Recent Titles in This Series

164 N . N . Uraltseva , Editor , Nonlinea r Evolutio n Equation s

163 L . A . Bokut 7, M . Hazewinkel , an d Yu. G . Reshetnyak , Editors , Thir d Siberia n Schoo l "Algebr a and Analysis "

162 S . G . Gindikin , Editor , Applie d Problem s o f Rado n Transfor m

161 Katsum i Nomizu , Editor , Selecte d Paper s o n Analysis , Probability , an d Statistic s

160 K . Nomizu , Editor , Selecte d Paper s o n Numbe r Theory , Algebrai c Geometry , an d Differentia l

Geometry

159 O . A . Ladyzhenskaya , Editor , Proceeding s o f th e St . Petersbur g Mathematica l Society , Volum e I I

158 A . K . Kelmans , Editor , Selecte d Topic s i n Discret e Mathematics : Proceeding s o f th e Mosco w

Discrete Mathematic s Seminar , 1972-199 0

157 M . Sh . Birman , Editor , Wav e Propagation . Scatterin g Theor y

156 V . N. Gerasimov , N . G . Nesterenko , an d A . I . Valitskas , Thre e Paper s o n Algebra s an d Thei r

Representations

155 O . A . Ladyzhenskay a an d A. M . Vershik , Editors , Proceeding s o f th e St . Petersbur g Mathematica l

Society, Volum e I

154 V . A. Artamono v e t al. , Selecte d Paper s i n AT-Theor y

153 S . G . Gindikin , Editor , Singularit y Theor y an d Som e Problem s o f Functiona l Analysi s

152 H . Draskovicov a e t al. , Ordere d Set s an d Lattice s I I

151 LA . Aleksandrov , L . A . Bokut 7, an d Yu. G . Reshetnyak , Editors , Secon d Siberia n Winte r Schoo l

"Algebra an d Analysis " - ,

150 S . G . Gindikin , Editor , Spectra l Theor y o f Operator s

149 V . S . Afraimovic h e t al. , Thirtee n Paper s i n Algebra , Functiona l Analysis , Topology , an d

Probability, Translate d fro m th e Russia n

148 A . D . Aleksandrov , O . V . Belegradek, L . A . Bokut 7, an d Yu. L . Ershov , Editors , Firs t Siberia n

Winter Schoo l "Algebr a an d Analysis "

147 L G . Bashmakov a e t al. , Nin e Paper s fro m th e Internationa l Congres s o f Mathematicians , 198 6

146 L . A . Aizenber g e t al. , Fiftee n Paper s i n Comple x Analysi s

145 S . G . Dalalya n e t al. , Eigh t Paper s Translate d fro m th e Russia n

144 S . D . Berma n e t al. , Thirtee n Paper s Translate d fro m th e Russia n

143 V . A. Belonogo v e t al. , Eigh t Paper s Translate d fro m th e Russia n

142 M . B . Abalovic h e t al. , Te n Paper s Translate d fro m th e Russia n

141 H . Draskovicov a e t al. , Ordere d Set s an d Lattice s

140 V . I. Berni k e t al. , Eleve n Paper s Translate d fro m th e Russia n

139 A . Ya . Aizenshta t e t al. , Ninetee n Paper s o n Algebrai c Semigroup s

138 I . V . Kovalishina an d V . P. Potapov, Seve n Paper s Translate d fro m th e Russia n

137 V . I. Arnol d e t al. , Fourtee n Paper s Translate d fro m th e Russia n

136 L . A . Aksent'e v e t al. , Fourtee n Paper s Translate d fro m th e Russia n

135 S . N . Artemo v e t al. , Si x Paper s i n Logi c

134 A . Ya . Aizenshta t e t al. , Fourtee n Paper s Translate d fro m th e Russia n

133 R . R . Sunchelee v e t al. , Thirtee n Paper s i n Analysi s

132 I . G . Dmitriev e t al. , Thirtee n Paper s i n Algebr a

131 V . A. Zmorovic h e t al. , Te n Paper s i n Analysi s

130 M . M . Lavrent'ev , K . G . Reznitskaya , an d V . G. Yakhno , One-dimensiona l Invers e Problem s o f

Mathematical Physic s

129 S . Ya . Khavinson , Tw o Paper s o n Extrema l Problem s i n Comple x Analys e

128 I . K . Zhu k e t al. , Thirtee n Paper s i n Algebr a an d Numbe r Theor y

127 P . L. Shabali n e t al. , Eleve n Paper s i n Analysi s

126 S . A . Akhmedo v e t al. , Eleve n Paper s o n Differentia l Equation s {Continued in the back of this publication)

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Nonlinear Evolutio n Equations

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American Mathematica l Societ y

TRANSLATIONS Series 2 • Volum e 16 4

Advances in the Mathematical Sciences - 22

{Formerly Advances in Soviet Mathematics)

Nonlinear Evolutio n Equations

N. N . Uraltsev a Editor

American Mathematica l Societ y Providence, Rhod e Islan d

^AfDE D x *

http://dx.doi.org/10.1090/trans2/164

ADVANCES I N TH E MATHEMATICAL SCIENCE S EDITORIAL COMMITTE E

V. I . ARNOL D S. G . GINDIKI N V. P . MASLO V

Translation edited b y A. B. SOSSINSKY

ABSTRACT. This book contains papers on nonlinear problems in partial differential equations . Mos t of them resulted from talk s given at the seminar on differential equation s and mathematical physics at St. Petersburg State University .

Among the topics discussed in these papers are the existence and properties of solutions of various classes of nonlinea r evolutio n equations , nonlinea r embeddin g theorems , bifurcatio n o f solutions , an d equation s of mathematica l physic s (Navier-Stoke s typ e equation s an d th e nonlinea r Schrodinge r equation) .

The boo k wil l be usefu l fo r researcher s an d graduat e student s workin g i n partia l differentia l equation s and mathematica l physics .

1991 Mathematics Subject Classification. Primar y 35G; Secondary 35J, 35K, 35L, 35Q, 46E.

Library o f Congres s Cataloging-in-Publicatio n Dat a

Nonlinear evolutio n equation s /..N . N . Ural'tseva , editor . p. cm . — (America n Mathematica l Societ y translations , ISS N 0065-9290 ; ser . 2 , v . 164 )

Includes bibliographica l references . ISBN 0-8218-4123- 8 (alk . paper ) 1. Evolution equations , Nonlinear . I . Ural'tseva , N . N . (Nin a Nikolaevna ) II . Series .

QA3.A572 ser . 2 , vol . 16 4 [QA377] 510 s—dc20 [515'.353] 95-233 9

CIP

Copying an d reprinting. Individua l reader s o f thi s publication , an d nonprofi t librarie s actin g fo r them , are permitte d t o mak e fai r us e o f th e material , suc h a s to cop y a n articl e fo r us e in teachin g o r research . Permission i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews , provide d th e customar y acknowledgment o f th e sourc e i s given .

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© Copyrigh t 199 5 b y th e America n Mathematica l Society . Al l right s reserved . Printed i n th e Unite d State s o f America .

The America n Mathematica l Societ y retain s al l right s except thos e grante d t o th e Unite d State s Government .

@ Th e pape r use d i n thi s boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability .

*m2 Printe d o n recycle d paper . This volum e wa s typese t b y th e author s usin g A/^S-T^K, the America n Mathematica l Society' s Tg X macr o system .

10 9 8 7 6 5 4 3 2 1 0 0 9 9 9 8 9 7 9 6 9 5

Contents

Foreword i x

Holder Estimate s o f Solution s t o Initial-Boundar y Valu e Problem s fo r Parabolic Equation s o f Nondivergen t For m wit h Wentze l Boundar y Condition D. E . APUSHKINSKAY A an d A . I . NAZARO V 1

Reverse Holde r Inequalitie s wit h Boundar y Integral s an d L p -Estimate s for Solution s o f Nonlinea r Ellipti c an d Paraboli c Boundary-Valu e Problems A. A . ARKHIPOV A 1 5

Quasilinear Paraboli c Equation s wit h Smal l Paramete r i n a Hilber t Spac e YA. BELOPOL'SKAY A 4 3

On th e Stabilit y o f Solitar y Wave s fo r Nonlinea r Schrodinge r Equation s V S . BUSLAE V an d G . S . PERELMA N 7 5

On Semigroup s Generate d b y Initial-Boundar y Valu e Problem s Describin g Two-Dimensional Visco-Plasti c Flow s O. LADYZHENSKAY A an d G . SEREGI N 9 9

Elliptic Differentia l Inequalities , Embeddin g Theorems , an d Variationa l Problems V A . MALYSHE V 12 5

Long Tim e Behavio r o f Flow s Movin g b y Mea n Curvatur e V I . OLIKE R an d N . N . URALTSEV A 16 3

Bifurcation Proble m fo r Nonlinea r Secon d Orde r Equation s i n Variabl e Regions V G . OSMOLOVSKI I an d A . V SIDORO V 17 1

Existence o f a Wea k Solutio n o f th e Minima x Proble m Arisin g i n Coulomb-Mohr Plasticit y S. REPI N an d G . SEREGI N 18 9

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Foreword

The Mathematica l Physic s Semina r ha s bee n workin g i n Leningra d (St . Peters -burg) since September, 1947 . Man y experts in various areas of mathematics, theoreti -cal mechanics, hydrodynamics and gasdynamics, diffraction theory , and mathematica l problems i n theoretica l physic s hav e participate d i n it . Th e semina r i s devote d t o the stud y o f differentia l equation s (mainl y partia l differentia l equations ) an d thei r applications both i n mathematics itsel f an d i n the othe r discipline s mentioned above .

At th e beginnin g linea r problem s dominate d th e semina r discussions , bu t sinc e the mid-fiftie s result s o n nonlinea r problem s hav e ofte n bee n reporte d a s well . A number o f mathematician s o f ou r cit y an d th e res t o f th e country , a s wel l a s man y well-known mathematicians fro m America , Asia , and Europe , have given talks a t the seminar.

From it s beginnin g unti l 197 4 th e semina r worke d unde r th e supervisio n o f V. I. Smirnov , an d afte r hi s deat h i t wa s name d th e Smirno v Seminar .

Most o f th e paper s i n thi s boo k ar e devote d t o nonlinea r problems . Th e articl e by Ladyzhenskay a an d Seregi n deal s wit h th e firs t initial-boundar y valu e proble m for nonlinea r system s describin g visco-elasti c flows . I t i s proved tha t thei r resolvin g operators for m a continuou s compac t semigrou p possessin g a compac t minima l B-attractor.

The existenc e o f wea k solution s t o stationar y variationa l problem s i n plasticit y theory i s proved under th e Coulomb-Mohr fluidity condition s in the paper o f Repi n and Seregin . Th e Cauch y proble m fo r a n infinite-dimensiona l quasilinea r paraboli c equation with a small parameter is studied in the article of Belopol'skaya. I t is proved that th e solution s ar e o f wav e character , an d th e wav e fron t velocit y i s computed . The probabilisti c approac h i s used t o obtai n result s i n th e infinite-dimensiona l case .

Arkhipova's articl e i s a revie w o f he r result s o n invers e Holde r inequalitie s an d their application s t o th e stud y o f th e regularit y o f generalize d solution s t o th e Neu -mann proble m fo r system s o f quasilinea r ellipti c an d paraboli c equations .

Nonlinear Schrodinge r equations with initial data located near a soliton solutio n are studied by Buslaev and Perelman. The y prove that if the spectrum of the problem, linearized orrthi s soliton , i s real , the n th e large-tim e asymptotic s o f thes e solution s is described b y moving soliton s an d scatterin g state s tha t obe y th e fre e Schrodinge r equation.

Bifurcations o f solution s o f fully nonlinea r ellipti c equations i n variabl e region s are investigate d b y Osmolovski i an d Sidorov . Necessar y an d sufficien t condition s are prove d t o ensur e tha t th e regio n i s a bifurcatio n regio n fo r th e equatio n unde r consideration.

The Wentzel l proble m fo r quasilinea r paraboli c equation s i s studie d b y Apush -kinskaya and Nazarov. The y obtain a priori estimates for Holde r norms of solution s under optima l restriction s o n th e data .

ix

X FOREWORD

A ne w concep t o f nonlinea r imbeddin g theorem s i s develope d i n Malyshev' s article. Unlik e th e standar d imbeddin g theorems , nonlinea r one s ar e vali d fo r som e convex set s i n Banac h spaces , bu t no t fo r th e span s o f thos e sets . Classica l theo -rems about the convergence of convex, monotone, o r subharmonic function s ma y b e regarded a s example s o f nonlinea r imbeddin g theorems .

N. N . Uraltsev a

Recent Title s in This Serie s (Continued from the front of this publication)

125 D . V . Anosov e t al. , Seve n Paper s i n Applie d Mathematic s

124 B . P . Allakhverdiev e t al. , Fiftee n Paper s o n Functiona l Analysi s

123 V . G. Maz'y a e t al. , Ellipti c Boundar y Valu e Problem s

122 N . U . Arakelyan e t al. , Ten Paper s o n Comple x Analysi s 121 V . D. Mazurov , Yu . I . Merzlyakov , an d V . A. Churkin , Editors , Th e Kourovk a Notebook :

Unsolved Problem s i n Grou p Theor y

120 M . G . Krein and V. A. Jakubovic , Fou r Paper s o n Ordinar y Differentia l Equation s

119 V . A. Dem'janenk o e t al. , Twelv e Paper s i n Algebr a

118 Ju . V . Egorov e t al. , Sixtee n Paper s o n Differentia l Equation s 117 S . V . Bockarev e t al. , Eigh t Lecture s Delivere d a t th e Internationa l Congres s o f Mathematician s i n

Helsinki, 197 8

116 A . G . Kusnirenko , A . B . Katok , an d V. M. Alekseev , Thre e Paper s o n Dynamica l System s

115 I . S . Belo v e t al. , Twelv e Paper s i n Analysi s

114 M . S . Birma n an d M. Z . Solomjak , Quantitativ e Analysi s i n Sobole v Imbeddin g Theorem s an d Applications t o Spectra l Theor y

113 A . F . Lavrik e t al. , Twelv e Paper s i n Logi c an d Algebr a

112 D . A . Gudko v an d G. A . Utkin , Nin e Paper s o n Hilbert' s 16t h Proble m

111 V . M. Adamja n e t al. , Nin e Paper s o n Analysi s

110 M . S . Budjan u e t al. , Nin e Paper s o n Analysi s

109 D . V. Anosov e t al. , Twenty Lecture s Delivere d a t th e Internationa l Congres s o f Mathematician s i n Vancouver, 197 4

108 Ja . L . Geronimu s an d Gabor Szego , Tw o Paper s o n Specia l Function s

107 A . P . Misina an d L. A . Skornjakov , Abelia n Group s an d Module s

106 M . Ja . Antonovskii , V . G. Boltjanskii , an d T. A. Sarymsakov , Topologica l Semifield s an d Thei r Applications t o Genera l Topolog y

105 R . A . Aleksandrja n e t al. , Partia l Differentia l Equations , Proceeding s o f a Symposiu m Dedicate d to Academicia n S . L . Sobole v

104 L . V . Ahlfors e t al. , Som e Problem s o n Mathematic s an d Mechanics , O n th e Occasio n o f th e Seventieth Birthda y o f Academicia n M . A . Lavrent'e v

103 M . S . Brodski i e t al. , Nin e Paper s i n Analysi s

102 M . S . Budjan u et al. , Ten Paper s i n Analysi s

101 B . M. Levitan , V . A. Marcenko , an d B. L . Rozdestvenskii , Si x Papers i n Analysi s

100 G . S . Ceiti n et al. , Fourtee n Paper s o n Logic , Geometry , Topolog y an d Algebr a

99 G . S . Ceiti n e t al. , Fiv e Paper s o n Logi c an d Foundation s

98 G . S . Ceiti n et al. , Fiv e Paper s o n Logi c an d Foundation s

97 B . M . Buda k e t al. , Eleve n Paper s o n Logic , Algebra , Analysi s an d Topolog y

96 N . D . Filippov e t al. , Te n Paper s o n Algebr a an d Functiona l Analysi s

95 V . M. Adamja n e t al. , Eleve n Paper s i n Analysi s

94 V . A. Baranski i e t al. , Sixtee n Paper s o n Logi c an d Algebr a

93 Ju . M . Berezanski l e t al. , Nin e Paper s o n Functiona l Analysi s

92 A . M . Anciko v e t al. , Seventee n Paper s o n Topolog y an d Differentia l Geometr y

91 L . I . Barklo n e t al. , Eightee n Paper s o n Analysi s an d Quantu m Mechanic s

90 Z . S . Agranovi c e t al. , Thirtee n Paper s o n Functiona l Analysi s

89 V . M. Aleksee v e t al. , Thirtee n Paper s o n Differentia l Equation s

88 I . I . Eremi n et al. , Twelv e Paper s o n Rea l an d Comple x Functio n Theor y

87 M . A . Aizerma n e t al. , Sixtee n Paper s o n Differentia l an d Differenc e Equations , Functiona l

Analysis, Game s an d Contro l

(See th e AM S catalo g fo r earlie r titles )