Nonlinear Evolution Equations — Global Behavior of Solutions

323 Pages·1981·2.297 MB·English

by  Alain Haraux (auth.)

Lecture Notes ni
Mathematics
Edited by .A Dold and .B Eckmann
148
Alain xuaraH
Nonlinear noitulovE Equations-
Global Behavior fo snoituloS
galreV-regnirpS
Berlin Heidelberg New York lg81
Author
Alain Haraux
Analyse Numerique
Tour 55-65, 85 etage
Universite Pierre et
Marie Curie
4, Place Jussieu
?5230 Paris Cedex 05
France
AMS Subject Classifications (1980): 34G10, 34G 20, 35-02, 35A05,
35A07, 35 B10, 35B15, 35B35, 35 B40, 35B65, 47 H05, 47H20
ISBN 3-540-10563-8 Springer-Verlag Berlin Heidelberg NewYork
ISBN 0-387-10563-8 Springer-Verlag NewYork Heidelberg Berlin
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PREFACE
These four chapters, divided into thirty five one hour
lectures, contain the basic material used in my two semester
"Seminar Course" at Brown University, during the academic year
1979-1980.
I am particularly endebted to H. Brezis for allowing me to
rely in an essential manner on his lecturenote book.
I wish to express my thanks to my colleagues of the Mathe-
matics Department and the "Students" of my seminar for the
favorable atmosphere in which these notes were written; more
especially, I express my gratitude to C. M. Dafermos and
W. A. Strauss for their unfailing encouragement during the
preparation of this work.
I thank very much Miss Sandra Spinacci for her efficience
in the material realization of the manuscript which represented
quite a heavy task.
May, 1980 Alain Haraux
ELBAT FO STNETNOC
Introduction ....................................... XI
,A EHT YHCUAC MELBORP
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
I. Generalities and Local Theory ...................... 1
Lecture i: Generalities, the continuous and
linear cases ........................... 1
Lecture 2: Quasilinear evolution equations ........ i0
II. The Global Existence Problem ....................... 19
Lecture :3 Generalities, first integrals and
Liapunov functions ..................... 19
Lecture 4: Methods relying on the Gronwall lemma .. 27
Lecture 5: A singular generalized "Gronwall lemma"
and application to a special nonlinear
Schrodinger problem .................... 32
YHPARGOILBIB ROF RETPAHC A-I,II .............................. 83
III. Theory of Monotone Operators and Applications ...... 39
Lecture 6: General properties, Minty's theorem,
Yoshida's regularization ............... 40
Lecture :7 Examples of maximal montone
operators .............................. 47
Lecture 8: Sums of maximal monotone operators ..... 54
Lecture 9: The range of a maximal monotone
operator ............................... 60
Lecture I0: Quasi-autonomous systems generated by a
maximal monotone operator .............. 70
I. Nonlinear semi-group generated by -A ......... 70
2. Quasi-autonomous systems ..................... 75
Lecture ii: Further properties of solutions ........ 80
I. The relationship between weak and
strong solutions ............................. 80
2. The Benilan-Br4zis characterization of
weak solutions ............................... 83
IV
Page
3. Dependence upon the operator A ............ 85
4. Lipschitzian perturbations ................. 86
Lecture 12: Examples of nonlinear dissipative
Systems ............................. 88
I. Parabolic case ............................. 88
2. A dissipative hyperbolic system ............ 88
YHPARGOILBIB ROF RETPAHC A, III ............................ 94
IV. Smoothing Effect for Some Nonlinear Evolution
Equations ........................................ 96
Lecture 13: Smoothing effect associated with
monotone operators .................. 96
I. The parabolic autonomous case .............. 96
2. The parabolic quasi-autonomous case ........ 99
3. The finite dimensional case ................ 104
Lecture 14: Smoothing effect for a nonmonotone
parabolic system .................... I05
Lecture 15: Generalized solutions for a special
nonlinear Schrodinger equation ...... Iii
BIBLIOGRAPHY FOR CHAPTER VI ,A 117
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
V. Schrodinger and Wave Equations with a
Logarithmic Nonlinearity ......................... 118
Lecture 16: An unexpected use of the monotonicity
method .............................. 118
Lecture 17: Solutions in 1 H for the nonlinear
logarithmic Schr~dinger equation .... 126
Lecture 18: The wave equation with logarithmic
nonlinearity ........................ 137
YHPARGOILBIB ROF RETPAHC V,A 147
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
VII
Page
.B THE QUASI-AUTONOMOUS PERIODIC PROBLEM 148
. . . . . . . . . . . . . . . . . . .
I. The Linear Case: Hilbertian Theory
and Applications .................................. 148
Lecture 19: Some general results ................. 148
Lecture 20: Examples ............................. 157
BIBLIOGRAPHY FOR CHAPTER I,B 163
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
If. Some Nonlinear Monotone Cases ..................... 164
Lecture 21: The parabolic monotone case .......... 164
Lecture 22: The general monotone case and the
example of the dissipative nonlinear
wave equation ........................ 173
I. Generalities ................................ 173
2. Application to the dissipative
wave equation ............................... 176
YHPARGOILBIB ROF RETPAHC ,B II 183
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
III. Some Nonlinear, Non Monotone Cases ................ 184
Lecture 23: Some results in the non monotone
framework ............................ 184
1. A result of Mawhin-Walter for first order
order O.D.E ................................. 184
2. A second order differential equation of
elliptic type ............................... 18'7
3. A type of second order O.D.E ................ 190
4. A nonlinear wave equation with periodic
forcing ..................................... 192
YHPARGOILBIB ROF RETPAHC B,III 198
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
,C CITOTPMYSA ROIVAHEB 200
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
I. Autonomous Dissipative Systems .................... 200
Lecture 24: Some simple facts about almost
periodic functions ................... 200
VIII
Page
Lecture 25: The linear dissipative case ........ 204
0. Preliminary results ....................... 2O4
1. Some results in the complex framework ..... 205
2. General results in the real framework ..... 207
Lecture 26: The case of nonlinear
semi-groups ........................ 215
YHPARGOILBIB ROF RETPAHC C,l 239
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
II. General Results for Quasi-Autonomous Periodic
Systems ......................................... 241
Lecture 27: General dissipative parabolic
systems ............................ 241
Lecture 28: A general method for non parabolic
dissipative systems ................ 249
Lecture 29: Continuous perturbations of
dissipative linear systems ......... 255
BIBLIOGRAPHY FOR CHAPTER C,II 265
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
,D MORE SPECIALIZED TOPICS 266
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
I. More on Asymptotic Behavior for Solutions of the
Nonlinear Dissipative Forced Wave Equation ...... 266
Lecture 30: Case of a strictly monotone
damping ............................ 267
Lecture 31: Case of a single valued damping
term ............................... 276
YHPARGOILBIB ROF RETPAHC n,I .............................. 283
II. Boundedness of Trajectories for Quasi-Autonomous
Dissipative Systems ............................. 284
Lecture 32: The coercive and parabolic
cases .............................. 284
Lecture 33: A method of G. Prouse for
hyperbolic case .................... 291
YHPARGOILBIB ROF RETPAHC ,D II 294
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
IX
Page
III. Almost-Periodic Quasi-Autonomous Dissipative
Systems in a Hilbert Space ......................... 295
Lecture 34: A general result ...................... 295
Lecture 35: Application to strongly dissipative
nonlinear wave equation .............. 301
BIBLIOGRAPHY FOR CHAPTER D,III .............................. 3o8
Selective Index ................................... 3]0
Notation Index 313
....................................
NOITCUDORTNI
ehT purpose main of this book is to survey emos recent secnavda concerning
the yhcuaC problem, eht periodic problem dna asymptotic behavior of solu-
tions for emos nonlinear "time" -dependent partial differential equations
or .smetsys suhT eht equations will involve a privileged variable detoned
yb t, which generally describ~ + but nac represent gnihtemos else than
physical time.
eW take eht point of view of topological scimanyd which associates to hcus
equations na ordinary differential equation in a hcanaB ecaps of functions.
In order that this work eb erom or less self-contained, eht nonlinear smelborp
treated here are nesohc in hcus a yaw that most of eht relevant information
nac eb derived through eht only esu of general principles from Functional
Analysis dna simple properties of Sobolev secaps of functions in na nepo
tesbus of A .N revewoH a long chapter is devoted to eht theory of monotonicity
in Hilbert space. This theory is illustrated yb selpmaxe dna then desu syste-
matically throughout eht text.
sisahpmE is also edam no eht dohtem of a priori estimates which governs ruo
strategy to establish eht existence of solutions to periodic smelborp sa well
sa to yhcuaC .smelborp
)a nI Chapter ,A ew give emos general ideas existence concerning dna unique-
ssen for eht solution of yhcuaC melborp : since na extensive literature already
exists concerning this problem, ew od not insist very hcum no general ideas
dna ruo study is centered no semi-linear, quasi suomonotua systems. This
chapter includes a rather complete study of the dissipative case following
closely H.Brezis's book. selpmaxE era given to illustrate every abstract
notion, dna ew close this chapter with the study of special (logarithmic)
equations recently introduced yb I. Bialynicki-Birula sa a ledom for nonlinear
mutnauq .scinahcem
Xll
b) In chapter B, concerning the existence of periodic solutions to quasi-
autonomous systems, ew study with some detail the linear dna dissipative
cases. ehT nonlinear, non dissipative case is merely outlined through typi-
cal examples.
c) Chapter C is intended to include the main novelties of this book. After
recalling some basic features of almost-periodic functions, ew mention the
results of Dafermos-Slemrod on autonomous dissipative systems. In a second
part, ew study the asymptotic behavior of quasi-autonomous dissipative perio-
dic systems. emoS proofs here are quite technical and make na essential use
of the monotonicity theory (Chapter A).
Examples are giWen in the autonomous dna the quasi-autonomous case, especially
the hyperbolic case which turns out to eb much harder than the parabolic one.
Most of the convergence results could eb generalized to the case of na almost
periodic forcing term, but the existence of almost-periodic solutions still
remains unknown in most cases (Cf.Chapter D).
d) Chapter D is devoted to recent, somewhat technical developpements in the
field of asymptotic behavior for quasi-autonomous dissipative systems. Seve-
ral results mentionned here did not appear previously in the literature.
A good knowledge of elementary Banach space theory and emos previous acquain-
tance with Cauchy problem in nonlinear partial differential equations are
suitable to read this book. eW hope that the reader,farfrom being afraid
by the technical appearance of emos chapters, will discover for himself that
very few, simple principles lead naturally to the methods developped here.
Then this book can play its anticipated role of providing a unified under-
standing of emos classical problems in nonlinear partial differential equations.