f'(0) > — A r , (10.39) ,

in which f is a locally Lipschitz continuous function: 1[8 satisfying the growth condition

If(u)I
-4

JR with f (0) = 0

a.e.onJR

withr>0arbitraryifN=1or2and03. We obtain the following result.

(10.40)

I



164 Stability of stationary solutions Under the above hypotheses, the stationary solution Proposition 10.3.6. (10.38) is exponentially stable in X = Ho (S2) x L 2 () in the (0, 0) of (u, v) exist 6 > 0 and R = R(6) such that, for all x E X with there following sense: such that u(0) = x is global and satisfies of (10.38) IIxMI < R, the solution u Vt > 0, JJu(t)II < M(6)^JxJIexp(-6t). (10.41) Proof. It is well known, and this can be deduced from the proof of Theorem 8.4.5 and Lemma 8.4.6 with H = 0, that the contraction semigroup T0 (t) generated in X = Hp (S2) x L 2 (cl) by the equation + 1l, utt—L\u+.\ut =0 inRx {u=0 onR+xOR satisfies (10.10). The method of proof clearly applies to the slightly more general equation utt — Au + f'(0)u + Au t = 0 in R+ x S2,

I

u=0 onR+x852.

(10.42)

In order to apply Theorem 10.2.2 with T(t) the semigroup generated by (10.42), I all we need to check is that the function F(u, v) = —(0,1(u) — f'(0)u) satisfies (10.11) with a = 0 and v arbitrarily small. But this is immediate, since the function cp(s) = f (s) — f'(0)s is o(Is) near the origin and, by (10.40), we have I(s) J <_ C(Jst"') for s large. Therefore, for each 6 arbitrarily small, we have < 6^s +C(6)Is, globally on R. The result then follows immediately from ❑ the Sobolev embedding theorems.

•

10.4. Stability and positivity

We consider the semilinear parabolic equation (10.16), where f is a locally Lip' schitz continuous function on the reals. If u is solution of (10.16) uniformly bounded on ][8+ x SI, the solution u(t,.) asymptotes towards the set of stationary solutions as t - oo. In particular, the existence of a bounded trajectory implies the existence of at least one solution to the elliptic problem (10.20). Roughly speaking, the stability of a solution cp of (10.20) rests on the sign of the first eigenvalue 77 = A 1 (—L + f'(cp)I) in the sense of Ho (1); more precisely, if 77 > 0,

0.

Stability and positivity 165

A typical example where the exponential stability of the positive solution occurs is when f (u) = cl u^ a u — Au

for some constants c, a > 0 and A > ) (1l). When N = 1, 52 = (0, L), and for .A > (x/2) 2 = .11(51), non-trivial stationary solutions appear as pairs of opposite functions that have a finite number (n + 2) of zeroes equally spaced on [0, L] with n < (L/ir)A 1 / 2 — 1, built from positive solutions of the same problem on (0, L/(n + 1)). A new pair of solutions appears when the increasing positive parameter A crosses an eigenvalue (kir/L) 2 = .\k(0, L) of the operator (—u xy ) in Ho (0, L). It has been known for some time (cf., e.g., Chafee and Infante [1]) that the only stationary solutions that are stable in the sense of Liapunov are the positive and the negative solution. In higher dimensions the situation seems much more intricate, but it is still of interest to investigate the relationship between stability and the absence of zeroes. For instance, in the case of Neumann boundary conditions, a result of Casten and Holland [1] asserts that if S2 is star-shaped, any non-constant solution is unstable. Counter-examples of stable solutions changing sign in f1 are known for both Dirichlet and Neumann boundary conditions in non-convex domains. On the other hand, even for fI convex, there is no general instability result for solutions changing sign in the case of Dirichlet boundary conditions. 10.4.1. The one-dimensional case Consider, as a motivation, the one-dimensional semilinear heat equation u t — u yx + f (u) = 0 in lR + x (0, L),

(10.43)

u(t, 0) = u(t, L) = 0 on R + ,

where f is a C 1 function: H —# R. Any solution u of this problem which is global and uniformly bounded on ][8+ x (0, L) converges as t —+ oo to a solution cp of the elliptic problem (p E Ho (0, L),

—

(10.44)

coxx + f ( p) = 0. ,

Proposition 10.4.1. If cp is a stable solution of (10.44), then cp has a constant sign on (0, L). Proof. Indeed, if cp is not identically 0 and vanishes somewhere in (0, L), the function w := cp' has two zeroes in (0, L) and satisfies WE

C 2 ([0, L]) f1 Ha (0, L),

—w + f'(^p)w = 0

in (0,L).

166 Stability of stationary solutions

In particular, if 0 < a < ,3 < L are such that w(a) = w(0) = 0, w 54 0 on (a, 3) and if we set w = (a, Q), we clearly have ) 1 (w; — + f'(^p)I) = 0, where A l (w; —^ + f'(cp)I) denotes the first eigenvalue of —,L + f'(cp)I in the sense of Ho (w). We introduce the quadratic form J defined by Vz E Ho (0, L) J(z) :=

f

{zx + f'(^p)z 2 }dx.

Let

in

:= inf{J(z), z E Ho (0, L),

z 2 dx = 1}.

Let us also denote, by (, the extension of w by 0 outside w. Because J() _ f^{fix+f'(p)^ 2 }dx = f^ {+f'(p)C 2 }dx = f^ {wx+f'(^p)w 2 }dx = 0, we clearly have 71 = A (1l; —. + f'(^P)I) < 0. Assuming that 71 = 0 means that a multiple AS = 1i of ( realizes the minimum of J and therefore is a solution of E C 2 ([ 0 , L]) n Ho ( 0 , L), — 0xx + f'(^P)V) = 0.

This is impossible since V) is not identically 0 and yet vanishes on (0, a). Therefore 77 < 0. The deduction of instability will now follow in a few lines: assume that cp is stable in the sense of Liapunov in Co(S2) x Ho(1l) and let u E be the solution of (10.43) with u,(0) = cp + E0, where V) is a positive solution of ^b E C 2 n Ho ({ 0 , L]), — V)x. +f'() = rl0 in (0, L). Since ij> > 0, the order preserving property implies that u F > V. Now let W e =u E —cc>0. Because w E --> 0 uniformly as E —+ 0, we have f (uE) = f(cc) + f'(cc)we + 6(E)I wEI , with 6(E) —^ 0 as e --> 0. On the other hand, we have

d

f

w,V)(x) dx = j 0(uExx — f (u E ))dx

=J

O(Pxx + f V)xxw£ dxJ — .f (cc)') dx

s^

= —77

—

s^

ro

Jn f'(cp)weV)dx

—

Js^ 6 (E)1weIV)dx

J w,V)(x)dx — fn b(E)jweI0dx

(1771/ 2 ) f w,O(x) dx,

Stability and positivity 167

for all t >_ 0 and e > 0 small enough. This inequality, combined with positivity of z() and w E together with boundedness of w E , implies that

0

=f

,b(z)w(0, x)dx = e

^

J 1p2(x)dx.

I

^

Thus Proposition 10.4.1 is proven by contradiction. 10.4.2.

❑

The multidimensional case

It is interesting to remark that the above instability result does not require hypotheses on the shape of f. Even the differentiability condition can in fact be relaxed, since the important point is just boundedness of the potential f'(cp(x)). In higher dimensions, at present we have no such general results; however, the previous technique can be extended to some particular cases of special interest. First, we establish a basic lemma. Let S2 be any bounded open domain of RN, and let w be any Lemma 10.4.2. open sub-domain which is not dense in Il. Then, for any potential p E L(1l),

we have Ai (I; —A + p(x)I) < A i (w; —A + p(x)I).

Proof.

We introduce the quadratic form J, defined by dz E Ho (1),

J(z) :_ f {^Vzl 2 + p(x)z2}dx.

I

By definition,

rl := inf{J(z), z E Ho (0, L),

z 2 dx = 1} =

—A + p(x)I).

Let w E Ho (w) denote a normalized eigenvector of —A + p(x)I in Ho (w) associated with A, (w; —A + p(x)I) and let us consider the extension Ho (Sl) of

w by 0 outside w. Since

J(S) = f {IV I 2 +p(x)(2}dx = / {IV 1 2 +p(x)1 2 }dx

=

f

w

fw

{IVwI 2 +p(x)w 2 }dx=. 1 (w; —A+p(x)I),

we clearly have ?) < A, (w; —A + p(x)I). Assuming that 77 = a l (w; —A+p(x)I) means that a real multiple of ( realizes the minimum of J and therefore —AC + p(x) = 0. This is impossible, since C is not identically 0 and yet vanishes on a non-empty open subset of Q. ❑

I

I

168 Stability of stationary solutions An especially interesting special case is when ci is a rectangle in R 2 , and if f is a C' odd function on the reals. For instance, if f (u) = cu 3 — Au with a large, in addition to the unique positive solution, (10.20) will have more complicated solutions arising by odd extensions from the positive solutions in cellular subdomains which divide the domain into m x p equal rectangular regions. Such solutions, which are the analogue of odd-periodic extensions in the one-dimensional case, are also Liapunov-unstable. More precisely, we have the following. Proposition 10.4.3. Let cp be a solution of (10.20) in a rectangle S2 = (0, a) x (0, b) and assume that there is a sub-rectangle R = (0, a/p) x (0, b/q) with p, q integers > 1 and max(p, q) > 1 such that cp has a constant sign in R and cp = 0 on 8R. Then cp is unstable.

Proof. Assume, for instance, that p > 1. It is clear that the trace of (p on R' = (a/p, 2a/p) x (0, b/q) coincides with the odd reflection of cc with respect to the line x = a/p, and in addition cp^ R is even with respect to the line x = a/2p. Therefore, w := &,o/ax vanishes on the boundary of the sub-rectangle

w = (a/2p, 3a/2p) x (0, b/q) and satisfies w E C 2 (w) n Ho (w), —Aw + f' (p)w = 0 in w.

In particular, since w does not vanish inside w, we have A 1 (w; —A+f'(p)I) _ 0. Then Lemma 10.4.2 gives 1\1(52; —A + f'(1P)I) < ,A 1 (w; —A + f'(cc)I) = 0. Therefore cp is unstable.

u

There is another quite interesting result for the case in which S2 is a sphere in R N : in Comte, Haraux, and Mironescu [1] the following result is obtained. Proposition 10.4.4. If S2 is a ball in 1R', N >_ 2, and cp is a Liapunov-stable solution of (10.20), we have either cp = 0, or (p is spherically symmetric with constant sign in Ii. Notes. Theorem 10.2.2 is also valid when (T(t)) t > o is a general Co-semigroup; see Haraux [5]. The exponential stability property (10.21) can also be established in a non-autonomous framework; see Haraux [6].

Further remarks. We have not addressed the Navier—Stokes equation. The reader may consult Constantin and Foias [1], Foias and Temam [1], Fujita and Kato [1], Giga [1, 2], Heywood [1], Ito [1], Kato [1], Ladyzhenskaya [1], Leray [13], Temam [1], and Lions [3, 4]. For the Korteweg—De Vries equation, see, for example, Gardner, Greene, Kruskal, and Miura [1], and Kato [2].

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Index a priori estimate 58, 135-7 adjoint 22-3 attractor 152

behaviour, asymptotic 120-2, 142 blow-up 72-6, 87-90, 114-21 blow-up alternative 57, 58 Bochner's Theorem 7 boundary regularity 28, 62, 124, 134, 146 compactness 124-41, 143-4,.151 complete metric space 56, 124, 147 conservation of energy 78, 83, 92, 100 continuous, absolutely 13, 14, 15, 53, 55, 139, 141, contraction semigroup 39-41 strict 56 convexity 6, 148 damping 50-55, 134-41 dependence, continuous 59, 100 differentiable, differentiability 38, 51, 60, 62, 78 differential inequality 72, 73, 75, 87, 88, 116, 117, 120, 125, 131-3, 137, 139 distribution 2 vector-valued 10 domain bounded 27, 43, 46, 62, 68, 74, 124, 134, 141, 146

t

of an operator 18 dynamical system 142, 143, 145 eigenfunction 72 embedding, compact 3, 151 energy, see conservation of energy equation inhomogeneous 50-55 non-autonomous 50-55,134-41 parabolic, see heat equation with second member, see equation, non-autonomous equilibrium point 144, 148, 152 estimate, uniform 67, 68, 84, 112-13, 130, 136, 139 existence global 58, 65-71, 76, 83-6, 112-14 local 56-9, 64-5, 76, 82-3, 100-12 extrapolation 163 forcing 50-55,134-41 function integrable 7-8 measurable 4-6 functional analysis 1 Gagliardo-Nirenberg's inequality 3, 112 generator 39 graph, closed 18, 20 Gronwall's lemma 55, 125

186 Index

growth condition 65, 70, 84, 86, 112, 119, 127, 130, 133, 138, 149 heat equation 42-7, 62-77, 124-30, 134-7, 146-9, 158-63, 164-8 Hilbert complex space 25-6 Hille-Yosida-Phillips Theorem 40 invariance principle 143-4 isometry group 37, 41, 47-9, 61, 78, 91, 137, 149 Klein-Gordon equation, see wave equation Lax-Milgram Theorem 1, 26-7 Liapunov function 143-4, 147, 150 strict 144, 147, 150 Lipschitz condition, local 55, 57, 62, 63, 79-81, 100, 145-6, 149 maximum principle, 65-7 w-limit set 142, 143, 145 operator closed 18, 20 dissipative 19 m-dissipative 18-32, 33-5, 38-41 self-adjoint 24, 26, 32, 35 skew-adjoint 24, 26, 29-32, 37, 61, 78

Pettis' Theorem 5 point fixed 57 hyperbolic 156 regularity 28, 33, 35-7, 39, 41-3, 47, 48, 60-1, 62, 78, 100 Schrodinger equation 47-9, 91-123 set, connected 143 smoothing effect 37 Sobolev embeddings 3 Sobolev space 2, 13-17 solution bounded, see estimate, uniform global, see existence, global maximal 57 stationary 157 stability 154 asymptotic 154 exponential 158-64 Strichartz' estimates 96-100 trajectory 142 bounded, 144-7 relatively compact 143 uniqueness 56, 106 variation of the parameter formula 50 wave equation 47, 78-90, 130-3, 137-41, 149-52, 163-4