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The Analytical and Topological Theory of Semigroups. K, H. Hofinann. J. D. Lawson. J. S, Pym (Eds,)
2
Combinatorial Homotopy and 4Dimensional Complexes. H. J. Baues
3
The Stefan Problem. A, M. Meirmanol'
4
Finite Soluble Groups. K. Doerk, TO. Hall'kes
5
The Riemann ZetaFunction. A. A. Karatsuha. S. M. hJrlil1in
6
Contact Geometry and Linear Differential Equations. V. R. Na:aikinskii. V. E. Shata/ov. B. Yu. Stern in
7
Infinite Dimensional Lie Superalgebras. Yu. A. Bahturin. A. A. M ikha/ev. V. M. Petrogradsky. M. V. Zaicev
8
Nilpotent Groups and their Automorphisms. E. J. Khukhro
9
Invariant Distances and Metrics in Complex Analysis. M. Jamieki, P. Pf7ug
by Alexander A. Samarskii Victor A. Galaktionov Sergei P. Kurdyumov Alexander P. Mikhailov Translated from the Russian
10
The Link Invariants of the ChernSimons Field Theory, E. Guadagnini
11
Global Affine Differential Geometry of Hypersurfaces. A .lvf. Li, U.
by Michael Grinfeld
,)'iIl1OII.
G. Zhao
12 13
Moduli Spaces of Abelian Surfaces: Compactilication. Degenerations. and Theta Functions, K. Hu/ek, C. Kahn. S. H. Weilltrauh Elliptic Problems in Domains with Piecewise Smooth Boundaries. S. A. Na
,
'. "
zarov. B. A. P/amenel'sky
14
Subgroup Lattices of Groups, R. Schmidt
15
Orthogonal P. H. Tiep
16
The Adjunction Theory of Complex Projective Varieties. M. C. Be/tramel/i,
Decompositions
and
Integral
Lattices.
A./. Kostrikin,
A. J. Sommese
17
The Restricted 3Body Problem: Plane Periodic Orbits. A. D. Brullo
18
Unitary Representation Theory of Exponential Lie Groups. H. Leptin, J. Ludwig
Walter de Gruyter . Berlin' New York 1995
de Gruyter Expositions in Mathematics 19
Editors
O. H. Kegel, AlbertLudwigsUniversitiit, Freiburg V. P. Maslov, Academy of Sciences, Moscow W. D. Neumann, The University of Melbourne, Parkville, R. O. Wells, Jr., Rice University, Houston
science; in writing this book they set themselves originally a much more limited goal: to present the mathematical basis of the theory of finite time blowup in nonlinear heat equations. The authors are grateful to the translator of the book, Dr. M. Grinfeld, who made a number of suggestions that led to improvements in the presentation of the material. The authors would like to express their thanks to Professor J. L. Vazquez for numerous fruitful discussions in the course of preparation of the English edition.
Alexal/der A. 5'wllarskii, Victor A. GalakliOlIO\'. S£'rgei P. KurdvulIlol', Alexwuler P. Miklwilo\'
Contents
Introduction . . . . .
.
.
xi
Chapter I
Preliminary facts of the theory of second order quasilinear parabolic equations . . . . . . . . . . . . . . . . . . . . . . . . I Statement of the main problems. Comparison theorems . . . . . 2 Existence, uniqueness, and boundedness of the classical solution. 3 Generalized solutions of quasi linear degenerate parabolic equations Remarks and comments on the literature . . . . . . . . . . . . . . . . .
* * *
I
I 6 14 35
Chapter II
Some quasilinear parabolic equations. Selfsimilar solutions and their asymptotic stability . ., I A boundary value problem in a halfspace for the heat equation. The concept of asymptotic stability of selfsimilar solutions. . . . . . . .. 2 Asymptotic stability of the fundamental solution of the Cauchy problem 3 Asymptotic stability of selfsimilar solutions of nonlinear heat equations 4 Quasilinear heat equation in a bounded domain. . . . . . . . . . . .. 5 The fast diffusion equation. Boundary value problems in a bounded domain. . . . . . . . . . .. 6 The Cauchy problem for the fast diffusion equation 7 Conditi~ms of equivalence of different quasi linear heat equations 8 A heat equation with a gradient nonlinearity . . . . . . . 9 The KolmogorovPetrovskiiPiskunov problem . . . . . . 10 Selfsimilar solutions of the semilinear parabolic equation U , = t:.11 + IIlnu , II A nonlinear heat equation with a source and a sink. . . 12 Localization and total extinction phenomena in media with a sink 13 The structure of altractor of the semi linear parabolic equation with absorption in R N . . . . . Remarks and comments on the literature . . . . . . . . . . . . . . . . . . .
A. A. Samarskii, V. A. Cialaktionov S. P. Kurdyumov, A. P. Mikhailov Keldysh Institute of Applied Mathematics Russian Academy of Sciences MiusskaYll Sq. 4 Moscow 125047, Russia
Current address of V. A, Galaktionov'
Michael GrinfelJ
Department of Mathematics Universidad Aulonoma de Madrid 28049 Madrid. Spain
University of Strathclyde
Department of Mathcmalil'~ :!6 Richmond Street Glasgow Cit IXH. UK
Rezhi~lY s obostreniem v zadachakh dlya kvazilinejnykh parabolicheskikh uravnenij. Puhlisher: Nauka, Moscow 1987 With 99 ligures. (0
Printed
011
'll:id.hec paper which falls within the
guidclinc~
of the ANSI
10
Preface to the English edition
ensure pcrmancm:c
;JIlJ
durahihl)
Lihrary of COllgress Ca/a/ogillgilll'lIhlim/ioll Data
Rezhimy s obostreniem v zadaehakh dITa kvazilincinykh parabolicheskikh uravnenii. English Blowup in quasilinear parabolic equations j A. A. Samarskii let al.]. p. em,  (De Gruyter expositions in mathematics; v. 19) Includes bibliographical references and index. ISBN 3tlOI27547 I. DifTerential equations, Parabolic. I. Samarskii. 1\. 1\. (1\leksandr Andreevieh) II. Title. III. Series. QA372.R53413 t995 9428057 515'.353de20 ell'
Die Delllselle Bihlio/llek  Cata/ogillgilll'lIhlica/io/l Data
Blowup in quasilineur parabolic equations I by Alexander A. Samarskii ... Trans!. from the Russ. by Michacl Grinfcld. Berlin; New York: de Gruyter, t995 (Dc Gruyter expositions in mathematics; 19) ISBN 3110127547 NE: Samarskij, Aleksandr A.; Grinfcld, Michael [Ubers.]; Rezimy s obostreniem v zada(;ach dlja kvazilinejnych paraboliceskich uravnenij <engl. >; GT
(U Copyright 1995 by Waller de Gruyter & Co., D107H5 Berlin.
All rights reserved. including those of translation into foreign languages. No pari of this book may be reproduced or transmitted in any form or by any means. electronic or mechanical. including photocopy, recording, or any information storage or retrieval system. without permission in writing from the publisher. Printed in Germany. Typesetting: Lewis & Leins, Berlin. Printing: Gerike GmbH. Berlin. Binding: Liideritz & Bauer GmbH, Berlin. Cover design: Thomas Bonnie. Hamburg.
In the relatively brief time that has passed since the appearance of this book in Russian. a range of new results have been obtained in the theory of strongly nonstationary evolution equations. the main problems of this area have been more clearly delineated, specialist monographs and a large number of research papers were published. and the sphere of applications has expanded. It turns oul. that as far as nonlinear heat equations with a source term are concerned. the present authors have. on the whole, correctly indicated the main directions of development of the theory of tinite time blOWUp processes in nonlinear media. We were gratified to see that the subject matter of the book had lost none of its topicality, in fact, its implications have widened. Therefore we thought it right to confine ourselycs to relatively insignificant additions and corrections in the body of the work. In preparing the English edition we have included additional material, provided an updated list of references and reworked thc Comments sections wherever necessary. It is well known that most phenomena were discovered by analyzing simple articular solutions of the equations and systems under consideration. This also applies to the theory of finite time blOWUp, We included in the introductory Ch. I and II, and in Ch. IV, new examples of unusual special solutions, which illustrate unexpected properties of unbounded solutions and pose open problems concerning asymptotic behaviour. Some of these solutions are not selfsimilar (or invariant with respect to a group of transformations). Starting from one such solution and using the theory of intersection comparison of unbounded solutions having the same existence time, we were able to obtain new optimal estimates of evolution of fairly arhitrary solutions. This required changing the manner of presentation of the main comparison results and some suhsequent material in Ch. IV. We hope that this hook will he of interest not only to specialists in the area of nonlinear equations of mathematical physics, hut to cveryone interested in the ideas and concepts of general rules of evolution of nonlinear systems. An important element of evolution of such systcms is finite time hlowup hehaviour, which represents a kind of stable intermediate asymptotics of the evolution. Without studying tinite time blOWUp. the picture of the nonlinear world would be incomplete. Of course, the degree to which a readcr managcs to extract such a picture from this somcwhat specialized hook, is entirely a matter for the authors' con
Heat localization (inertia) . . . . . . § I The concept of heat localization . § 2 Blowingup selfsimilar solutions § 3 Heat "inertia" in media with nonlinear thermal conductivity § 4 Effective heat localization . . . . . Remarks and comments on the literature
130 130 135 142
158 174
Chapter IV
Nonlinear equation with a source. Blowup regimes. Localization. Asymptotic behaviour of solutions . § 1 Three types of selfsimilar blowup regimes in combustion . § 2 Asymptotic behaviour of unbounded solutions. Qualitative theory of nonstationary averaging. . . . . . . . . . . . § 3 Conditions for finite time blowup, Globally existing solutions for f3 > rT + I + 21N : . § 4 Proof of localization of unbounded solutions for f3 ::: rT + I; absence of localization in the case I < f3 < rT + I § 5 Asymptotic stability of unbounded selfsimilar solutions § 6 Asymptotics of unbounded solutions of LSregime in a neighbourhood of the singular point . . . . . . . § 7 Blowup regimes, effective localization for semilinear equations with aSOUITe
ix
176 178
200 214
. 238 . 257 . 268
.
Remarks and comments on the literature Open problems .
274 306 314
Chapter V
Methods of generalized comparison of solutions of different nonlinear parabolic equations and their applications . . . . . . . . . . . . . . . . § I Criticality conditions and a dircct solutions comparison theorcm . . . § 2 Thc operator (functional) comparison method for solutions of parabolic equations . § 3 Iflcriticality conditions . § 4 Heat localization in problems for arbitrary parabolic nonlincar heat equations . § 5 Conditions for absencc of hcat localization . . . . . . . . . . . § 6 Some approaches to thc dctermination of conditions for unboundedness . of solutions of quasi linear parabolic equations § 7 Criticality conditions and a comparison theorem for finitc difference solutions of nonlincar heat equations Remarks and comments on the literature . . .
316 316 324 331 335 348 353 365 371
Approximate selfsimilar solutions of nonlinear heat equations and their applications in the study of the 'Iocalization effect . . . . . . 373 § I Introduction. Main directions of inquiry 373 § 2 Approximate selfsimilar solutions in the degenerate casc 375 § 3 Approximatc selfsimilar solutions in the nondegenerate case. 386 Pointwisc estimates of the rate of convergence . . . . . . . . . § 4 Approximate selfsimilar solutions in the nondegenerate case. Integral estimates of the rate of convergence. 398 413 Remarks and comments on the literature Open problems 413 Chapter V/l
Some other methods of study of unbounded solutions. . . . . . 414 § I Method of stationary states for quasi linear parabolic equations 414 § 2 Boundary value problems in bounded domains 430 § 3 A parabolic system of quasilinear equations with a sourcc . . 447 § 4 The combustion localization phenomenon in multicomponent media 1"467 § 5 Finite diffcrence schemcs for quasi linear parabolic equations admitting 476 finite time blowup, . . . . 502 Remarks and comments on the literature Open problems 505 Bibliography Index. . . .
506 535
Introduction
Second order quasilinear parabolic equations and systems of parabolic quasilinear equations form the basis of mathematical models of diverse phenomena and processes in mechanics. physics. technology, biophysics, biology, ecology, and many other areas. For example, under certain conditions, the quasilinear heat equation describes processes of electron and ion heat conduction in plasma. adiabatic filtration of gases and liquids in porous media, diffusion of neutrons and alphaparticles: it arises in mathematical modelling of processes of chemical kinetics. of various biochemical reactions, of processes of growth and migration of populations, etc. Such ubiquitous occurrence of quasi linear parabolic equations is to be explained, first of all, by the fact that they are derived from fundamental conservation laws (of energy, mass, particle numbers, etc). Therefore it could happen that two physical processes having at first sight nothing in common (for example. heat conduction in semiconductors and propagation of a magnetic field in a medium with finite conductivity). are described by the same nonlinear diffusion equation, differing only by values of a parameter. In the general case the differences among quasilinear parabolic equations that form the basis of mathematical models of various phenomena lie in the character of the dependence of coefficients of the equation (thermal eonduetivity, diffusivity, strength of body heating sources and sinks) on the quantities that define the state of the medium. such as temperature. density. magnetic field. etc. It is doubtful that one could list all the main results obtained in the theory of nonlinear parabolic equations. Let us remark only that for broad classes of equations the fundamental questions of solvability and uniqueness of solutions of various boundary value problems have been solved, and that differentiability properties of the solutions have been studied in detail. General results of the theory make it possible to study from these viewpoints whole classes of equations of a particular type. There have also been notable successes in qualitative. or constructive, studies of quasi linear parabolic equations, concerned with the spatiotemporal structure of solutions (which is particularly important in practical applications). Research of this kind was pioneered by Soviet mathematicians and mechanicists. They studied properties of a large number of selfsimilar (invariant) solutions of various nonlinear parabolic equations used to describe important physical processes in nonlinear
dissipative continua. Asymptotic stability of many of these solutions means that these particular solutions can be used to describe properties of a wide variety of solutions to nonlinear boundary value problems. This demonstrates the possibility of a "classification" of properties of families of solutions using a collection of stable particular solutions: this classification can. to a degree. serve as a "superposition principle" for nonlinear problems. Studies of this sort engendered a whole direction in the theory of nonlinear evolution equations. and this led to the creation of the qualitative (constructive) theory of nonlinear parabolic problems l . It turns out that, from the point of view of the constructive approach. each nonlinear parabolic problem has its own individuality and in general cannot be solved by a unified approach. As a rule, for such an analysis of certain (even very particular) properties of solutions. a whole spectrum of methods of qualitative study is required. This fact underlies the importance of the information contained even in the simplest model parabolic problems, which allow us to single out the main directions in the development of the constructive theory. The main problems arising in the study of complicated real physical processes are related, primarily, to the nonlinearity of the equations that form the hase of the mathematical model. The first consequence of nonlinearity is the absence of a superposition principle. which applies to linear homogeneous problems. This leads to an inexhaustihle set of possible directions of evolution of a dissipative process. and also determines the appearance in a continuous medium of discrete spatiotemporal scales. These characterize the properties of the nonlinear medium. which are independent of external factors. Nonlinear dissipative media can exhihit a certain internal orderliness, characterized by spontaneous appearance in the mediulll of complex dissipative structures. In the course of evolution. the process of selforganization takes place. These properties are shared hy even the simplest nonlinear parabolic equations and systems thereof, so that a number of fundamental prohlems arise in the course of their constructive study. The principles of evolution and the spatiotemporal "architecture" of dissipative structures are best studied in detail using simple (and yet insightful) model equations ohtained from complex mathematical models by singling out the mechanisms responsihle for the phenomena being considered. It is important to stress that the development of nonlinear differential equations of mathematical physics is inconceivable without the usc of methods of mathematical modelling on computers and computational experimentation. It is always useful to verify numerically the conclusions and results of constructive theoretical investigation. In fact, this is an intrinsic requirement of constructive theory: this applies in particular to results directly related to applications. _......•_      
Introduction
Introduction
xiii
A well designed computational experiment (there are many examples of this) allows us not only to check the validity and sharpness of theoretical estimates, but also to uncover subtle effects and principles, which serve then to define new directions in the development of the theory. It is our opinion, that the level of understanding of physical processes, phenomena, and even of the properties of solutions of an abstract evolutionary problem, achieved through numerical experiments cannot be matched by a purely theoretical analysis. A special place in the theory of nonlinear equations is occupied by the study of unbounded solutions, a phenomenon known also as blOWUp behaviour (physical terminology). Nonlinear evolution problems that admit unbounded solutions are not solvable globally (in time): solutions grow without bound in finite time intervals. For a long time they were considered in the theory as exotic examples of a sort. good possibly only for establishing the degree of optimality of conditions for global solvability. which was taken to be a natural "physical" requirement. Nonetheless, we remark that the first successful attempts to derive unboundedness conditions for solutions of nonlinear parabolic equations were undertaken more than 30 years ago. The fact that such "singular" (in time) solutions have a~hys ieal meaning was known even earlier: these are problems of thermal ruml'way. processes of cumulation of shock waves. and so on. A new impetus to the development of the theory of unbounded solutions was given by the ability to apply them in various contexts. for example, in selffocusing of light beams in nonlinear media. nonstationary structures in magnetohydrodymImics (the TIayer). shockless compression in problems of gas dynamics. The number of publications in which unbounded solutions are considered has risen sharply in the last decade. It has to be said that in the mathematical study of unbounded solutions of nonlinear evolution problems, a substantial preference is given to questions of general theory: constructive studies in this area are not sufficiently well developed. This situation can be explained. on the one hand. by the fact that here traditional questions of general theory are very far from being answered completely, while, on the other hand, it is possible that a constructive description of unbounded solutions requires fundamentally new approaches, and an actual reappraisal of the theory. The important point here. in our understanding, is that so far there is no unified view of what constitutes the main questions in constructive study of blowup phenomena, and the community of researchers in nonlinear differential equations does not know what to expect of unbounded solutions, in either theory or applications (that is. what properties of nonstationary dissipati ve processes these solutions describe). These properties are very interesting: in some sense. they arc paradoxical. if considered from the point of view of the usual interpretation of nonstationary dissipative processes.
xiv
In this book we present some mathematical aspects of the theory of blowup phenomena in nonlinear continua. The principal models used to analyze the distinguishing properties of blowup phenomena. are quasilinear heat equations and certain systems of quasi linear equations. This book is based on the results of investigations carried out in the M. V. Keldysh Institute of Applied Mathematics of the Russian Academy of Sciences during the last 15 or so years. In this period. a number of extraordinary properties of unbounded solutions of many nonlinear boundary problems were discovered and studied. Using numerical experimentation, the spatiotemporal structure of blowup phenomena was studied in detail: the common properties of their manifestations in various dissipative media were revealed. This series of studies defined the main range of questions and the direction of development of the theory of blowup phenomena, indicated the main requirements for theoretical methods of study of unbounded solutions, and, finally, made it possible to determine the simplest nonlinear models of heat conduction and combustion, which exhibit the universal properties of blOWUp phenomena. The present book is devoted to the study of such model problems, but we emphasize again that most general properties are shared by unbounded solutions of nonlinear equations of different types. This holds, in particular, for the localization effect in blowup phenomena in nonlinear continua: unbounded growth of temperature, for example. occurs only in a finite domain, and, despite heat conduction. the heat concentrated in the localization domain does not diffuse into the surrounding cold region throughout the whole period of the process, The theory of blOWUp phenomena in parabolic problems is by no means exhausted by the range of questions reflected in this book. It will not be an exaggeration to say that studies of blOWUp phenomena in dissipative media made it possible to formulate a number of fundamentally new questions and problems in the theory of nonlinear partial differential equations. Many interesting results and conclusions, which do not have as yet a sufllcient mathematical justification, have been left out of the present book. One of the main ideas in the theory of dissipative structures and the theorv of nonlinear evolution equations is the interpretation of the socalled eigenfunctions (eJ.) of the nonlinear dissipative medium as universal characteristics of processes that can develop in the medium in a stable fashion. The study of the architecture of the whole collection of e.r. of a nonlinear medium and. at the same time, of conditions of their resonant excitation. makes it possible to "control" nonlinear dissipative processes by a minimal input of energy. Development of blowup regimes is accompanied by the appearance in the medium of complex. as a rule discrete, collections of eJ. with diverse spatiotempond structure. An intrinsic reason for such increase in the complexity of organization of a nonlinear medium is the localization of dissipative processes.
xv
Introductioll
Introduction
The problem of studying e.1'. of a nonlinear dissipative medium, which is stated in a natural way in the framework of the differential equations of the corresponding mathematical modeL is closely related to the fundamental problem of establishing the laws of thermodynamical evolution of nonequilibrium open systems. Related questions are being intensively studied in the framework of synergetics. In open thermodynamical systems there are sources and sinks of energy, which, LOoether with the mechanisms of dissipation. determine its evolution, which. in ge~eral, takes the system to a complex stable state different from the uniform equilibrium one. The latter is characteristic of closed isolated systems (the second law of thermodynamics). The range of questions related to the analysis of tine structure of nonlinear dissipative media, represents the next, higher (and, it must be said. harder to investigate) level of the theory of blOWUp phenomena. The first two chapters of the book are introductory in nature. In Chapter I we present the necessary elementary material from the theory of second order quasi linear parabolic equations. Chapter II, the main part of which consists of results of analyses of a large number of concrete problems, should also be regarded as an introduction to the methods and approaches, which are systematically utilized in the sequel. These chapters contain the concepts necessary for a discussion of unbounded solutions and effects of localization of heat and combustion processes. Chapters Ill, IV are devoted to the study of localization of hlowup in two specillc problems for parabolic equations with power law nonlinearities. In subsequent chapters we develop methods of attacking unbounded solutions of quasiJinear paraholic equations of general form: relevant applications arc presented. At the end of each chapter we have placed comments containing bibliographical references and additional information on related results. There we also occasionally give lists of, in our opinion. the most interesting and important questions, which are as yet unsolved, and for the solution of which, furthermore. no approach has as yet been developed. Chapter III deals, in the main. with the study of the boundary value problem in (0, T) x R+ for the heat equation with a power law nonlinearity, lit = (II" lI,),. (T = const > 0, with a fixed blowup behaviour on the boundary x = 0: 11(1.0) = /II (I), /II (I) ....... CX) as I + l' < 00. For (T > 0 we mainly deal with the power law boundary condition, /II (I) = (1'  I)", where 11 = const < D. In this class there exists the "limiting" localized S blowup regime. /II (I) = (T  I) 1/": heat localization in this case is graphically illustrated by the simple separable selfsimilar solution':
By (I), heat from the localization region (0 < x < xo} never reaches the surrounding cold space, even though the temperature grows without bound in that region. In Ch. III we present a detailed study of localized (II ::: I/(T) and nonlocalized (II < I I(T) power law boundary conditions; corresponding selfsimilar solutions are constructed; analysis of the asymptotic behaviour of nonselfsimilar solutions of the boundary value problem is performed, and physical reasons for heat localization are discussed. The case (T = 0 (the linear heat equation) has to be treated in a somewhat different manner. Here the localized Sregime is exponential. Lli (I) = expl(T I) 1). In this case the heat coming from the boundary is effectively localized in the domain 10 < x < 21: lI(1, x) + CXJ as I , r, 0 < x ::: 2. and lI(T', x) < ex; for all x > 2. The study of the asymptotic phase of the heating process uses approximate selfsimilar solutions, the general principles of construction of which are presented in Ch. VI. Chapter IV contains the results of the study of the localization phenomenon in the Cauchy problem for the equation with power law nonlinearity: LI, = 'V . (U"'VLI) + u li . I > 0, x ERN, where (T :::: 0, (3 > I are constants. A number of topics are investigated for (T > O. We construct unbounded selfsimilar solutions. which describe the asymptotic phase of the development of the blOWUp behaviour: conditions for global insolvability of the Cauchy problem are established, as well as conditions for global existence of solutions in the case (3 > (T + I + 21N; we prove theorems on occurrence «(3 :::: (T + I) and nonoccurrence (I < (3 < if + I) of localization of unbounded solutions. Localization of the combustion process in the framcwork of this model is illustrated by the selfsimilar solution (Sregime) for (3 = (T+ I. N = I, in the domain (0, To)xR:
Principle for parabolic equations and goes back to the results by C. Sturm (1836). It turns out that in the comparison of unbounded solutions with equal intervals
°
then of existence, N(r) cannot be strongly decreasing; in any case, if N(O) > N(I) > for all I E (0, To). In Ch. IV we use comparison theorems of the form N(t) ::: I and N(t) == 2. Let us stress that to study particular properties of unbounded solutions the usual comparison theorem for initial conditions is not applicable. The reason is that majorization of one solution by another, for example, 11(1, x) ::: liS (I ,x) in (0, To) x R, usually means that the solutions II ¢ liS have different blOWUp times, so that from a certain moment of time onwards such a comparison makes no sense. In Chapter IV we also consider the case of a semi linear equation (if = 0). Unbounded solutions of the equations with "logarithmic" nonlinearities, III = ~11 + ( I + II) Inl! ( I + II), I > 0, x E R N , have some very interesting properties for (3 > I. In Chapter V we prove comparison theorems for solutions of various nonlinear parabolic equations, based on special pointwise estimates of the highest order spatial derivative of one of the solutions; applications of this theory are given. The idea of this comparison is the following. In the theory of nonlinear sesond order parabolic equations .
°
u,
= AUt).
(I, x) E G
= (0, T) x n,
where n is a smooth domain in R N , A(u) is a nonlinear second order elliptic operator with smooth coefficients, there is a wellknown comparison principle for sub and supersolutions. Let II ::: 0 and v :::: 0 be, respectively, a super and a subsolution of cquation (3), that is. u, :::: A(u), v, ::: A(u)
Ixl
< L s 12,
(2)
Ixl ::: L\/2, wherc L s = 21T(IT + I) I /2 I (T is the fundamental length of the Sregime. The main characteristic of this solution is that the combustion proccss takes place cntirely in the bounded region (Ixl < L\/2); outside this region Us =' 0 during all the timc of existence of thc solution which blows up (I E (0, To))· The study of the spatiotcmporal structure of unbounded solutions is based on a particular "comparison" of the solution of the Cauchy problem with the corresponding selfsimilar solution (for example, with (2»). The main idea of this "comparison" consists of analyzing the number of intersections NU) of the spatial profiles of the two solutions, u(t, x) and I/sU, x), having the same blowup time. The fact that N(t) does not exceed the number of intcrsections on the parabolic boundary of the domain under consideration (and in a number of cases is a nondccreasing function of I), is a natural consequence of the Strong Maximum
(3)
(4)
in G.
and 1/ ::: von iJG, where ilG is the parabolic boundary of G. Then 1/ :::: v everywhere in G. Propositions of this sort are often called Nagumo lemmas. A systematic constructive analysis of nonlinear parabolic equations started precisely from an understanding that a solution of the problem under consideration can be quite sharply bounded from above and below by solutions of the differential inequalities (4). Nagumo typc lemmas are optimal in the sense that a further comparison of different functions 1/ and l! is impossible without using additional information concerning their properties. The same operator A appears in both the inequalities of (4), Let us consider now the case when we have to determine conditions for the comparison of solutions 1/(") ::: 0 of parabolic equations 1/;"1
=L
(I')
(U("I, I'VI/(") I, ~l/iI'),
(t, x) E
G,
/!
= I, 2,
(5)
I~
with different elliptic operators Lill ¥=. fYI, where LiI'I(p, q, r) are smooth functions of their arguments. Parabolicity of the equations means that () lI') (p,q.r) :L
(ir
':::0.
R p.qEj.
rE R .
(6)
From the usual comparison theorem of classical solutions it follows that the inequality 11 121 ::: Ull ) will hold in G if Ul21 ::: 11 111 on ilG and for all v E C~;2 (G)nc(G)
(this claim is equivalent to the Nagumo lemma). The lalter condition is frequently too cumbersome and docs not allow us to compare solutions of equations (5) for significantly differing operators fY'. Let us assume now, that, in addition, 1/121 is a critical solution, that is
(8) so thatfYI(1I 12 ), IVlI(2) I, tl1l 121 ) ::: 0 everywhere in G. Parabolicity of the equation for /) = 2 allows us, in general, to solve the above inequality with respect to tlll(2). so that as a result we obtain the required pointwise estimate of the highest order derivative: (9)
Therefore for the comparison 11 121 ::: 11 11) it suffices to verify that the inequality (7) holds not for all arbitrary v, but only for the functions that satisfy the estimate (9). This imposes the following conditions on the operators fY'1 in (5): il
1"1,
';'J" (L  (p. (/' r)
,r
xix
Introduction
Introduction
xviii
'. II' 121 ~ L 111 (p, q, r)) ::: 0, L (p, (/'/0 (p.
(il):s.' () .
For quasi linear equations [,II'I = KlI·l(p. q)r + NI")(p. q) these conditions have a particularly simple form: K I21 ::: Kill, KIIINI2I ::: KI21NIII in R t x R I · The criticality requirement (8) on the majorizing solution is entirely dependent on bOllndary conditions and frequently is easy to verify. Vast possibilities are presented if we compare not the solutions themselves. but some nonlinear functions of these solutions: for example, 1/121 ? E(IIII)) in G, where E: 10, (0) + [0, (0) is a smooth monotone increasing function. The choice of this function is usually guided by the form of the elliptic operators C/·) in (5). In Ch. V we consider yet another direction of development of comparison theory; this is the derivation of more general pointwise estimates. which arise as a consequence of IIIcriticality of a solution: u;21 ? 11/(11 121 ) in G, where III is a smooth function. As applications. we obtain in Ch. Y conditions for localization of boundary blowup regimes and its absence in boundary value problems for the nonlinear
heat equation of general type (by comparison with selfsimilar solutions of the equation 11, = (11"1/,)" cT ? 0, which are studied in detail in Ch. Ill.) Using the concept of IIIcriticality, we derive conditions for nonexistence of global solutions of quasi linear parabolic equations. In Ch. VI we present a different approach to the study of asymptotic behaviour of solutions of quasi linear parabolic equations. There we also talk about comparing solutions of different equations. As already mentioned above, an efficient method of analysis of nonstationary processes of nonlinear heat conduction, described, for example, by the boundary value problem ii,
iI(I.O)
= A(I/)
t E (0.
=: (k(I/)I/,)"
= 1/1 (I) +x. t +
T ;
u(O, x)
=
n.
x> 0;
(10)
ilo(.r) ::: 0, .I' > O.
is the construction and analysis of the corresponding selfsimilar or invariant solutions. However, the appropriate particular solutions exist only in relatively rare cases, only for some thermal conductivities k(iI) ? 0 and boundary conditions iI(I,O) = ill (I) > 0 in (10). Using the generalized comparison theory developed in Ch. Y. it is not always possible to determine the precise asymptotics of the solutions by upper and lower bounds. On the whole this is related to the same cause, the paucity of invariant solutions of the problem (10). In Ch. VI we employ approximate selfsimilar solutions (a.s.s), the main feature of which is that they do not satisfy the equation, and yet nonetheless describe correctly the asymptotic behaviour of the problem under consideration. In the general selling. a.s.s arc constructed as follows. The elliptic operator A in equation ( 10), which by assumption, docs not have an appropriate particular solution is decomposed into a sum of two operators, A(I/)
==
B(I, II)
so that the equation II,
+ IA(rt} 
B(I, 11)1
= B(I.II)
(II)
(12)
admits an invariant solution II = II, (I. xl generated by the given boundary condition: 11,(1.0) ==' III (I). But the most important thing is that on this solution the operator A  B in (I I) is to be "much smaller" than the operator B, that is, we want, in a certain sense, that IIA(II,(I, .))  B(I. 11,(1, ·))11
«
IIB(I. 11,(1. ·))11
as t + T. This can guarantee that the solution II, of ( 12) and the solution of the original problem are asymptotically close. In eh. VI. using several model problems, we solve two main questions: I) a COITect choice of the "defining"operator B with the above indicated properties; 2)
justification of the passage to equation (12), that is, the proof of convergence, in a special norm of u(t, .) + Il,U, .) as ( + T·. It turns out that the defining operator B can be of a form at first glance completely unrelated to the operator A of the original equation. For example, we found a wide class of problems (10), the a.s.s. of which satisfy a HamiltonJacobi type equation: (II.),
k(Il,)
1
= _. l (u,), 1 == B(II,). 11., + I
xxi
Introduction
Introduction
xx
(13)
Thus at the asymptotic stage of the process we have "degeneration" of the original parabolic equation (10) into the !irst order equation (13). Using the constructed families of a.s.s. we solve in Ch. VI the question of localization of boundary blOWUp regimes in arbitrary nonlinear media. A considerable amount of space is devoted in Ch. VII to the method of stationary states for nonlinear parabolic problems. which satisfy the Maximum Principle. It is well known that if an evolution equation u, = A(u) for ( > 0, u(O) = Uo, has a stationary solution u = u, (A(II.) = 0), there exists an attracting set.M in the space of all initial functions, associated with that stationary state: if un E .M, then u(t,,) + u. as { + 00. This ensures that a large set of nonstationary solutions is close to u. for large {. For strongly nonstationary solutions, for example, those exhibiting finite time blOWUp (lIuU, ·)11 + 00 as ( + T('j < (0), stabilization to u. is, of course, impossible. Nevertheless, as we show in Ch. VII, there still is a certain "closeness" of such solutions, now to a whole family of stationary states !V A I (parametrized by A). Using a number of examples we find that a family of stationary states IV Al (A(U A) = 0), continuously depending on A, contains in a "parametrized" way (in the sense of dependence A = AU)) many important evolution properties of the solutions of the equation. Since to use the method we need only the most general information concerning the family IV A\. this fact allows us to describe quite subtle effects connected with the development of unbounded solutions. In addition, in Ch. VII we analyse blOWUp behaviour and global solutions of boundary value problems for quasi linear parabolic equations with a source. In the last section we consider difference schemes for quasi linear equations admitting unbounded solutions. In the !irst two introductory chapters we use a consecutive enumeration (in each chapter) of theorems, propositions and auxiliary statements. In the following, more specialized chapters, theorems and lemmas are numbered anew in each section. In each section formulas are numbered consecutively as well. The number of references to formulas from other sections is reduced to a minimum; on the rare occasions when this is necessary, a double numeration scheme is used, with the first number being the section number. The authors are grateful to their colleagues V. A. Dorodnitsyn, G. G. Elenin. N. V. Zmitrenko, as well as to the researchers at the M. V. Keldysh InstilUte of
Applied Mathematics of the Russian Academy of Sciences, Applied Mathematics Department of the Moscow PhysicoTechnical Institute, and the Numerical Analysis Department of the Faculty of Computational Mathematics and Cybernetics of the Moscow State University. who actively participated in the many discussions concerning the results of the work reported here. We are also indebted to Professor S. I. Pohozaev and all the participants of the Moscow Energy Institute nonlinear equations seminar he heads for fruitful discussions and criticism of many of these results.
." ' "
"
,;\ ,~..,to.... 1"\.
.'
\
' .: I
Chapter I
Preliminary facts of the theory of second order quasilinear parabolic equations
In this introductory chapter we present wellknown facts of the theory of second order quasilinear parabolic equations, which will be used below in our treatment of various more specialized topics. The main goal of the present chapter is to show, using comparatively uncomplicated examples, the wide variety of properties of solutions of nonlinear equations of parabolic type and to give the reader an idea of methods of analysis to be used in subsequent chapters. In particular, we want to emphasize the part played by particular (selfsimilar or invariant) solutions of equations under consideration, which describe important characteristics of nonlinear dissipative processes and provide a "basis" for a description, in principle, of a wide class of arbitrary solutions. This type of representation is dealt with in detail in Ch. VI. In this chapter we illustrate by examples the simplest propositions of the theory of quasilinear parabolic equations. A more detailed presentation of some of the questions mentioned here can be found in Ch. II; subsequent chapters develop other themes.
§ 1 Statement of the main problems. Comparison theorems I Formulation of boundary value and Cauchy problems
•
In the majority of cases we shall deal with quasi linear parabolic equations of the following type: nonlinear heat eqllations, lI,=A(u)
= 'V. (k(lI)'Vu), 'V(.)
= grad, ('),
xER
N
,
(I)
or with nunlincur hcal cqualiollS wilh sOllrce (sink), II,
I Preliminary facts of the theory of second order quasilinear parabolic equations
I Statement of the main problems. Comparison theorems
3 N
Here the function k(u) has the meaning of nonlinear thermal conductivity, which depends on the temperature u = u(t, x) ::': O. We shall take the coefficient k to be 2 a nonnegative and sufficiently smooth function: k(lI) E C «0. 00)) n C([O. 00)). If 1I > 0 is a sufficiently smooth solution, then (I) can be rcwritten in the form II,
where
/l
2 k(II)/l1l+k'(1I)1V'1I1 .
= A(II):=
(I')
is the Laplace of!eralor.
Wc are looking for a solution in the class of functions bounded uniformly in x E R for I E [0. T). In the above statement of the problems we omittcd some dctails, which need to be clarified. First of all. it is not made clear in what sense the solution 1I(1. x) is to satisfy the equation. and the boundary and initial conditions. This question is easily solved if we are looking for a classical solulio/l U E Cr'/«D. T) x n)nC([O, T) x?l), which has all the derivatives entering the equation, and which satisfies it in the usual sense. Naturally, for a classical solution to exist, we must have a compatibility condition between the initial and boundary conditions in the first boundary value problem: 1I0(X)
Equation (I) is equivalent to the equation II, c/J(u)
= A
/'1/
=
(It) :=
(I")
/lcjJ(u).
k(ry) dry.
II::': O.
. Il
Thc function Q(u) in (2) describes the process of heat emission or combustion in a medium with nonlinear thermal conductivity if Q(u) ::': 0 for II ::': D. or of heat absorption if Q(II) ::: o. Unless explicitly stated otherwise, we shall consider the function Q(u) to be sufficiently smooth: Q(II) E CI([O, 00)). In most cases we assume that thcre is no heat emission (absorption) in a cold medium, Q(O) = O. In the following, we shall mainly deal with the tirst boundary value problem and with the Cauchy problem for thc equations (I), (2). In thc firsl hOll/lllary value problem wc havc to lind a function u(r. x), which satisfics in (0, T) x n, N where T > 0 is a constant and n is a (possibly unbounded) domain in R with a smooth boundary an, the equation under consideration. togethcr with the initial and boundary conditions 11(0. x)
= lIo(x)
11(1, x)
::':
0,
x En;
= 111(1. x) III E
::':
110 E c(n).
O. IE (0. T),
C
an).
sup 110 < 00;
x E
SUpll1
an;
(3) (4)
< 00.
The function 110(.r) in (3) can be considered as the initial temperature perturbation. The condition (4) describes the exchange of heat with the surroundings on the boundary an of the domain. The condition sup 110 < 00 is of importance in the case of unbounded n. The solution of problems (I), (3), (4) or of (2)(4) is then also sought in the class of functions bounded uniformly in x E for I E lO, T). Apart from the tirst boundary valuc problem, we shall also consider the Cal/ch,' problem in (0. T) x R N with the initial condition
n
1/(0. x)
= I/o(x)::,: O.
x ERN; I/o E C(R"'j, sup I/o < 00.
(5)
= UI (0. x).
x E
an.
In this case conditions (3). (4) or (5) are satisfied in the usual sense. Secondly. the coefficients k. Q were defined only for U ::': D. Therefore the formulation of the problems assumes that the solution u(t. x) is everywhere nonnegative. This is cnsurcd by the Maximum Principle, which plays a fundamental part in practically all aspects of the theory of nonlincar parabolic equations.
2 The Maximum Principle and comparison theorems The Maximum Principle characterizes a kind of "monotonicity" property of solutions of parabolic equations with respect to initial and boundary conditions. We shall not present here the Maximum Principle for linear parabolic equations, which serves as the basis of proof of similar assertions for nonlinear problems. It is extensively dealt with in many textbooks and monographs (sec, for example, [282, 101, 378, 338. 357, 320. 22. 361, 365, 42 I). There the reader can also find the necessary restrictions on the smoothness and the structure of the boundary an (they is unbounded). Therefore we move are especially important when the domain on directly to assertions pertaining to the nonlinear problems discussed above. Assertions of this kind arc known under the heading of Maximum Principle, since they all share the same "physical" interpretation and are proved by broadly the same techniques. which are frequently used in the course of the book. The comparison theorems we quote below are proved in detail. for example. in [101, 338. 356, 401. We state the theorems in the case of boundary value problems. but they apply without changes also to the case of Cauchy problems.
n
Theorem 1. LeI 1/( II alld 1/(21 be Ilollnegmil'e classical SOlllliolls (~r eql/atioll ill (0. T) x n. sl/ch Ih(lI. moreol'l'r.
(2)
(6)
4
§ I Statemcnt of the main problems. Comparison theorcms
1 Prcliminary facts of the thcory of second ordcr quasilincar parabolic equations
Theil
(8)
The theorem can be easily explained in physical terms. Indeed, the bigger the initial temperature perturbation, and the more intensive the boundary heat supply, the higher will be the tempeniture in the medium. The proof of the theorem is based 121 on the analysis of the "linear" parabolic equation for the difference z. = 1I  III II and in essence uses the signdefiniteness of the derivative 6.;: at an extremum point of the function ;:. As a direct corollary of Theorem I we have the following Proposition l. LeI Q(O) (2)(4).
71lCII
= 0 alld leI II(X. r)
he a classical sollllioll o( Ihe prohlem
1I ::: 0 ill 10, T) x U.
Indeed, 1/(11 == 0 is a solution of equation (2). Then by setting 11 12 } = II. we sec that conditions (6), (7) hold, so that 11 121 ::: 11(11 == 0 everywhere in [0, T) xU. The comparison theorem makes it possible to compare different solutions of a parabolic equation and thus enables us, by using some fixed solution. to describe the properties of a wide class of other solutions. However. the fact that this theorem involves only exact solutions significantly restricts its applicability. The following theorem has much wider applications in the analysis of nonlinear parabolic equations [101, 377, 338, 3651· Theorem 2. LeI he defilled Oil [0, T) x U a classical SOllllioll II(X.I) ::: 0 of" Ihe problem (2)(4), as 11'1'1/ as IhefillletirlllS 111(1, x) E C~;2((O, T) x U) n C(IO. T) x
11).
which sa/i.I:/)' Ihe ineqllalilies
allt/al ?:
B(lII)'
all /al:::
B(I/
)
ill
(0, T)
x
n.
(9)
5
Statements similar to Theorems 1,2 hold also for nonlinear parabolic equations of general form, in particular. for essentially nonlinear (not quasi linear) equations ( 13)
1I 1 = F(lI, \Ill, 6.lI,I, x),
where F(p. q. r. I. x) is a function which is continuously differentiable in R r x RN x R x [0. T) x The parabolicity condition here has the form
n.
aF(p. q. r. I, x)/ar :::
(13')
n.
If we take for F the operator in (I) or (2). then condition (I:\') becomes the inequality kIp) ::: 0 for p ::: o. Under some additional restrictions on the domain U and its boundary, these assertions also hold for the secolld hOlilldar\' vallie prohlem, in which instead of (4) we have on au, for example. the Neumann condition of the following type: all/all =
112(1. x). I E
(0, T). x
E
au;
112 E
C. sup //2
< DC.
(14)
where a/an denotes the derivative in the direction of n. the outer normal to au. Condition (14) makes sense if the partial derivatives II,. are continuous in 10, T) x Then a new compatibility condition arises:
n.
allo(x)/an
= 112(0. x),
x
E
aU,
and then we can talk about a classical solution of the second boundary value problem. In this case in Theorem instead of the inequality (7) we must have the inequality
( 14')
and .fil/·Ihermore II. (0, x)
//
<
1I0(X) ::: III (0, x),
(I. x)::: 1I1(1, x)::: III (I, x),
t
E
[0.
Theil II .. ::: II ::: 11(
in [0, T) x
x E
T),
n.
n; X E
( 10)
au.
(II)
Since the product k(lI)iJII/an equals the heat flux on the boundary. (14') has a simple physical meaning. Correspondingly. in Theorem 2 the inequalities (II) are replaced by the inequalities all. /an ::: all/an:::
( 12)
Let us emphasize that here we are talking about comparing a solution of the problem not with another solution or the same problem. as in Theorem 1. but with solutions of the corresponding differential inequalities (9). This extends the possibilities for analysis of properties of solutions of nonlinear parabolic equations. 'since it is much simpler to find a useful solution or a differential inequality than it is to lind an exact solution of a parabolic equation. The functions IIi and II • which satisfy the inequalities (9)( II) are called. respectively, a .I'lIpl'rsollltioll and a slliJ.I'ulutioll of the problem (2)(4).
alit /all, I E [0,
T), x
E a!!
( 15)
(in this case additional smoothness conditions have to be imposed on super and subsolutions II±). With the required changes, the theorems still hold if we have more general nonlinear boundary conditions or the third kind on an. such as all/iill
*2 Existence, uniqueness, amI boundedness of theclassical solution
I Preliminary facts of the lheory of second order quasilinear parabolic equalions
§ 2 Existence, uniqueness, and boundedness of the
7
if they are IIlliformly parabolic. This means that
classical solution (3')
Questions of existence and uniqueness of classical solutions of boundary value problems for nonlinear heat equations are studicd in detail in the wellknown monographs 1282, lOl, 361], where a wide spectrum of methods is used. Below we consider some important restrictions on coefficients, that are necessary for existence and uniqueness of a classical solution. We shall bc especially interested in questions of conditions for global solvabililY of boundary value problems, when the solution 1/(1, x) is defined for aliI::: 0, and, conversely, in conditions for global illsolwlbilily or illsolvabilily ill Ihe large. In other words, we want to know when a local solution 1/(1. x), defincd on some interval (0, 7'), can be extended to arbitrary values I > 0, and when it cannot. Local solvability (solvability in the small) holds for a large class of quasilinear equations with sufficiently smooth coefficients without any essential restrictions on the nature of the nonlinearity of these coefficients. Such restrictions arise in the process of constructing a global solution. For equations with a source,
n,
for arbitrary I E [0. T), x E p ::: 0, q, r ERN, where the continuous functions and p(p) are strictly positive. Condition (3') means, in particular, that the second order elliptic operator in (3) is nondegenerate and that the matrix I\aij II is positive definite. Local solvability has been established also for a wide class of more general equations of the form (1.13)1 (sec [261,69)). In this case the uniform parabolicity condition has the form
Il(p)
J!(p) :::: aF(p. C/, r, I. x)/ilr:::: p(p).
For equations of the form (I) the uniform parabolicity condition has a particularly simple form.
Proposition 2. LeI Ihe jilllCliolls k(II). QUI) be sl(tJicielllly smoolh fiJI" Q(O) =
= V'.
II f
(k(II)'\7I1)
+ Q(lI),
n
n
f
'/11
k(lI) :::
= 00,
n
(2)
IEII
which makes it impossible to continue the solution to values of I > To. Questions related to the loss of requisite smoothness of a bounded solution are discussed in 3.
*
1 Conditions for local existence of a classical solution This question is now well understood [260, 282, 363, 101, 2131. Classical solutions of boundary value problems and of the Cauchy problem exist locally for smooth boundary data and under the necessary compatibility conditions for quite arbitrary quasi linear parabolic equations with smooth coefficients of the form N
II,
=
L i,J'7!
a ,J (II. '\711.1. X)II"I,
II
0.
If Ihe cOlldilioll
( I)
the existence of a global solution is equivalent to its boundedness in on an arbitrary interval (0,7'). Namely: a global solution is defincd and bounded in for all I ::: 0, while an unbounded solution is defined in on a finite interval (0, Til), such that moreover 'lim sup lI(I. x)
o.
+ a(lI, '\711, I. xl.
Ell
= const > Ofor
1/
>
O.
(4)
holds. Ihell Ihere exisls a local classical sollllioll of Ihe bOll/lllary \'{IllIe problem ( I. 2)( 1.4) .. IIl11reover. if 1/11 ¥= 0 ill or if II I (0. x) ¥= 0 Oil thell 1/(1, x) > 0 ill
n
n for all admissible f
an.
> O.
A nonnegative solution of a uniformly parabolic equation (I) is strictly positive everywhere in its domain of definition, In other words, in heat transfer processes described by such equations, perturbations propagate with infinite speed. If, for example, in the Cauchy problem, the initial function I/o ¥= () has compact support and possibly is nondifferentiable, the local solution will still be a classical one for I > O. Moreover, for all sufficiently small I > 0 the function 11(1. x) will be strictly positive in R N . Under appropriate restrictions on the coefficients of the equation in any admissible domain (0, 7') x n it will possess higb order derivatives in I and x. If condition (4) does not hold, a solution of the Cauchy problem with an initial function 110 of compact support. may also have compact support in x for all I > 0, and as a result even its first derivatives in I and x can be nq~ defined at a point where it vanishes. We sball treat generalized solutions in more detail in 3, where we state a necessary and sufficient condition for existence of a strictly positive (and therefore classical) solution.
*
(3) I In
this way we refer to formulae from previous sectiolls; in this case it is from
* I.
8
*2 Existence, uniqueness, and boundedness of theclassical solution
I Preliminary facls of the theory of second order quasi linear parabolic equations
and let (6) be violated, that is,
2 Condition for global boundedness of solutions First of all let us observe that in the boundary value problem (1.\), (1.3). (1.4) without source, Q == O. the question of boundedness of solutions does not arise. This follows directly from Theorem I (§ I). Setting in that theorem UI21 ((.X)
== M =const:::: max{supllo,SUplll}'
l)
(5)
I
x
d."
  < 00
Q(.,,)
.1
(8) .
where Q(u) > 0 for u > O. The solution of the problem will then be spatially homogeneous: lI(t. x) == lI(t), where lI(() satisfies (7) and the condition lI(O) = Ill, that is,
!
"lfI_~=I. Q(.,,)
. III
we see that conditions (1.6) and (1.7) hold, so that u(l. x) ::: M, that is, u is bounded in for all t E (0, T), where I' > 0 is arbitrary. It is easy to verify that the same is true for equation (I) with a sink. when Q(lI) ::: 0 for all u ::: O. For equations with a source the situation is different.
is defined on a finite time interval (0. To) where d." = ./'''' Q(r/l 111
< 00:
lI(t) , 00. I , I'll'
o
(6)
is l/ Ilecessury ulld sllfficielll cOlldilioll Fir glohal hOllluledlle.l's or UIlY .l'ollllil!l1 of Ihe prohlem (\.2)( 1.4). Proof Sufficiency. Let us use Theorcm I. As u 121 (1, x) let us take the spatially homogeneous solution ,pi (I) of (I):
where thc constant M satisfies (5), The function equation
lI121(1)
is determined from the
.L"·'(fI ~i~) = /.
where, morcover, by (6) lI121(1) is defined for aliI E (0. by setting III I I == II we obtain that lI(I. x):": lII21(!), IE (0.
(0).
Then from Theorem I.
n. x E n.
that is, II is globally bounded. Necessity. This follows from the following simple example.
Example 1. In the Cauchy problem for ( I ) let uoCr) ==
111
= const > O.
N
x E R .
Proposition 3 reflects one aspect of the problem of unboundedness of solutions. In a number of problems with specific boundary conditions, the existencc of a global upper bound for the classical solution depends on the interplay of the coefficients k. Q, functions cntering the statement of the boundary conditions, as well as the spatial structurc of the domain n, In the gcneral setting the problcm of unboundedness is quite a complicated one. For some classes of equations this problem will be analyzed in subsequent chaptcrs (somc examples arc givcn below). Lct us observe that the necessary and sufficient condition (6) of global boundedness of all classical solutions arises in an analysis of an ordinary differcntial equation. In Example I we constructed an unbounded solution which grows to infinity as I + I'll on all of the space R N at the same time . What happens if we consider a boundary value problem in a bounded domain n, such that. furthermore. on an the solution is bounJed from above uniformly in (I Can such spatially inhomogeneous solutions be unbounded in the sense of (2)" The following example gives a positive answer.
Example 2. Let us consider a bounJary value problem for a semi linear equation,
= illl + Q(lI),
n.
x E
n.
in a boundeJ Jomain n E R N with a smooth boundary in n, 110 E nIi), 110 'f=. 0, and
(let us note that if Q(II) » II as II  f 00 this condition is the same as (8)). Let us also assume that Q E C 2(R,l is a convex function: (14)
> D.
I
> 0; E(O)
= Eo' :: 80 .
E(I)
> Eo for all
> I,
I>
d7/
HII
o. II
III/II dX) = Q(E)
I
> O. and conse
quently (\ 3)
Q"(II) ::
Q( .IllI'
Q(U)I/II dx ::
Hence under our assumptions we have that
> O. and furthermore
II
(for this estimate to hold it is essential to have I{II > 0 in n and for I/J to be normalized by (12)). so that from (15) we have the inequality
( 12)
Let Q( u)
2 Existence. uniqueness. and boundedness of theclassical solution
Furthermore, from Jensen's inequality for convex functions [2111 we obtain
> 0 the first (smallest) eigenvalue of the problem
'//1 (x) the first eigenfunction. which is known
n.
§
or the theory of second order quasilinear parabolic equations
I Preliminary facts
Then for any initial functions uo(.d :: 0 sllch that
Q(7/)  AI 7/ 
O.
Therefore by (13) E(I)  f 00 as 1> T,' ::: T., and since E(I) ::: sup, 11(1. x), the solution u(I. x) satisfies (2) for some To ::: T I and is unbounded. The interest of this example lies in the fact that for sufficiently "small" initial data lIoCd this boundary value problem has a global solution defined for all I$>O (see eh. VB, § 2). For "large" 110 it grows unboundedly as t + Til' To < 00. One can then pose the question: in what portion of the domain n does it become unbounded as I > Ti;? This question. of localization of unbounded solutions, is considered in subsequent chapters. We close the discussion of global boundedness conditions by an elementary example of a second boundary value problem.
the solution of the problem is unbounded and exists till time Example 3. Let n be a smooth bounded domain. n ERN. Let Q(II) be a function convex for II> O. which satisfies (8). For (\), let us consider the second boundary value problem with noflux Neumann boundary condition. To prove this. let
LIS
(16) aU/ail = O. t > O. x E an, with initial perturbation u(O, x) = uo(x) :: 0 in n. Let us show that any nontrivial
introduce the function E(I)=
1'II(1.X)I{I\(x)dx.
.Ill
Then E(O) = Eo and furthermore. as follows from (9). dE(t) _.
dt
.
l'
6I1(1,X)I/'j{X)dx+
II
l'
.Ill
E(I)
satisfies the equality
Q(II(1,X))'/II(X)dx.
. II
Integrating by parts and taking into account (10) and
I'
¢ 0) solution of the problem is unbounded. Assuming sufficient smoothness of the solution. let us integrate equation (I) over the domain n. Then. if we introduce the energy (II
(\ 5)
H(t)
11(1, x) dx,
t ::
.Ill
(II),
we obtain
o.
and integrate by parts. taking (16) into consideration. we have dH(I)
611(1, x)l{ll(x)dx =
= I'
dl
H(O)
(
Q(u(I. x)) dx,
t > 0;
.Ill
(17)
= H o = I'
.Ill
lIoCr) dx > O.
12
*2 Existence, uniqueness, and houndedness of theclassical solution
I Preliminary facts of the theory of second order quasilinear paraholic equations
Using the Jensen inequality
1·
Q(II(t,
l!
.I
x))dx == (rneasn)
::: (meadl) Q
. l!
~
It is clear that the problem (18), (19) has the trivial solution u(t. x) == 0, However, in addition it has an infinite number of other spatially homogeneous solutions 1I(t, x) == 11(1), which satisfy the ordinary differential equation
1 Q (II(t, x))dx:,: 'l meas ~
(r.Ill ~lll(l, meas
11'(1)
x)
dX)
== (Incas!}) Q (
13
= lI"(I),
I >
0;
u(O) :::::
o.
(20)
H(t)'l)'
meas ~
Solutions of this problem are the functions
we obtain from (17) the inequality (21)
~~H(I) til
:::
(meas n) Q (~~) , meas!}
I>
O.
Therefore by (8) it follows that the energy H(t) (and therefore and bounded only on a bounded interval (0, T I), where 'X f ( T) < ] ' I < '7' *' == ,/ II,,/meas l! Q( T))
and therefore lim sup,
lI(I, x) ::::: 00, I , Til:::
1/(1,
x))
is defined
where T ~ 0 is an arbitrary constant. Therefore, due to nondifferentiability of the source for II = 0, there appear from the zero initial condition (19) nontrivial solutions that grow at the same rate on the whole space. Let us note that for a E (0, I) all the functions lJ 7 (t) are classical solutions. since lJ E C I ([0, (X))). It is not hard to see that similar nontrivial solutions of the Cauchy problem can be constructed in the case of arbitrary sources Q(II) > 0, 1I > O. if T
00,
TI• ·1
1
,0
3 lJniqueness conditions for the classical solution
tiT)   <"
Q(r/)'
00
(22) .
Hence we obtain the condition Under the assumption of sufficient smoothness of the coefficient Q in (I), the local classical solution is always unique. This follows directly from Theorem I of I. Indeed, if we assume that there exist two different solutions 1I< and II, of equation (I) corresponding to the same initial and boundary conditions, then it follows from Theorem I. by first setting 11 11 ) ::::: II', 11 121 ::::: II, and then exchanging lI' and 1I" that we have at the same time lI' ::: lI, and II' ::: lI., that is. II' == u,. It remains to check how essential is the smoothness requirement on the coefficient Q, which is a nonnegative function. In case ofa heat sink (Q(II) < 0, II > 0), it is not hard to verify that uniqueness of the solution holds without any restrictions on the smoothness of Q(II), Thus, let a continuous function Q(II), <0(0) = 0; QUI) > 0, II > 0) be nondifferentiable for II = 0, Q E C l «0, (0)). The following example shows what this
*
can lead to. Example 4. Let us consider the Cauchy problem for the equation ( IX)
where a E (0, I) is a constant. Here
Q(II) ::::: II". Q(O)
11(0, x) ::::: 1I0C,)
==
= 0, N
0, x E R .
Q'(O+) :::::
t
(22')
.10 which is at least neas,wry Jilr Ihe IIniqlleness oI Ihe SO/III ions 01" Ihe Cal/chy pm/JIem.
This example is entirely based on an analysis of spatially homogeneous solutions, which satisfy an ordinary differential equation. What if we consider a problem with houndary conditions that do not allow the solution to grow at the boundary'l It turns out that in this case also lack of sufficient smoothncss of thc source for II ::::: may cause the solutions to he nonunique.
°
Example 5. Let!l he a hounded domain, n eRN, and let AI > 0, 1//1 (x) > 0 in n, be, respectively, the first eigenvalue and the corresponding eigcnfunction of the problem (II). Let us consider in R t X !l a boundary value problem for the equation (23) with the conditions
I Preliminary facts of the theory of second order quasilinear parabolic equations
Let a E (0. I); then the source 1/J:~"(x)u". which depends not only on the solution II, but also on the spatial coordinate x, is nondifferentiable in II for u = 0+ everywhere in n. It is not hard to see that the problem (23), (24) has, in addition to the trivial solution II == O. the family of solutions
3 Generalized solutions of quasilincar dcgencratcparabolic equations
1 Examples of generalized solutions (finite speed of propagation of perturbations, localization of boundary blOWUp regimes and in media with sinks) Example 6. (tinite speed of propagation of perturbations) Let us consider equation (1.1) in the onedimensional case:
= (k(II)U,),
ul
where lJT(t) are the functions defined in (21), To conclude, let us observe that nonuniqueness is rebted to the particular formulation of a problem, If, for example. we take in the Cauchy problem for (18) an initial condition uo(x) ::: Do > 0 in R N , then its solution will be classical and unique. since by Theorem 2, the solution will satisfy the condition u(t; x) ::: Do in RN. In the domain II ::: Do the coefficients of the equation are sufticiently smooth. which ensures uniqueness of the solution. Simibrly, if in the problem (23). (24) Uo > 0 in n, then its solution will also be unique.
IIS(t.
x)
=:
In this section we consider equations (1,1), (1.2) which do not satisfy the uniform parabolicity condition, As above. we shall assume that the functions k and Q are I sufficiently smooth: k E C"«O. 00)) n C([O. 00)), Q E C ([0.00)) (as was shown in 2 this last condition is necessary for the uniqueness of the solution), k(lI) > 0 for II > 0, and furthermore
*
k(O)
= O.
Is(~). ~
(I)
(3)
At,
r\') + A''dl's = 0 dl; ,
d d, ( k(I )'' d~\ d~
dis
+ AI, ..\' = c.
(3')
Selling C = 0 (what this corresponds to will be made clear in the following), we obtain the equality k Us) (~fs (4)     , =A Is dg . Let us assume that I [
.0
k(r,)
(5)
d'T/ < 00. 'T/
so that the function (1)(11)
that is, the equation is degenerate, Formally this condition means that the second order equation (1,1') that is equivalent to (1,1) degenerates for II = 0 into a first order equation (if k'(O) i= 0 and 1I(t. x) has two derivatives in x). Before we move on to examples that elucidate certain properties of generalized (weak) solutions, we shall make a remark. When we dealt with classical solutions II E C~;", there was no need to require continuity of the heat flux W(t. x) = k(u(t. X))\7I1(I. x), This condition. as well as continuity of the solution itself (temperature), is a natural physical requirement on the formulation of the problem, In the present case we shall constantly have to monitor this property of generalized solutions,
= x 
where A > 0 is the speed of motion of the thermal wave. Substituting the expression (3) into (2). we obtain for Is(~) ::: 0 the equation
k(rs)"dg
parabolic equations
(2)
and let us construct its particular selfsimilar solution of travelling wave type:
or, which is the same,
§ 3 Generalized solutions of quasilinear degenerate
15
=
II [
, 0
k('T/)
 dT], r1
u ::: 0; <1>(0)
= O.
(5')
makes sense, Then it follows from (4) that
Let
go = 0,
then
where <1>1 is the function inverse to (it exists by monotonicity of OJ identically by zero; it follows from (3')
~
16
§ 3 Generalized solutions of quasi linear degenerateparabolic equations
I Preliminary facts of the theory of second order quasilinear parabolic equations
that continuity of the heat flux kUs )fl' will still hold at the point ~ == 0 for C = O. As a result we obtain the following selfsimilar solution:
r
Us
'~
(t, x)
~ ~d7J
= «>"[,1(,11  x)j·l. I> 0, x E R.
<
00
D Tj
~_.__
tI,\(t, x)
17
.
(6)
where we have introduced the notation (K)+ = (K. if K ::: 0 and 0, if K < 01. Let us set To ::::: «>(00)/,1" :: 00. Then (6) can be considered as the solution in (0, To) x R j of the first boundary value problem for equation (2) with the conditions OL_'
Thus if condition (5) holds, the problem (2), (7) has a solution with everywhere continuous heat flux, which has compacl supporl in x for each I E (0, To): liS (I , x)
== O.
.l._ _l.._ _....1.:;;.X
Fig. 1. Travelling wave in the case of linite speed of propagation of perturbations Illt,:l:)
x> AI, IE (0. To).
Therefore equation (2) describes processes wilh finile speed or propagalion 11 perturbations. At the point where liS > 0, the solution of the problem is a classical one and it is not necessarily sufficiently smooth at the front (the interface) of the thermal wave, xl(l) = AI, where it vanishes. For a more detailed study of the behaviour of the solution at the points of degeneracy, let us consider the case k(tI)
== II".
IT
= const
> O.
o

_:::=~====:=====. .1.'
Then <»(11) = II"/(T. «> '(tI) = «(TII)'/", To = 00 and the travelling wave solution has an especially simple form
Fig. 2. Travelling wave in the case of inlinite speed of propagation of perturbations
IIs(I,x)=llrA(AI~X)II'/", I >(). x>O.
Condition (5) is necessary and sufficient for the existence of a compactly supported travelling wave solution. If it is violated, that is if
Let us check again that the heat flux is continuous at the points xf(l) Indeed. W(I. x) = tl:~(lIsl\ = (T'/" ,11"1 '1/"1(,11  xl 1'/".
=
At.
j
that is, W(I. xr(l)) = W(I. x; (I)) == W(I. xI(l)l = 0 for aliI> O. At the same time. if (T ::: I. at the degeneracy points x = xI(I) the derivatives tl r , II"II" arc not defined, In the case (T E [1/2, I) the derivatives II,. II, exist, hut the derivative u,,(I. xf(l)) is not defined. If, on the other hand, IT E (0.1/2), II,. It,. II" arc defined everywhere (that is. the compactly supported solution (8) is a classical one), however, higher order derivatives do not exist at the front points. These are the main differentiability properties of the generalized solution we have constructed. The function (8) is schematically depicted, for different times, in Figure I. This solutioll represents a thermal wave moving over the unperturbed (zero) temperature background with speed A,=, dXr(r)/dl.
/
., ~~1J.2 dT] = 00.
.0
•• •• •• •• ••• •• •• •• •• ••• ••
(9)
T]
then, as follows from (4), the function Is(~) is strictly positive for all admissible ~ E R and therefore (3) represents a positive classical solution of the equation (2) (see Figure 2). It is obvious that in the case k(O) > O. that is, for uniform parabolicity of the equation (see Proposition 2, ~ 2), condition (9) holds. However. among coefficients k(tI). k(O) = 0, there are some for which (9) holds. This is true, for example, for the function k(tI) = Ilnlll '. tI E (0,1/2), k(tI) > 0 for tI::: 1/2. Then the travelling wave solution is strictly positive and therefore classical. Moreover. if k(lI) E C''(R,l. then II can be differentiated in I and x in the domain (0. To) x R, any numher of times.
I·1.:
•• •• •• •• •• •
•• ••• •• •• •
•• •• • •• •• •• •• •• ••• •• •• •• •• •• •
'.
18
~ 3 Generalized solutions of 4uasilinear degenerateparabolic e4uations
I Preliminary facts of the theory of second order 4uasilinear parabolic equations
19
Us (t.x)
It is interesting that the condition (5), which was obtained without any diffi
culty, is not only sufficient, but also necessary for finite speed of propagation of perturbations in processes described by equation (1.1). A travelling wave type solution has another exceptional quality: it demonstrates in a simple exa,mple localization in boundary heating regimes with blOWUp, The study of this interesting phenomenon in various problems occupies a substantial part of the present book. Example 7. (localization in a boundary blOWUp regime) Let k(lI)
= lIexpl1I1.
II :::.
D.
Then it is not hard to see that the solution (6) has the following form: , _ {  Inll  1.( At  x) 1, D.
11.1'(1, .\) 
D ::: x ::: AI.
( I ())
x> At,
Fig. 3. Travelling wave selfsimilar solution (10) localized in the domain (0. 1/ A)
which is dehned for a bounded time interval [0, To), where To boundary condition at x = 0 corresponding to (10) has the form 11.1'(1,0)
= III (I) = In( I 
2
A t). () <
I
1/1. 2 . The
(II)
< To.
k that satisfy
and therefore II I (I) c> 00 as I ~ TIl' However, though the temperature at the houndary blows up, heat penetrates only to a finite depth L = 1/ A, that is. IIS(t. x) == 0 for all x:::. L for all the times of existence of the solution, I E (0. To) (see Figure 3). Here we have that everywhere apart from the boundary point x is bounded from above uniformly in I:
= 0, the solution
D < x < 1/ A,
x:::.
It is clear thal the same comparison argument and thus the same result on localization of arbitrary boundary blowup regimes, also holds in the case of coefficients
1/1..
and it grows without bound due to the boundary blowup regime at the single point x = D. The limiting curve II = 11.1' (T!) ,x) is shown in Figure 3 by a thicker line. Let us note the striking difference between this halted thermal wave and the usual temperature waves shown in Figures I, 2. It is easy to see that in this case every boundary blOWUp regime leads to localization. Indeed, for any boundary function III (I) c> 00 as I c> To (for simplicity we set 1I0(X) == 0), we can compare the solution 11(1. x) with the "shifted" selfsimilar solution 11.1'(1. x  1/ A), which is delined for x > xo(l) = AI. We have that II ::: Us = 00 for x = xo(l) for all I E (D. To)· Therefore by the comparison theorem II ::: 11.1' in (D. To) x Ix > xo(l)l and linally II(T o. x) ::: IIs(T o • x  I/A) < 00 for x > I/A.
I
, I
""
k(1/)   d1/ < 00. 1/
This follows immediately from the representation (6) of the corresponding travelling wave solution generated by the blOWUp regime. Let us consider an example of a generalized solution of the heat equation in the multidimensional case. Example 8. Let us find a solution of the Cauchy problem for an equation with a power law nonlinearity II,
= '\7. (11"'\711).
I>
D. x
ERN.
( 12)
having constant energy
I' .IRS
11(1. x) dx
= Eo = const > D
( 13)
(this is a solution of the instantaneous point source type). We shall look for it in the selfsimilar ansat7. (14)
20
~ 3 Generalized solutions of quasi linear degenerateparabolic equations
I Preliminary facts of the theory of second order quasi linear parabolic equations
where U', (3 are constants, and where O(g) :::: 0 is a continuous function. Substituting (14) into (12), we obtain the following equation: crr,,I{!  (3(,,1
~ L
Its solutions have the form
o
_
(1]) 
ilfJ. (:
'j>: S,
= ("IIT+II "1{3'V." ., . ((}IT'Vc. 0). <
Here
From here we have the necessity of the equality aI = a(l' + 1)  2(3; then the terms involving time can be cancelled. Furthermore, using the identity
I' 11(1. x) dx == I' ("0 (~) .IR' .IRS (I'
I' {}(O d{; .IR' (it is assumed that 0 E LI(R'v)), by (13) we have that a + N(3 = O. Hence we obtain a unique pair of parameters a = N/(Nrr + 2). (3 = 1/(NlT + 2), that is. dx ==
I lIT
[
If'
2(NlT
2_
+ 2)(1]0
2 1])t
t',!Nf3
1]0
1] ;;:
J
(15)
I (""
21
(20)
O.
is a constant, which is determined from the condition (18): _
1]0(l:0)
=
{
.. NI1[2(NlT 7T
.
+ 2)J I!IT
l'(N/2
IT
I'
+ I + l/lT) }IT!INIT+21 + I) 'eO L'
(i/IT
Thus the required selfsimilar solution with constant energy has the form
ll'( x
.d. ) (
(T
___ V/IN,r+ 1 1
'.

[
2(NIT
+ 2)
Ixl,
.,
) ) lilT
'
( 1]0
(2/INIT+21
+
(21)
.
the desired solution has the form (16)
For any ( > 0 it has compact support in x. while as ( , 0+. it goes to a rSfunction: Eoo(x), 1 > 0 1 • Everywhere except on the degeneracy surface R+ x (1.1'1 = 1]0(I/INITt21} it is classical (and infinitely differentiable), while on the surface of the front (on the interface) it has continuous heat flux. Differentiability properties of the solution (21) are the same as those of the particular solution of travelling wave type considered in Example 6. Since equation (12) is invariant under the change of 1 to T + I, where T = const > 0, lls(1 + T. x) will also be a solution with constant energy. In the following example we use the solution constructed above to illustrate an intriguing property of a quasi linear degenerate parabolic equation with a sink.
lls(1. x) >
Then it follows from (15) that the function (} ;;: 0 satisfies the following quasilinear elliptic equation: tT
'V f . ({) 'VfO) .
N +  I  ~iJO L  , {;, +    0 = 0, {; N1T+2
as well as the condition
I'
I
.InN
d{;;
NlT+2
E
v
R' ,
OWd{; = Eo.
( 17)
(18)
Let the function 0 be radiallr S\'IIIII1('lri1', that is, let it depend only on one dinate: 0 = H( 1]), 1] = Igi ;;: O. Then equation (17) takes the form
COOl"
Example 9. (localization of heat in media with absorption) Let us consider the equation
_1_( N .IOITO')' 1]N . I1]
.j _ _1._ 0'1]
+
NlT+2
_N __ O = O.
1]
> 0;
=0
+ '(01],v), = O.
1] >
where
O.
II
I (tiT()'
+ __I_{}1]N = O. Nlr + 2
1]
> 0; ()iTO'(O)
(22)
 yll. I> 0, x ERN.
= O.
= exp(yl}v(l. X),
is a new unknown function. Then the equation for
N1T+ 2
Integrating it, and setting the integration constant equal to lero (this, as is easily verified, is necessary for the existence of a solution with the required properties), we :lITive at the lirst order equation 1]N
(1/" 'VII)
11(1. x)
holds. Equation (19) is equivalent to the equation (r/" I (}iTO')'
= 'V.
where y > 0 is a constant. Compared with ( 12), this equation has a linear sink of heat. Let us see how this is reflected in the properties of the generalized solution. In equation (22) let us set
moreover, by symmetry we havc to require that the condition (}iT(t'(O)
§ 3 Generalized solutions of quasi linear degenerateparabolic equations
22
23
I Preliminary facts of the theory of second order quasilinear parabolic equations
rile pro!Jlem (1.2)( 1.4) if the generalized derivative 'V(p(lI) == k(u)'Vu exists, is square integrable in any bounded domain Wi C (0, T) x n, and if for every continuously differentiable in (0, T) x n function f(l· x) with compact support. which is zero for (I, x) E 10. T) x and for r == T. we have the equality
(I(
we obtain for v
= V(T,
x) the equation liT
= 'V. (v"'Vv),
an
which we considered above; its particular solution we already know. Choosing as lI. for example. the function IIs(l + T. x) (see (21)). and inverting all the changes of variable, we obtain the following solution of equation (22): 11(1, x)
=
= exp( _y' llg(l) 1 N/{N" I 21 r
(T
l2(N(T
where g(1)
== I + T(I). IXf(l)1
+ 2)
This solution has the degeneracy surface
= TID
l
1+
I  exp( yeT'
11
1 /{N
' r 2: 0,
yeT
(23)
on which the flux is continuous. But this is not its main distinguishing feature. As in Example 8, the support of the generalized solution grows monotonically. however here we have L
==
I)
lim IXJ(I)I
'.'C>.)
= TID (1+. yeT
I !(N
<
00,
that is. heat perturbations arc localized due to the action of the sinks of energy in a bounded domain in the spaL:e. a ball with radius L.
2 Definition and main properties of generalized solutions The examples we considered in subseL:tion I allow us to demonstrate many of the properties of generalized solutions of quasi linear degenerate parabolic equations. Let us note again that a generaliz.ed solution does not necessarily have everywhere defined derivatives, but at points of degeneracy it possesses a certain regularity: the heat nux is continuous. At all other points where the equation is nondegenerate (and is. therefore. uniformly parabolic in a neighbourhood of these points). the solution is. as is to be expected. classical. Let us give a definition of a generalized solution, which takes into account all the indicated properties. Let us consider in (0. T) x n the first boundary value problem (1.2)( 1.4) for an equation with coefficients k(u). Q(II) sufficiently smooth for u > O. such that, furthermore. k does not satisfy the uniform parabolicity condition. that is k(O) = O.
Definition. A nonnegative continuous bounded function u(T,
l l ( a, .
the boundary conditions (1.3), (1.4) will be called a gCllerali,.ed (weak) sollllioll
dx d'
+
l
1Io(x)!(0, x) dx ::= O.
(24)
.!I
{I
Let us note that formally the equality (24) is obtained by multiplying equation (1.2) by f and integrating over the domain (0, T) x n. Integration by parts (in the variable x) is then justified if the function k(II)'VII is continuous in n. This requirement is not contained in the definition. where weaker restrictions are imposed on the derivative 'VeP(lI) (existence in the sense of distributions and the condition 'VeP(1I) E L?nc((O. T) x n), for which the integrals in (24) make sense). However for a wide range of degenerate equations the above restrictions are sufficient in order to prove continuity of k(II)'V1I (we deal with this in more detail below). Naturally. it is necessary to define a solution in the generalized sense in the oose when the solution 11(1. x) has degeneracy points in (0, T) x n. where 11(1, x) == O. In the opposite case. if. for example. tln(x) > 0 in and Q(1I) 2: O. then II(T. x) > 0 in (0, T) x n and the solution is a classical one, since the equation does not
n:
degenerate in the domain under consideration. Generalized solutions of quasi linear degenerate parabolic equations were studied in detail in a large number of works (see. for example, [319, 341. 86. 377,296]). Without entering into details, let us note one important point. As a rule. the generalized solution 11(1. x) of an equation with smooth coefficients is unique and can be obtained as the limit as II ~ 00 of a monotone sequence of smooth bounded positive solutions 11" (I, x) of the same equation. As a result, in a neighbourhood of all the points (I, x) E (0, T) x n. where 11 > O. the solution is classical. and it loses smoothness only on the degeneracy surface. which separates the domain (II > O} from the domain (II = O}. To prove continuity of the heat nux k(II)'VII. additional techniques must be mobilized (see. for example, 116\). Some additional information concerning differentiability and other properties of generalized solutions can be found in the Comments section of this chapter. Below we shall treat in a more detailed manner the restrictions, under which it is necessary to consider solutions of a parabolic equation in the generalized sense. This will be done using the example of the nonlinear heat equation
u, == 'V. (k(u)'Vu), , > 0, x
ERN.
(25)
for which we consider the Cauchy problem with an initial function of compact support
x). which satisfies
+
11;a( _ k(II)'V1I . 'V j.Q(II)! )
7
.0
24
*3 Generalized solutions of quasilincar degenerateparabolic equations
I Preliminary facts of the theory of second order quasi linear parabolic equations
so that
uo(X) ~ 0, [xl> I
= const
> O.
(27)
We return now to the condition we obtained in Example 6 concerning compact support of a travelling wave solution. It is quite general.
Proposition 4. COllver!iellce (d' the inte!iral
is a Ilecessary and s(~fJicient cOlldition FJr the soilltion or the Callchy pmhlem (25)(27) to have compact Sllpport ill x. N
In other words, if the integral in (28) diverges, then u(l, x) > 0 in R for all t > O. The proof of this assertion is based on the comparison theorems for generalized solutions, which are essentially similar to the ones quoted in § I. The second of these theorems is slightly different in the generalized setting.
Theorem I extends to the generalized case word for word. In the general case its proof is based on the analysis of integral identities of the form (24) for solutions u l l), ul21 or by comparing a sequence of positive classical solutions u;,! I, u;,21, which converge, respectively, to the generalized solutions //11, U(21. The statement of Theorem 2 has to be changed. In specific applications we shall use the following version.
n
Theorem 3. Let there be de(/illed in [0, T) x a nOIlne!iative !ienera!i:.ed solution of the houndllry value prohlem (1.2)( 1.4) liS (vel/as the.filllctiol1S u± E C(lO, T) x OJ, III E CI.2 e\'el)'\vhere in (0, T) x apart .limn a jinite Illllnher oIsmooth Ilonintersecting SllrrllCeS (0, T) x S,(I) on \vhich the jilllction '1c/J(u) == k(II)'11I is cOllfinllolls. Let the inequalities (1.9) hold everywhere in (0, T) x (n\(x E S,(I)}j, while Oil the paraholic hOlllldar." or the dOli/a in (0, T) x n \I'e h(/\.'e the condition.1
n
u. :'.: II :'.:
"+
in (0, 1') x
n,
4 Proof of Proposition 4 (concerning finite speed of propagation of perturbations) and some of its corollaries Sufficiency of the condition (28) follows directly from the analysis of selfsimilar solutions of travelling wave type. which was undertaken in Example 6. Let us place a bounded domain w = supp lin in a parallepiped P = I[xd < In. i = 1,2, .,., N} with sides parallel to the axes Xi so that w C P. Let us show that the speed of propagation of perturbations along the ith direction is finite. As in Example 6. let us construct a particular solution of travelling wave type. having the form
d,(l.
x)
= H(x, 10

AI),
11'\(0.
x)
==
()(x, 
In)
=0
for Xi = In. It is strictly positive in the left halfneighbourhood lin  f < Xi < InJ of the plane x, = In. However supp lin C P. so that there exists f > O. such that II()(X) = 0 for XI = In  f. By continuity of 11(1, x) for x, = In  f for some time t E (0, T). we shall have the inequality 11(1, x) :'.: 1I~,(t. x). and by the comparison theorem. Theorem L 11(1. x) :'.: 11'1'((' x) in the domain It E (0, T), x, > In  f}. Therefore 11(1, x) has compact support in x along an arbitrary direction Xi· As supp 11((, x) grows, the parallepiped /' becomes larger. and the same argument applies. To prove necessity we use a different selfsimilar solution of equation (25):
3 Comparison theorems for generalized solutions
(1.10), (1.11). 7111'n
25
(29)
The new element in comparison with Theorem 2 is just the fact that the generalized supersolution II" and subsolution u can have compact support, while on the degeneracy surfaces (0, T) x S,(I) the corresponding heat fluxes must be continuous. Thus, roughly speaking, we are imposing the same requirements on the functions IIj as on the generalized solution of the problem. If in (1.9) we replace the inequality signs by equality signs (in (0,1') x (H\\x E Si(l)})). then the functions 11+ will be simply different generalized solutions of equation (1.2).
(30)
where the function
f :::
0 satisfies the ordinary differential equation
I. eNl
s
'I
(tV Ik(nt) + 7I'~:::: 0.
(31 )
{; > O.

Lemma. COllditioll (211) is l/ IlI'CeSsell'." alld sufficil'lIt cOlldition .fiJI' existellce or a 1101llIegati,'1' gellerali::.ed sollltiOIl or 1'l{lll/tiOIl (31), which 1'1II1is!Jl's at a [Joillt ~ = ~n > 0, where the heat .flllx tV Ik(Ilt is l'IIlItin/lIIl1s. Proof The existence of a solution f = f(~). such that I(~n) = 0, (k(IlI')(~n) = 0, f(~) > 0 for all ~ E (0, ~o) is established by rcducing (31) in a neighbourhood of the point ~ = ~o to the cquivalent integral equation with respect to the monotonc decreasing function {; = ~(Il:
r ~~ O.
(32)
Local solvability follows from the Banach contraction mapping theorcm. If thc intcgral in (28) diverges, there is no solution with a finitc front point ~ = ~n. Indced, on the one hand lim ~(n = to < N, I .n···
* 3 Generalized solutions of quasi linear degenerateparabolic equations
I Preliminary facts of the theory of second order 4uasilinear parabolic equations
Proposition 5. LeI Q(II) ::': 0, Q E C I ([0, (0)) (/Ild leI the coefficielll k(u) s(/Ii.~h' cOlldilioll (34). Theil il' IIU(X) to. Ihe solutioll or Ihe Cauchy problem (1.2), (1.5) is slrictlv pOsili\'e ill RN lor £11/ admissible I :> O.
while on the other hand we obtain from (32) that k (7))   d7) = 00. 7)
In this case (31) has a monotone solution f = I(t), strictly decreasing in R+, such thal f(~) + 0 as t + 00 (sec e.g. 1337.24,327]). Necessity of condition (28) is also proved using Theorem 'I. Let the integral in (28) diverge. Let us show that 11(1. x) > 0 in R N for all I > O. Let us take the solution I(t) of the lemma and set (33) The function f(~) can be undefined for ~ = 0, but that is not essential. For us it is important that by (33) 11 111 (I, x) , 0 as I + 0' uniformly in any domain > oj, 0 = const > O. Without loss of generality let E supp lIu. Let us pick 0 > 0 small enough, so that llxl ::: oj c supp 11(1. Then, obviously, there exists T > 0, such that 11(11(1. x) < 11(1, x) for Ixl = 6, I E (0, T). Let LIS usc now the fact that the solution of the problem (25), (26) can be
lIxl
27
°
obtained in the form
Proot: By comparison Theorem 3 the generalized solution of the problem under consideration (denoted by IPI (I. x)) is everywhere not smaller than the solution II == III J 1 of the Cauchy problem (25), (26) for the equation without a source: N (1 11 121 ::': 11 11 ). However. from Proposition 4 it fDlIows that 11 ) > 0 in R for I > 0; therefore this also hDlds for 11'21. 0
For an equation with a sink the situation is more complicated. Here even if k(O) :> 0, the solution 11(1. x) can have compact support. However, for that to happen the sink must be very powerful for low temperatures II > 0 and the function Q(II) must be nondifferentiable ilt zero. Otherwise, as shown in the example below. the solution will still be positive and a classical one. Example 10, Let us consider the Cauchy problem for a semilinear equation ';¥ith
a sink:
(35)
with an initial function lIo( x) t 0 with compact support, 0 ::: 110 ::: M. supp Ill) C I Ilxl < 1 1. Let Q(II) :> 0 for II > 0, Q(O) :::: 0 and Q E C (\0. (0)). Let us show 0
11(1, x)
=
lim
11.(1, x), I> 0, x
that 1I(t. x) > 0 in R N for I:> O. First of all we immediately obtain from Theorem I that
ERN.
l .. _",()I
where II. arc classical solutions of equation (25), which correspond to the initial conditions II. (0, x) = E + lIuL,), x E R N . But, as is easily seen. for every E > 0 111 we can always tind I. E (0, T) (I. > 0 as E + 0+), such that 11 (1, x) < 11.(0, x) in llxl :> oj for I E (0, I. \. while by construction Df the family lll.) we have £1(1)(1 + I., x) ::: 11.(1, x) for Ixl = 0, I E (0, T  I.). Therefore from comparison Theorem I we obtain the inequality ulll(l+I" x) ::: 11.(1, x) for x E RN\llxl ::: 0). I E (0. T  I.). Passing in this inequality to the limit E + 0., we obtain that 11 111 (1, x) ::: 11(1, x) for x E RN\llxl ::: 01, I E (0, T). which by (33) implies strict positivity in R N of the solution of the problem (25). (26) for all arbitrarily small I > O. This concludes the proof of Proposition 4. 0 Therefore if the cDndition
1I(t.
x) ::: M. r ::': 0, x E R
N
Next. taking into account the restrictions on the coefficient Q we deduce that Q(lI) ::: CII. liE
[0. Mj; C == const » O.
Then. using Theorem 3 to compare the solutions of equation (35) and of the equation V, ::::
~v  Cv.
I >
n, x
E R
N •
which satisfy the same initial condition. we convince ourselves that (36)
· i 1
,u
k(7))
  dTJ
= 00
(34)
However. v » 0 for
I :>
O. Indeed, setting
7)
holds. there is no need to detine the solution of the problem (25). (26) in generalized (weak) sense: any nontrivial solution is strictly positive and therefore a classical one. Naturally, this will also be com::ct for any equation (1.2) with a source.
0:::
1'=
(37)
expI Cllw
we obtain for II' the heat equation w, = ~lI!. w(O. x) = lIo(x) ::': 0 in R , Ill) to. and therefore w :> 0 in R N for I > O. The required result follows from (37), (36). N
28
*:I Generalized solutions of quasilinear degenerateparabolic equations
I Preliminary facts of the theory of second order quasi linear parabolic equations
The next example shows that in a medium with a strong sink the thermal wave can be not only compactly supported. but also localized.
Example 11. Let us consider the first boundary value problem for a heat equation with a sink in the onedimensional setting:
29
Here the constant C 2. which is determined from the first of conditions (42). has the form C 2 :=: I J2/ (a + I) and therefore I 
v( x) = [
IX
,J2(a+
(/ _
Xl] 2/1 I
"I
]
2" I
0 <
<
X
I.
I)
By construction the function ._. II". I
ll, :=: tI"
> O. x > O.
(38) 1I'1 (.1') :=:
I
l

J2(a
= 110(.\) ?: O. x > O. 1/(1.0) = III (I) ?: 0, I > 0,
11(0. x)
(39)
(40)
where a E (0, I) is a constant. so that the function II" is nondifferentiable for II = 0 (strong absorption). Let the initial perturbation 110 have compact support: 110(.1') = 0 for all x > 10 > O. while the external heat supply is bounded: 111 (I) :'S M < 00 for all I ?: O. Let us show that under these conditions the solution always has compact support (even though k(lI) == I > 0) and is, moreover. localized in a bounded domain. Both these assertions are proved by comparing thc solution of thc problcm (38)(40) with the stationary solution v:::: vex) of the samc cquation V"
_.
v"
= 0,
(41 )
which is determined in the following fashion. Let us fix I > 0 and consider for (41) the Cauchy problem in the domain (0 < x < II with the conditions v(/)
= 0,
v'(/)
= O.
IX
+
(/ 
\)
"I
X
I
> O.
(43)
I)
is a classical stationary solution of equation (38) and has for x > 0 continuous derivatives II,. lI u (let us note that at a front point x = I higher derivatives do not necessarily exist). Let us choose now I = I. > 0 large enough. so thall/o:'S WI.(X) for x > 0 and furthermore WI ((})
.
Then IIdl) < WI. (0) for all have the estimate 11(1,
l
I a
= J 2(a + I) I. I
]2/11',,1 > M.
?: O. Therefore by the comparison Theorcm I we
x) <
WI.(X),
I?: 0, x
E
R
I ·
Thus, first. the function II has compact support in x for all I > 0 and. second. heat is localized in the domain (x E (0,1.)1 at all times I E (0, (0). Let us stress that these properties are possible only in the case a E (0, I); for a ?: I, as shown in Example 10, the solution is strictly positive for I > O. Absence of nontrivial solutions of the stationary problem (41), (42) with finite I > () in the case a ?: I also testifies to that.
(42)
5 Conditions of local and global existence of the generalized solution Onc solution of the pi'oblem (41). (42) is the trivial one. However. it is easily vcrified that there is another solution. which is positive on (0, I). Any solution of (41) satisfies the identity I 1 I II (v,t  v'" 2 a + I
=el ,
where the constant C 1 must be l.ero. which follows from (42). Then
and therefore
*
On the whole. all the assertions stated in subsection 2 of 2 concerning classical solutions. are valid here. Local existence of the generalized solution follows from thc ability to construct it as a limit of a sequence of classical solutions defined on a finite interval (0, T). Naturally, Proposition 3 is also valid. since the condition entering it has been obtained in an analysis of dassical solutions. Analysis of unbounded classical solutions in Examples 2. 3 applies also to generalized solutions. Let us consider the following example (in a morc gencral selling such problems are considered in 2. Ch. VII).
*
Example 12. Let n be a bounded domain in R N with a smooth boundary iT > 0 is a fixed constant. For a degenerate equation
§ 3 Gcncralized solutions of quasilincar degcnerateparabolic cquations
I Prcliminary facts of thc thcory of sccond ordcr quasilincar parabolic cquations
Using (47) and taking into account the fact that !J.I/I I from (46)
let us consider the boundary value problem with the conditions
= uo(x) ::: o. x E n, = o. I > O. x E an.
11(0, x) 11(1, x)
110 E
C({l).
Let us denote by 1111 (x) > 0 in n, 1I'1',III.'lltl = I. the first eigenfunction of the problem !J.III + AliI =: 0 in n, III = 0 on an, and by A, > 0 the corresponding eigenvalue. We shall show that for A\ < (T + I every nontrivial solution of the problem exists only for a finite time. We shall proceed as in Example 2. Let us form the scalar product in /}(,{l) of the equation (44) with r/J" Introducing the notation E(I) =: (u(t, x), I111 (x»), we obtain dE (1)=
l
I/II(X)'\I'(II
dl.ll
"'"
I vll)iX+
l ,{ ( '" 'I'I x)u
I {x. ,
(46)
,1I
r ~/iCx)'\I. (1/"'\111) dx = _1r u"+1 6.1//1 dx. + I .Ill
(47)
n
r
I111 '\I. (u"'\III)dx
( I  AI) I 1/'1),  (11"+, (T + I
I
AI~/I in
n, we obtain (49)
> O.
If A, < (T + I, then using Jensen's inequality (u"+ I is a COnvex function for we arrive at the estimate
AI) (II, IPI )"+1 ~ (I A  ,_1 ) E"+I.
dE ( 1  _::: dl
+I
(T
Hence it follows that
E(I)
(T
=
I
I
0),
> O.
(and thus u(t. x») remains bounded for time not greater
than
T,
+
II :::
(T+
+I
I
AI
~[r
.Ill
(T
1I 0 C\,)III\(X)dX]'"
< 00.
that is, there exists To E (0. T. j, such that !irn sup, 1I(t, x) =: x, I > To· Let us note that for AI ::: (T + 1 it follows from (49) that E(I) is bounded for all I ::: O. This can be considered as evidence of global boundedness of the solution (see § 2, Ch. VII). To conclude, let us give some simple examples of unbounded generalized solutions which illustrate the property of heat localization in nonlinear media with volumetric energy sources.
(T
If I/o > 0 in then u(l, x) is a classical solution. Let supp Uo C iL Let us denote by iJw(t) the degeneracy surface of equation (44) in this problem, that is. the boundary of the support of the solution w(t) == suppu(t, x). Then U(I, x) == 0 in fl\lrJ(t), and by Green's formula
.Ill
=
((
(T
Here, as in the case of a classical solution, we can integrate by parts the first term in the righthand side of (46). Let us show that
.Ill
dE ,(I)
(45)
=:
31
==
_1_/' + (T
I.
IIII!J.//"+' dx
Example 13. The equation
+ 1/""1.
1/, = (u"l/x),
has in the domain U(I, x)
=
(OC,
= (To 
I) II"Os(x)
all,
(48) where we denoted by iJ/illl, the derivative in the direction of the outer normal to ilw(t). However, u",·\ =: () on illlJ(t) and by continuity of the heat flux ilu"+ I/all, == '\1//,,+1 . II, = 0 for x E iJw(t) , Therefore the last two integrals in (48) are zero, which leads to the equality (47). It must be said that in the analysis above we did not consider the question of regularity of the surface (lw(1) (in particular, the existence of the derivative au,,+I/(lIl/ on (lw(l); for certain classes of equations this problem is quite well understood (sec the Comments section). In this particular case this is not necessary; the definition of a generalized solution implies that integration by parts is justified and allows us to prove the equality (47).
==
'I cos , II)
2(", (, (T( (T,_ '1)
_ ('" () _ t )1/" =
, .} (.t.
= canst> 0,
To) x R the following selfsimilar separable solution:
'"1/1
1Jr/J\u ,,+1 
(T
{
0,
,2
1T
,\
Ixl ::: Ls /2.
II"
Ixl
o<
<
L s /2.
I <
(50)
To,
where L s = 2rr«(T + I) 1/2 /iT and To > 0 is an arbitrary constant. Let us indicate the main features of this solution. First of all, it has compact support in x and is a generalized solution; at the points of degeneracy x =: ±Ls /2 the heat flux is continuous. Secondly, it exhibits finite time blOWUp: 11(1. x) > 00 as t > To for any Ix[ <
L.\/2.
Thirdly, its support, supp 11(1, x) = {Ixl < Ls/2l, is constant during the whole time of existence of the solution. It is localized; the heat from the localization domain {[xl < Ls/2l does not penetrate into the surrounding cold space (see Figure 4). even though at all points of the localization domain the temperature
32
§ 3 Gcncralized solutions of quasilincar dcgcneratcparabolic equations
I Preliminary facts of thc thcory of second ordcr quasilincar paraholic cquations
h± (0) = 0
and the exact solution satisfies in the generalized sense the singular initial condition u,(O, x) = Eoo(x) in R,
u(t,X)
! I
33
t5
ttl;~tj't;t5
where O(.r) stands for the Dirac delta function, and the constant Eo depends only on To and (T. More precisely, for small I > 0 we have the representation
Fig. 4. Localization of a finitc timc blOWUp combustion process in the Srcgimc (sclfsimilar solution (50))
grows without bound as I > Til' The halfwidth xc! (I) > 0 of this fast growing heat structure, that is, the coordinates of the point at which u(l, x,., (I)) = 11(1, 0) 12 are also constant in time. Example 14. The equation III = (II" II,) \ + 11",1. (T > 0, also has quite an unusual exact nonselfsimilar solution of the following form:
= {It)(l)lfll(l)+cos(27TxILs)!,}'/''::: 0 Ls12) and 11,(1, x) = 0 for x E R\(L sI2, LsI2),
II.(I,X)
for x E (L s I2, function 111(1) E (I. I) satisfies for I > 0 the equation III
and 4){I)
,
= Coil
I .~
= (T«(T +
I)" (oil  fir)
 1112 (I) 1
1 "1 21/2
1
,(T
11
,I>
0;
/II(())
where the
+ (T/2,
1/2),
then it is not hard to see by integrating the equation for I/J(I) that the solution u.{I. x) will blow up at time To: II.{I, 0) > 00 as I > Til' It is easy to see that the fronts of the generalized solution II, are at the points
as t
>
{Ixl
< Lsi2/. It is interesting to note that since ,11(0)
Til and the solution grows without bound only in the localization domain = I, we have the equality
u,(I, x)~(.r) dx > Eo~(O) as I > 0
for any smooth compactly supported test function ~(x). From these asymptotics it 2 follows immediately that Eo = aobi/ B(1 + II(T. 1/2). As far as the behaviour of the solution 11.(1, x) close to the blOWUp time is concerned, it is not hard to check, by computing the asymptotics of the functions 1jJ(I) and 4)(1) as I > Til' that this exact solution converges asymptotically to the simpler selfsimilar solution (50). which we considered in Example 13. Below (see § 5. eh. IV) we shall show that precisely this selfsimilar solution describes the asymptotics of a wide variety of unbounded solutions close to the blOWUp time. Finally, we observe that the above solution II •• which is not selfsimilar. can be treated as follows. Setting u" = v yields an equation with quadratic nonlinearities,
= I,
If the constant Co is chosen in the form
Co=Co(T o)= «(T+ 1)(TIT()'B(1
._,,
1'1
= A(v) ==
vv"
I
,
'
+ (T (v,)" + lTV".
The I/onlil/('ar operator A admits the following twodimensional lincar il/l'tlriallt slIhs{J(/('('
W2
=
Cfll. eos(Axll ==
==
(w(x) :
3 Co, C, E R, such that w(x)
= Co + C I cos(Ax) 1,
where A = 27T I Ls (.'1'1' l denotes the linear span of given functions). This means that A(W 2 ) ~ W 2 . Therefore substituting 1'(1, x) = C o(1) + C,(I)cos(Ax) E W 2 into the equation gives us a dynamical system on the coeflicients Ko(l), C, (I)}. which is precisely the parabolic equation on W2,
6 Examples of nonuniqueness of the generalized solution Obviously, the requirement of smoothness of the source Q ::: () in equation (1.2), which is necessary for unique solvability of the Cauchy problem (see subsection 3 of § 2) in the class of smooth functions, is still in force in the generalized setting. Example 4 applies in this case without changes. In addition, it is easy to give an example of a degenerate equation, constructed as in Example 5, which has in a bounded domain a nonunique spatially nonhomogeneous solution. For example, the problem
:=; 0 in
n
for I:=;O and in R f x iH!; (f E (0, 1), (T > 0 arc constants; the rest of the notation is the same as in Example 5 of § 2, has the family of nontrivial solutions
It
[((I, x)
,I/(IT'I) :=; vr(l)liJI (x). I >
()
,x E
()
H.
Let us consider an example which demonstrates explicitly that if uniqueness conditions do not hold, the comparison theorems for generalized solutions are no longer valid.
Example 15. Let us fix an arbitrary x > O. the equation
35
Remarks and comments
I Preliminary facts of the theory of second order quasilinear parabolic equations
Ii
"rt
,x)
o _ _____,.L...._
Fig. 5. Three differenl solutions of equation
(51),
M
which do nol satisfy the Maximum
Principle First, all these solutions, as solutions to a boundary value problem in R+ x R+, satisfy the same initial condition [(±(O. x) :=; [(*(0, x)
== 0,
x::: O.
Secondly, for A > 2 the boundary values satisfy the inequalities (T
(0, I) and let us consider for I > 0,
E
[(*(r.0) < [(M(I,O) < [(+(1.0), I> O.
[(, :=; ([(,r [( \),
+ [(I
(51 )
IT.
Here Q(It) :=; [('IT, so that Q(O) :=; 0, but Q'(()f) :=; 00. Let us find a travelling wave solution. Setting [((t, x) :=; Is(~), ~:=; x  At, A > 0, we obtain for Is ::: () the equation
Nonetheless, as seen from the position of these solutions relative to one another in Figure 5, they do not satisfy the comparison theorem. Let us note that already the existence of two solutions [(:± of travelling wave type with same speeds of motion and coinciding fronts, which correspond however to boundary regimes of different magnitudes, contradicts physical intuition.
which has for A > 2 two different solutions
Remarks and comments on the literature C 1 :=;
A±JA14)I/IT ( (T
2
> O.
Thcrefore the required selfsimilar solutions have the form [(J(I, x)
:=;
C ,1(,1.1  x),
[I/IT, I
> 0, x> O.
Let us compare these generalized solutions with the spatially homogeneous solution (Figure 5) [(*(1, x) :=; ((Till/IT, I ::: 0, x> O.
The necessary bibliographical references for most of the contents of § I, 2 are contained in the text. Concerning Propositions 2, 3 in § 2, sec [282, 320, 10 I, 3381: the restriction (6) in § 2 coincides with the Osgood criterion for global continuation of solutions of an ordinary differential equation [3541. The result stated in Example 2 was tirst obtained in 12431. Concerning Example 5, see 1243, 1161· Nonuniqueness of solutions of boundary value problems in a bounded domain for a semi linear equation with source concave in It was proved in 1116j (see also [114]). The generalized selfsimilar solution of Example 8 was constructed in 13851 (N :=; I) and 128, 386[ (N ::: I arbitrary). Asymptotic stability of the
36
I Preliminary facts of the theory of second order quasilinear parabolic equations
selfsimilar solution (21) of § 3 was established for N = I in 1234] and by a different method in 11871. The proof of stability in the multidimensional case was done in 11071 (qualitative formal results were obtained earlier, for example. in [5,384, 3861l; see also Ch. II. The localized solution of Example 9 is taken from [3021. The definition of a generalized solution in subsection 2 of § 3 for a degenerate equation of general type without a source was formulated in 1319, 341. 3421. These authors also prove existence and uniqueness theorems for generalized solutions for boundary value and Cauchy problems. For quasi linear parabolic equations with lower order terms such theorems are proved, for example. in 1377. 231. 21. 43. 203. 294, 344. 3451. where in a number of cases weaker generalized solutions are considered). Differentiability properties of generalized solutions of the equation tI, = (tI"+I),,, IT > 0, were studied in 116. 17. 18J; in particular. continuity of the heat flux (tI"+ I), was established, certain results concerning degeneracy curves were obtained, and Hblder continuity in x with exponent v = minI I. I/00l was proved. This implies Hblder continuity in I with exponent vl2 (see 1202, 258j). Under certain additional assumptions, it is shown in 175 J that the Holder continuity exponent in I is also equal to /) (from the form of the solution in Example 8 it follows that this is an optimal result). Later some of these results were extended to the case of more general degenerate equations 1230. 248. 203, 252, 2531· Properties of the degeneracy surface of the equation ((, = Llu'J+ I were studied in lIS, 5S, 59, 2521; there it is shown that starting from some moment of time it is differentiable (many of these results are summarized in 11031; see also [3281l. We shall discuss in more detail the properties of generalized solutions of degenerate equations in Ch. II, III, and in Comments to these Chapters. Sufficiency of condition (2S) in § 3 in Proposition 4 (tinite speed of propagation of perturbations) was established in 13191 for the onedimensional case; see also 133]. Necessity under some additional assumptions was proved in 1229J. In the proof of Proposition 4 we use a method that was employed in [327J for N = I. Concerning Theorem 3, see [231,232,2481. In the presentation of the result of Example II, we used the approach of 12311 (comparison with the stationary solution); in that paper conditions for localization in arbitrary media with volumetric absorption were obtained. For more details on localization in media with sinks see Ch. II and the surveys in 1233, 162]. In the analysis of the parabolic equation in Example 12 we used a generalization of the method of [2431 to the case of quasilinear problems 1120, 121, 1241 (see also 1225]. where the same method is used to study a quasilinear equation of a different type). The localized unbounded solution of Example 13 was first constructed in 1391. 353] (see Ch. IV). The localized solution of Example 14 was constructed and studied in 1134, 1761· There one can also find a method of constructing similar exact solutions for a large class of evolution equations and systems with quadratic nonlinearities. Let us note that this solution is not invariant with respect to Lie groups or LieBacklund groups;
Remarks and comments
37
see 1221, 3221. An example of this unusual kind of exact solution for a quasilinear equation with a sink was constructed in 1491 (see also a similar solution in [313]. Some general ideas on construction of finitedimensional linear subspaces that are invariant under a given nonlinear operator and of the corresponding explicit solutions via dynamical systems are presented in [1361 and 1139]. Example 15 is taken from I I221. In that paper were established conditions on the coefficients k(u), Q(u), under which a parabolic equation of general type admits at least two travelling wave type solutions. Existence of different travelling waves for an equation with power type nonlinearities, /I, = Llu'" + 1/", Ii < I < Ill. III + P ? 2. was established in [3231; see also the general results of [324] on "almost" uniqueness (for III + P < 2) and nonuniqueness, and 161 for the case III = I.
~ I A boundary value problem in a halfspace for the heat equation
Chapter II
Some quasilinear parabolic equations. Selfsimilar solutions and their asymptotic stability
In the present chapter, which, like the previous one, is of an introductory character. we briefly present results of analysis of specifie quasilinear parabolic equations. As can be seen from the title, one of the principal methods of investigation consists of construeting and analyzing selfsimilar (or, in the general case, invariant) solutIons of the problem being considered. Using various examples, we shall try to show, what role these particular solutions play in the description of general properties of solutions of parabolic equations of most diverse types. Here we also introduce the concept of approximate selfsimilar solutions (a.s.s.) of nonlinear parabolic equations. Usc of the construction of a.s.s as a tool in its own right will be considered in other chapters. The examples presented below eover a sufficiently wide spectrum of nonlinear equations. Comparatively simple and frequently well knowll examples illustrate many ideas and methods of analysis, which will be developed in subsequent chapters in a more explicit and detailed fashion. Many of the problems and questions considered below have been exhaustively researched; the corresponding references arc given at the end of the chapter. From all the available results we ehoose only those that are, first, constructive, that is, ones that make it possible to show explicitly certain properties of the solutions of a problem, and second, which is particularly important for an introductory chapter, those that can be proved in a relatively simple and brief manner at least Oil the formal level. Wherever this cannot be done, we restrict ourselves to short remarks on the proof, or discuss only the "physical meaning" of the result, which contains the ideas of a rigorous proof. For that reason, we do not aim at a great generality in our presentation; frequently other proofs of well known facts are given; these, in our view, either make explicit the "physical basis" of a phenomenon. or illustrate nmthematical methods to be used in the sequel. Let us note that this approach (frequently using similarity methods) makes it possible to obtain more optimal, ami even new results.
39
We want to emphasize in particular the concept of asymptotic stability of selfsimilar solutions of nonlinear parabolic equations with respect to perturbations of the boundary data of the problem, as well as with respect to perturbations of the equation itself. Selfsimilar (invariant) solutions are not simply particular solutions appearing serendipitously. In many cases they serve as a sort of "centres of gravity" of a wide variety of solutions of the equation under consideration, as well as of solutions of other parabolic equations obtained as a result of a "nonlinear perturbation" of the original equation. The sense in which the expression "centre of gravity" is to be understood, will become clear below. The specific form of selfsimilar solutions is to be determined from the conditions of invariance of an equation with respect to certain transformations. In the general case families of selfsimilar solutions are determined by a group classification of the equation. This allows us to find all classes of equations invariant with respect to a certain group of transformations (such as Lie groups of point transformations. or LieBacklund groups of contact transformations; see [221. 322]). We start with an analysis of a simple linear problem; however, as we show below, this analysis will allow us to determine properties of a whole family of nonlinear problems. $
§ 1 A boundary value problem in a halfspace for the heat equation. The concept of asymptotic stability of selfsimilar solutions For the linear equation 1/ 1
=
11., \. 1
> 0, X >
o.
(I)
let us consider the boundary value problem with boundary data 1/(0, x)
== I/o( x) 1/(1.0)
::':
==
O. x
:>
0; sup I/o
1/1(1):> 0.1:>
<. 00,
O.
It is assumed that the function I/o( x) is Lipschitz continuous in R +.
(2) (3)
Here we analyse the "dimensionless" equation with thermal conductivity coefficient ko = 1. This does not restric~ the generalit~ ,of the r~sults, sinc~ by scaling time 1 ~ kol (or the spatial coordmate x ~ ko / x) the Imear equatIon III = kou" reduces to the original one. Thus in equation (I) the variables I, x arc also dimensionless quantities. As we already mentioned, the problem (I )(3) models the process of heat action on a medium with a constant thermal conductivity. Our goal is to describe explicitly the evolution of the heating process, establish the law goveming the
* I A boundary value problem in a halfspace for the heat equation
II Some quasilinear parabolic equations
40
motion of the wave of heating, find how its depth of penetration (halfwidth) xe.r(t) depends on time l , and to determine the spatial profile of the wave. Solution of the stated problem can be written down explicitly in terms of heat potentials [2821:
41
Thus the problem of constructing a selfsimilar solution (6) of a partial differential equation has been reduced to the boundary value problem (7)(9) for a considerably simpler ordinary differential equation. The solution of the problem (7)(9) exists, is unique, monotone, and strictly positive: J f(1 +m) exp _: H (~) (10) 0( .\ S") = 2'//+ \ 71'1/2' 4 (l",·tll 2: .
{t,2}
+
x
J
2(7Tt)'/
lox [exp {(X  ~)2 
exp
41
.0
where H,.(z) is the Hermite function:
2
}
{(X + 0 }] 1I()(~)d~. " " .41
H,,(z)
(4)
I 10"" = r'
exp!I,  2zt}1
iI'! II
dl
( II)
(1'). ()
However it does not seem possible to glean directly from (4) the features of the process we are interested in. Therefore we proceed in a different way.
(a special function of mathematical physics [35, 317]). The function Ils(O decays rapidly as (; + 00: ( 12) Ol(~) ~ exp{e/4}. (; + 00.
1 A selfsimilar solution
The selfsimilar solution (6) constructed above has a simple spatiotemporal structure. From the form of the solution it is easy to determine the dependence of the depth of penetration (halfwidth) of the thermal wave on time:
Let us consider a special form (power law) boundary regime: 111(1)
= (1 + I)"',
where 111 > 0 is a constant. For such a boundary function equation suitable selfsimilar solution: 11.1(1. x)
= (I + I)"'{ls«(;),
Substituting the solution (6) into equation differential equation {",I'. 1
+ 2:I {'l\s"
(; (I),
 IIIfJ.s
= x/(I + 1)1/2.
(I)
has a
11\(1) ~
= I.

= o.
(9)
'The quantity x,. / (r) is determined for each timc r > () by the equality 1I(f. x,. j (f)) "" l/(r, 0)/2, that is, this is the point where the temperature is equal to half the temperature Oil the boundary.
By the comparison theorem, lis majorizes a large set of solutions of the problem
Proposition I. [,l'l
Then the solution liS satisfies the boundary condition (3), (5). Taking into consideration the condition of boundedness of liS as x + 00 (see (2)), we shall require the inequality fJs(oo) < 00 to hold. From equation (7) it is easy to deduce that sueh a solution has to satisfy the condition (}s(oo)
2 Comparison with other (nonsimilarity) solutions
(1)(3).
O.
Let (ls(O)
(13 )
= (;,./( 1+ r) 1/1 ,
where the constant (;,.j = (;", (III) is such that OS(~"J) = Os(O) /2 = 1/2. The function Os(l;) characterizes, for each I > 0, the spatial shape of the thermal wave.
(6)
we obtain for Os«(;) the ordinary
= (), "s >
.s '~"f(l)
(5)
I> O.
(I +1)'//. I> 0; 1I()(.~) ~ IJs(x),x > O.
( 14)
Then Ihe sollilio/l o( praMl'lII (I )(3) salis/ii'S rhl' ineqllalily
IIU, x) ~ (I +
1)//'OI(X/(
1+
1)1 0
). I> 0, x> O.
( 15)
Therefore if the inequalities (14) hold, we have an upper bound for the solution of the prohlem; this bound allows us to understand the form of the distribution in space of the heat coming in from the boundary. For example, let the boundary regime be of the selfsimilar form, 11,(1)
~ I A boundary value problem in a halfspace for the heat equation
II Some quasilinear parabolic equations
42
while the initial perturbation satisfies uoCt) x;'((t), that is, x"j(t):s: ~ ..{(II/)(I
:s: ()s(x)
in Rto Then by (15)
+1)1/2,
I> O.
x ..{(t)
:s:
Inequality (15) also gives us some information about the spatial profile of the nonselfsimilar thermal wave.
3 Asymptotic stability of the selfsimilar solution with respect to perturbation of the boundary data Let ~s. consider a different aspect of the problem. What would happen if the ~'estnctlons Ol~ the initial function uoCr) in (14) were not satisfied, for example. If lIo(x) == I 111 R, (then by the condition ()s(x) > 0 as x > x the inequality lIo(x) :s: ()sCt) does not hold for all sufliciently large x > 0). In this case the selfsimilar solution allows us to obtain sharp bounds on the spatiotemporal structure of the heating wave, but, naturally, only for sufficiently large I. Below we shall deal with asymptotic stability of the solution (6) with respect to perturbations of the initial function. Let equality (16) hold. Let us introduce the simi/urilY represelltalioll (simi/ariIY .. Irall.lfo 1'111 .. ) of the solution of problem (I )( 3), deli ned at each moment of tim~ in accordance with the form of the selfsimilar solution (6): O(t,~)
= (1
+ 1)'111//(t, t;( 1 + 1)1/2), 1 > 0, ~ >
o.
( 17)
T.his expression is arranged in such a way that the similarity transform of gives us exactly the function (J.o;(t;).
=
+ I)"'.
IIs(t, x)
The se/f~sill1i/ar sO/lIIioll (6) is asYlI/plOIically slab/e wilh respecI 10 a rbi Ira rv ( bO//llded) pI' rtll rim IiOlls of Ihe ill i lial
Proposition 2. LeI III (I)
.fill/Clioll: ji}r
lilly
(I
I
>
O.
//oCt)
Hence it follows immediately that
for all
~
o
::: O.
Thus for any initial function, the solution of the problem with a power law boundary regime after a certain time becomes quite close to the selfsimilar solution. From (18) it is not hard to derive, for example, the asymptotically exact expression for the depth of penetration of the thermal wave: (19)
which, for large I, is close to the selfsimilar one:
Here by (18) the similarity function correctly characterizes the profile of the heati~g wave at an advanced stage of the process. f This does not exhaust the properties of the constructed selfsimilar solution. It turns out that it is also stable with respect to small perturbations of the boundary regime. A general assertion concerning asymptotic stability of the selfsimilar solution (6) with respect to perturbations of the boundary data looks as follows (it is proved in exactly the same way as the previous one).
Proposition 3. LeI III (I)/(
I
+ I)'"
>
I,
I ~ 00.
(20)
Theil
IIHU,.)fJ,()IIClR.,=O! max lt lll ,llu\(t)/IIIIII]O,l>OO.
(21)
.
110(1,')  Os(·)lIcIR,
I
==
sUp~,(J 18(1,~)  ()\(t;)1
= ( 18)
= 0«1
+ 1)/1')
, 0, 1+00.
Pro!}/: It follows from the Maximum Principle. Let us sci satisfies the equation ;:,=::",I>O,X>O. and furthermore ::(1.0) son theorem we obtain
43
= 0,
I > 0, and sup, ,(J 1::(0, xli <
1::(1, x)1
:s:
M
= sup 1:.:(0, Xli, ,.(J
00.
= // 
//.1"
If (20) holds, we have the same exact estimate (19) for the depth of penetration of the wave. This result gives us an explicit form of the evolution of the heating process for arbitrary initial perturbations and for boundary regimes asymptotically close in the sense of (20) to a power law dependence.
Then 
4 Asymptotic stability of the selfsimilar solution with respect to small perturbations of the equation From the compariLet us show now lhat spatiotemporal structure of the selfsimilar solution is preserved for large I > 0 in the case of a "small nonlinear perturbation" of the original
I > O.
parabolic heat equation.
* I A boundary value problem in a halfspace for the heat equation
II Some quasilinear parabolic equations
44
Suppose a sufficiently smooth thermal conductivity coefficient is not constant: k == k(u) > 0 for II ) O. However it is close to a constant for large temperatures: k(u) ,
I. II
(22)
> 00.
Let us consider the same process of heating, but now in a nonlinear heat conducting medium: (23) II, :::: (k(u)II,)" I> O. x> O. where 11(1, x) satisfies the boundary conditions (2), (3). For convenience, let us introduce the function
45
the requirement that Uo E L 2 (R t ); by (12) the selfsimilar solution usU, .) is in L"(R+) for allm > O. We restrict ourselves to a formal analysis of equation (25). Let us take the scalar product of equation (25) with win 1}(R t ), and, having convinced ourselves that this product makes sense, let us integrate the righthand side by parts: this is allowed in view of uniform boundedness of the derivatives u, and (11.\), in R, x R+ and the condition ll' > 0 as x > 00. As a result we obtain 1£10
llwll"/'IR I:::: 2 dt ,.
(lJI, .. k(II)II,
 (us),)·
It is not hard to verify the identity (II,  (us),)(k(II)II,  (us),)::::
== _(k l / 2 (II)U,
 (liS),)"
+ (I
k 112 (l1)))"11 ,(us) ,

==
2
Proposition 4. LeI III :::: (I + I)"'. I > 0, Un E L (R t ); lin is lIOnincreasing in x and condilion (22) holds. Then Ihe se((similar solution (6) is stable wilh respecI to the indicated perturbations (~f the thermal conductivilY coefficienl, and we have using which the preceding equality takes the form
the estimate oc
I1OU,')  8s()II;.'IR,) == :::: () [(\ +
as t ,
1).2111 1/2
r l8U , t) .In
l (\
max { I,
8sWJ
2
dt == (24)
+ T)",II2Cd (I + T)'III dT }]
>
0
I £1 2 .1/2 2 llll'lI/'IR I == Ilk (U)II,  (lIs),lll'lR 1+ 2 dl '. .,
Let us note that convergence of fI(t . . ) to fl s (') as I > 00 in the L "(R1l norm implies, in particular, pointwise convergence almost everywhere. Proot: The function
).
*
If condition (22) holds, the righthand side of the estimate (24) does indeed go to zero as I + 00, which is not hard to see by evaluating the indeterminates
hm  1/2 ,o f">0.\
iJ
;Cdu). (liS), rJx
Under our assumptions on uo(x), II, (I, x) ::: 0 in R t for all I > 0 (this follows from the Maximum Principle; see I of eh. V). Taking into account in the last equality the fact that (liS), < 0 in R t x R+. we arrive at the estimate
00.
=
(
::::
CIs < 00. Then from (26) we obtain
I £1
R, the equation
(1(.1'),1"
while sup WI(t)1
,
2: til 11 1JI II i.'(lc 1_«/,\,(1+1)'" (25)
with w(t, 0) == 0, W(I. x) ..., 0 as x .} 00 (this follows immediately from the Maximum Principle) and w(O.·) E e(Rt ). The latter assertion is ensured by
*2 Asymptotic stability of the fundamental solution of the Cauchy problem
II Some quasilinear parabolic equations
47
§ 2 Asymptotic stability of the fundamental solution of the
from (27) we immediately obtain the estimate
Cauchy problem
+ 1)2/11
11"
+ 2qs(\ + I)
2/11
110(1,')  Os(·)II;'fR,1 ::: (\
1
11110(')  O,d·)II;l(R.1 1/2
r(\ +
.10
T)", 112
+
In this section we consider the Cauchy problem for the heat equation.
Cd (\ + T)/11 J dT.
II, :::;: U".
t > O. x
E
(\)
R.
(28)
o
which is the same as (24).
11(0.
x) :::;: 1I0(X) ::::
O.
x E
R;
110 E
(2)
C(R).
where the initial function has finite energy: Remark. The estimate (24) holds for sufficiently arbitrary (nonmonotone in x) initial functions 110 E /}(R,). such that () ::: 11(1 ::: 110 ::: II;~ in R j , . where II~ are monotone functions, lI(f (0) :::;: III (0). Then the same method can be used to derive estimates of the form (24) for the similarity representations o± (t. g) of the solutions II±(I. x), which satisfy the initial conditions 11+1/=0 :::;: LlG in R+. Therefore the stabilization 0(1. g) + (}s(g) as I , 00 will follow from the inequalities II" < U ::: 11+ (or, equivalently, 0' ::: 0+) in R j x R I · Thus. the selfsimilar solution (6) correctly describes for large t properties of solutions of a large set of quasilinear parabolic equations. The estimate (19) of the depth of penetration of the thermal wave also holds here, while the function (}s(g) determines its spatial form as t + 00. The function liS will be an approximate selfsimilar solution for the equation (23): Us does not satisfy that equation. but correctly describes asymptotic properties of solutions of this equation. Therefore, using just the selfsimilar solutions (6) we can describe asymptotic behaviour of solutions of boundary value problems corresponding to different boundary data I/o(x), III (I) and different equations (in this case. equations with different heat conductivity coefficients k(II». However, (I) admits also other selfsimilar solutions, such as, for example, a travelling wave type solution.
°::
(3)
Eo:::;: IllIolIll(RI <x. Eo > O.
Then the solution of the problem ( I). (2) will have the same property: its energy is constant in time:
I:
11(1. x)dx :::;:
(4)
Eo. t :::: O.
For simplicity we shall assume in the following that Lloer) == o(expllxI 2 }) as Ixl+ x, We set ourselves the same questions: how does the initial temperature profile spread. how do its amplitude and width change in time as t + oo? We stress that this problem in the above setting is very different from that considered in § I, Unlike the boundary value problem, here there is no "forgetting" of the properties of the initial condition, since the amount of energy Eo in (4) (which is a characteristic of the function lIo) plays an important role at the asymptotic stage of the process. This fact imposes additional restrictions on the methods of studying asymptotic properties of solutions of the problem (I). (2). Equation (\) has a wellknown selfsimilar (fundamental) solution in R c x R:
(5) IIS(t. x) :::;:
explt 
xl.
I >
O. x > O.
(29)
where
(6) for which III (I) :::;: explt). It is not hard to show that this solution is stable with respect to perturbations of initial and boundary functions, as well as to perturbations of the equation, which allows us to find asymptotically exact solutions for the class of boundary value problems with boundary regimes of exponential form. Finally, let us observe that the properties of selfsimilar solutions of equation (I) (for example. of form (6) or (19» are preserved also under perturbations of boundary regimes and the equation more drastic than those of (20) and (22) (sec § 4, Ch. VI).
It satisfies the conservation law (4),
Solution (5) will solve the problem (I). (2) only if the initial function uo(x) is also of a selfsimilar form. that is, if IIO(X)
==
IIS(O. x)
~.: :; 2~':)li exp { 
,:2 }. x E
R.
(7)
•
§ 2 Asymptotic stability of the fundamental solution of the Cauchy problem
II Some quasilinear parabolic equations
48
2 Stability with respect to nonlinear perturbations of the equation
1 Stability with respect to perturbations of the initial function The analysis of this problem is not very complicated, since there is a representation of the solution of the problem (I), (2) in terms of a heat potential [2821: 1I(t, x)
= 2(7T~).l:
1I0(Y) exp {  (x ~/)2} dr.
(8)
For convenience, let us introduce the similarity representation of the solution of the problem (I), (2) which corresponds to the spatiotemporal structure of the solution (5): (9)
(substitution of the solution
into
(5)
(9)
49
Here we use the selfsimilar solution (5) to study the nonlinear heat equation II,
= (k(II)U,).,
t > 0, x E R.
Since in the Cauchy problem (II), (2) the amplitude of the solution, 111/1(1) == SUP,II(1, x), goes to zero as t + 00, the asymptotic properties of the solution 1I(t. x) depend on the character of behaviour of the coefficient k(lI) for small values of the temperature II > O. Below we shall demonstrate stability of the selfsimilar solution (5) of the heat equation with respect to the following perturbations of the constant coefficient: k E C"«(O, (0)) n C([O, (0), k(lI) > 0, k'(II) > for II > 0,
°
Ik(u)lk'(II)l' + 00, II +
gives us the function f.\(~)).
Proposition 5. The se!fsimi/ar sO/lltion (5) is stahle with respect to lllNtmry pertllrbations (!f the se!t~sil1li/ar initilllfilllction (7). lI'hich preserve its energy: if (3) holds, we have pointwise convergence:
(II)
(\2)
0,
and furthermore, lim Ik(~II)/k(II)1 u···o
= I.
~ >
n.
( 13)
These conditions are satistied, for example by the coeflicient
(\0) k(lI) =
Proof Let us fix an arbitrary ~ transformations, we obtain I(t, ~)
+ t ) 112 exp = 2:I (I:;;:r x
=
xl(\ + 1)1 12 . Then, using (8), after elementary
{e} 4 /

x
.""'
lIo( \') exp
.,x
{e+\'"2~Y(I+1)1/2}  ::.'"'
.
4t
d\'.
Since 110 E L I (R) satisfies condition (3), the integral in the righthand side con0 verges to Eo as t ,. 00. This means that (10) holds. What are the consequences of this result'! First of all, it means that the amplitude of the thermal profile evolves for large times as .
. ~
Eo ,,1/2 SUPll(t,X)" l/2t ,t+oo, ,el{ _7T so that the width of the temperature inhomogeneity is
lin III". a
= const
> O. II E (0, 1/2),
(14)
°
which differs significantly as II + from the coefficient k == I. Nonetheless, asymptotic properties of solutions of equation (II) can be described using the fundamental solution (5) by transforming it in a convenient manner. Therefore the problem of stability of the selfsimilar solution with respect to nonlinear perturbations of equation (I) is considered here in a new selling (compared to § I). At the same time we shall prove stability of liS with respect to small perturbations of the thermal conductivity coefficient in the case k(lI) + I as II , 0·. In addition to (14), all the conditions are satisfied by the coefficients k(lI) = [In Ilnul!", ll' > 0; k(lI) = exp(Ilnlll"l, a E (0, I), and so forth. Equation (II) with an arbitrary nonlinearity has no selfsimilar solution describing the "spread" of the initial profile in the Cauchy problem. Therefore we shall look for an approximate .Iel(similar SO/lit ion u , which does not satisfy equation (II): I
11,(1. x)
== qJ(l)fs((), ?
=
r t/J'(I) ,
(15)
•• ••• •• •• •• •• •• •
•• •• •• •• •• •• •• •• •
.1
where t/J(1) is a monotone increasing positive function,
*2 Asymptutic stability of the fundamental solution of the Cauchy problem
11 Some quasilinear parabolic equations
50
The main problem is to determine the function (p(1) in (15), which depends on the behaviour of k(u) for low temperatures. It provides both the rate of amplitude decay: Eo
I
Tlte solwiOIl or lite problem (II), (2) COil verges 10 lite a.s.s. (15):
110(1.,)  fs(·)II" Proot:
um(l) :::: c/J(I) 27T111' I > 00.
Then" (/J(,u(l)
and
1I(,u(l). x)
It is convenient to carry out the proof of convergence of O(r. () to Is«() (which establishes similarity of asymptotic properties of the solution of the problem and a.s.s. (15» by considering u(r .. ) as an element of the Hilbert space hI(R). To this space belong functions W ELI (R), which satisfy the conditions
I:
Ii""
)o(x) dx
dx /"" w(y)
= O.
II,
Then the a.s.s.
I > ,u(r)
= (I + t) 1i2.
(15)
becomes the function
11'(1. x)
= 1I(,u(t). x)
==
that is,
(5),
as
us(l. x)
 IIs(,u(l). x).
1> 00.
Then
.""
/
.... '" w(r.x)dx=O, 1>0
/""
L"(R).
w(y) dy E
dvl < oo.ll~ /~ dx
11'(.1')
£lvl
(16)
(since by assumption II and 11.1' have the same energy) and 2: O. The function II' satisfies the equation
< 00.
= (v,
(d"ldx")
h
I (R)
for all
(17) WI
W)I
11' E
I
= 1,u'(I)(k(II)II\) 
(liS),],.
In the usual way we can introduce in this space the scalar product (v.
in the problem (II), (2).
= ,u'(t)(k(lI)lI,),.
IIS(,u(l). x)
Let us set
1>00.
satisfies the equation
Let us introduce, as usual, the similarity representation of the solution of the problem (II), (2),
= (/>(1)11(1. (c/J(I)).
O.
'(RI >
Let us make the change of variable
and the law governing the rate or change of the width of the temperature profile
O(r. ()
51
Taking the scalar product of this equation with parts in the righthand side. we obtain
1111 ).
where the function W = (d" I dx") 11IJ is the solution of the problem d 2 Wid x" = w, x E R; IW(±oo)1 < 00. It is not hard to verify that by (16) and (17) a solution of this problem exists. We shall denote by II . II" '(RI the norm in h I (R):
2
(d" I dx ) 111'
Id, (,u(l)(k(II)II,)(IIS)\. ' llwlli'IH)=
2 dl
I
and integrating by
(d)l) _1 m
•
(
1II.
(18)
X
It is easily veritied that (,u'(l)(k(II)II,)" (11.\),. (dldx) 1(11'_ us»)
==
I (~) I wll = II ('" d.\ 1.'i1~)' .\
Proposition 6. LeI colldiliolls (12), (1:1) hold. alld leI 11I0relll'er, 110(')  Is(') E h'I(R). Theil (p(l)
where F(II) k(T/) dT/. Since the first term in the righthand side is nonpositive, cstimating the second one lIsing the CauchySchwarz inequality, we obtain from (18) that
large I, Ivhere ,u ,·1 dell oIl'S IheJilllclioll illverse 10 Ihe 111(1/11110111' illcreasillgJilllClioll
r }t(1) =./0
1+1
dT
kl(l
+ T)
1111:::: kl(1
+ I)
~_
112,'
1>00.
2 For
..
the proof it is sufficient for this equality to hold for large r > O.
*3 Asymptotic stability of selfsimilar solutions of nonlinear heat equations
II Some quasilinear parabolic equations
52
Example 1. Let k(lI) = lIn 111" for small II > 0, a == const > O. As we already mentioned, conditions (12), (13) are satisfled. From Proposition 6 we obtain in this case that
Hence it follows (see the proof of Proposition 4) that
Ilwll"
I(RI
S Ilw(O, ·)11"
+
'iR)
+ (2qS)1/2 f'
(I
./n
qs
+
7) I12H I12 Us(O)(I
= sup If'\(()I
<
53
+
(21)
( 19) 7)1 12 ; 7) d7,
and therefore a.s.s. (15), to which the solution of the problem ( II ). (2) converges (I/o satisfles (3)), has the form
An estimate of the effective width of the inhomogeneous temperature profile for large times is given by (21). In the next section we move on to analyze selfsimilar solutions of nonlinear heat equations.
+
1/2 H I/2 Us(O)(1
+ 7)
1/2: 7)(17.
(20) Resolving consecutively all the indeterminacies that arise in the righthand using the equality
§ 3 Asymptotic stability of selfsimilar solutions of nonlinear
heat equations we obtain
Let us consider flrst the example of a sclfsimilar solution already encountered in Ch. I, which exists for arbitrary coeflicients k(lI) ~ O. lim (I
S 32q\ /
+ 1)1 12
,'X.
::::: 32qs
, hm
i
lIlO )(I+1I
,,2 1/L'(t)k(T])  II" dr]
.0
/'fIl
/'''':';.0
OI
1 A selfsimilar solution with constant temperature at the boundary
{kl((I +I), I12j  I }2 d( == O. kI (I + I) I/_j
Convergcncc to a.s.s. now follows from condition (13).
Remark.
=
o
If in addition to (12), (13) we also impose the condition I as II > 0, then asymptotically wc have
k(lI)jk(lIk l12 (II)) .....
This cxample helps us to emphasizc a fundamental property of selfsimilar solutions of nonlinear heat equations: their asymptotic stability with respect to perturbations of thc initial function. As in I, let us considcr the boundary valuc problcm in R) x R) for thc equation (I) 1/, = (k(u)u,),
*
(k(lI) >
0 for
II
>
0 is a sufllciently smooth function) with the initial and boundary
conditions 11(0. x)
This relation will hold, in particular, for the coefflcient (14).
*3 Asymptotic stability of selfsimilar solutions of nonlinear heat equations
II Some quasi linear parabolic equations
54
For arbitrary k(u) equation (I) admits a selfsimilar solution which satisfies condition (3): (4) US(I,X) = g.d() , (=x/(I+I)II".
55
the estimate (6) holds for u(t, x) as I + 00. In this case it is convenient to prove I asymptotic stability of the selfsimilar solution in the norm of the space h (R+). The Hilbert space h I(R+) is the space of functions v(x) E L1(R+), which satisfy the conditions
where g.d() solves the problem I
f
r
(k(gs)gs) +
I ,
'2 g.\( =
(7) 0,
(> 0,
gs(O) = I.
g\(oo) = 0;
(5)
this solution will or will not have compact support depending on whether equation (I) admits finite speed of propagation of perturbations, or docs not. Below we restrict ourselves to the analysis of the case when the coefficient k satisfies the condition for finite speed of propagation of perturbations: 1 [,
.0
The scalar product in h
I(R+)
(v. w)
(' vCr)
I
.Ill
where we have denoted by W
k(TJ)   dTJ < 00. TJ
d 2W/dx 2
and we take for gs(() a solution of the problem (5) with compact support. We shall assume that Uo in (2) also has compact support. Existence of a selfsimilar solution of the form (4) is related to in variance of equation (I) for arbitrary k(u) under the transformations I ~ I/a, x ~ x/a I12 ; a > O. Therefore, if u(t, x) is a solution. so will be u(t/a, x/cr l /2). Let us try to find a solution which is invariant under these transformations, that is, such that u(t, x) == u(t/a, x/a I12 ) for all (f > n. Setting in that equality (f = I, we obtain u(t, x) == u( I, X/I I /2). Denoting uO, (J by gs((J and using the change of variable I ~ I + I. which docs not affcct the form of thc equation, we obtain (4). Clearly, (4) is a solution of the original problem (I )(3) only if Uo == us(O, x) = gs(x). Below we shall show that for any perturbations of initial function with compact support UIl(X) the asymptotic behaviour of solutions u(t, x) for large I is described by the selfsimilar solution Us. Therefore the law of motion of the halfwidth of the selfsimilar tllerrnal wave, determincd from (4): (6)
rcmains valid as I '> 00 for other solutions of equation (I). Therefore the dependence of the wave speed on time is the same for equations (I) with a wide class of cocfficicnts k(u). Formulae (6) for different coefficients k(u) differ only by the magnitude of the constant (1'/ ' which, of course, depends on the form of k(u). Let us introduce the similarity representation of the problem,
[(_:!.~)
I
dx
= (_d 2 / d x 2 )
= U!, x>
I II!
0; W(O)
w1
(8)
(x) dx,
the solution of the problem
= O,IW(oo)1
< 00.
It is not hard to check that if (7) holds, a solution of this problem exists ana is lJllIque: W(x)
= (' .III
tty
IX 1J)(~) d~, x ::: O.
.\
The norm in h I(R+) is deflned using (8):
Ilwll' 'IR,) l
= (w,
1/2 w) I '
that is,
Ilwll" 'IR.I ==
1\ ( 
l;~r) Inl'IR.1 = II/X
w(y)
d.\l.'IR.I·
In the norm of h I (R+) convergence of g(t, .) to g.d·) is especially easy to prove (naturally, it also holds in stronger norms; see the bibliographic comments). Convergence in h I (R+) implies, in particular. pointwise convergence almost everywhere. Proposition 7. LeI uoCr) he a.tllllclioll wilh compacl supporl.
Proof The function ~
and show that g(t, () , gs(() as I ~ 00. This ensures that the main properties of the solutions u(t, x) and U.l'(t, x) are similar for large I, so that. in particular,
has the form
~f
::= II 
::=
Us
77Wl
satisfies the equation
lk(II)II,  k(us)(u.d,
J."
I >
0, x> 0;
(9)
~ 3 Asymptotic stahility of selfsimilar solutions of nonlinear heat equations
II Some quasi linear parabolic equations
56
moreover. z(t. 0) = 0, z(t, x) has compact support in x and z(t • .) E h I (R+) for all I :::: O. Taking the h· 1(R+) scalar product of equation (9) with z and integrating by parts. we obtain the equality I d
2:
J
dl II 7. II i, 'IR,
1
=(F(lI)  F(Lls). 1I  liS)·
== .
k(7]) d7]
is a monotone increasing function. Therefore we have from (10) that 'iR.l :::
Ilz(O. ·)11"
(F(lI)' F(lIs). 1I lis) ::::
==
(I
1I1 (t)
'IR. 1==
1I 1111(')
+ 1)'/41Ig(t,·) 
'i1~, I
 gs(·)II" 'fR.1
Obviously, there is no need to discuss here asymptotic stability of the selfsimilar solution (4) with respect to perturbations of the coefficient k, as to each k corresponds a different solution of the form (4).
2 The nonlinear heat equation with a power type nonlinearity
In this subsection we consider certain selfsimilar solutions of the boundary value problem for the quasi linear parabolic equation
lI(O, x)
= 1I11(X)::: O. 1I(t.0)
x>
0;
(T
= const
>
0,
x> 0; II;~+I E CI(R,).
= 1I1(1)
>
O.
I>
> O.
= (I + I)"'OS(~). ~ = x/( I + l)'li ""fin.
(\4)
(if 1I is invariant, that is, if 1I(t. x) == alllll(lla. xla(ltlllffl/C), then. setting a = I and then by the change of variable I .... I + I. we obtain (14)). The function Os(f) in (14) satisfies the following ordinary differential equation. obtained by substituting (14) into (II):
a')' (0\.0.1'
::: IIl(O. ,)11" 'iR.l( 1+ 1),/4
= (lI a u,).,. I > 0,
= const
( 15)
g,{·)II" '(R.I'
o
lI f
> 0; m
Then equation (II) has a selfsimilar solution of the following form:
we obtain the required estimate of the rate of convergence: IIg(t .. )  gs()II"
= (I + 1)111 ,I
0 and then
for all I > O. Since in view of (4) and the way we defined the similarity representation g(t. (), we have the identity 11;:(1. ·)II"'IR.I
As in § I, let
which can be related to its invariance with respect to the transformations
.11
1I;:(t. ')1111
1 A power law bOlllldary regime
liS (I , x)
l"
where F(lI)
(10)
57
O.
(II) (12)
, + I +m(T 2  0 \ t;
 mHs
= 0, t;
(\6)
> O.
where, as follows from the formulation of the problem and the spatiotemporal structure of the solution (14), the appropriate boundary conditions are (\ 7)
0.1(0) = l. Os( (0) = O.
A generalized solution of the problem (\ 6), (\ 7) exists, is unique and has compact support. This is not hard to see by transforming (16) into a first order equation (see eh. 111) or by first proving local solvability close to the point of degeneracy and then extending the obtained solution up to the point f = 0 ("shooting" to the first boundary condition in (17) is done by using the similarity transformation, which leaves equation (16) invariant). For 111 == lifT the problem (16), (17) has the obvious generalized solution Hs(~) = [(I  fT I12 t;l,t". In this case /Is = (I + 1)1/" Os(f), ~ = xl (I + I). and therefore the selfsimilar solution is just the travelling wave considered in Example 6 of eh. I. The depth of penetration of the thermal wave described by the selfsimilar solution (14) has the following dependence on time:
(13)
where thc boundary regime is strongly nonstationary: 1I1 (I) grows without bound with l. Some examplcs of generalized selfsimilar solutions of this problem were considered in the previous chapter. Let us make the prefatory remark that an equation with heat conductivity codficient k(lI) :;" kill/IT, where kll > 0 is a constant of. in general, physical dimensions (in (II) it is assumed that kll = 1), can be nondimensionalized by a change of variablc of the form I )0 kill.
·.1'(1)
.\ <'j
tell "' .. ./
+1)llt llllf l!2 • I~' /
O', 0·(1::) .\ C,d
=
1/2.
( IR)
The wave moves at a higher speed than in a medium with constant heat conductivity and the same boundary regime (§ I), since in (II) the thermal conductivity is an increasing function of temperature. This is also the speed of motion of the front . . I of. the therma 1 wave (l Ile pOlllt at wh'IC I1 /Is vallis les) X'(1"( I) = S,. I (I + I )11 ill/(fl/' , where ~./ = meas supp Os < 00. The evolution of this selfsimilar heating process
*J Asymptotic stability of selfsimilar solutions of nonlinear heat equations
II Some quasilinear parabolic equations
58
59
(if the perturbed equation admits finite speed of propagation of perturbations and is a function with compact support). Analysis of selfsimilar solutions discloses the physically reasonable principle: the more vigorous the boundary regime, the higher will be the speed of the resulting thermal wave. If the regime is of power type, then so is the depth of penetration; if the regime is exponential (as ( + oc the heating is more intense than for any power type regime), then the motion of the halfwidth is given by an exponential function. The following question arises: do there exist boundary regimes to which correspond "slower" moving thermal waves'? Such regimes exist, and to one of them corresponds a simple selfsimilar solution.
lIo(x)
c
Fig. 6. Evolution of the selfsimilar solution (14)
(Ill>
D. rr > I)
3 A pOIl'er type bOlllldary bl(}\I'lIp regime, Heat lo('a!i:a{io/l
is shown schematically in Figure 6. The trajectory of the halfwidth of the thermal wave is shown by the dashed line. As in I, this selfsimilar solution is asymptotically stable with respect to small perturbations of the functions lIo(x). 11\ (f), k(lI) entering the formulation of the problem (for the method of proof of such assertions see Ch. VI. 3. 4). Therefore the expression (18) for the halfwidth is asymptotically true for a large class of quasi linear equations (I) with coefficients k(lI) not of power type. which are close to lilT as II + 00.
*
*
Let the dependence of the temperature on the boundary x ::::;: 0 exhibit finite time blOWUp: (20) 1I,(f)::::;: (To  t) l/lT, 0 < { < To. where 0 < To < oc is a constant (blowup time), The boundary function in (~O) becomes infinite in tinite time: III (t) + oc as I + To this regime corresponds a selfsimilar solution of ( II) of an unusual form, a stalldillg {herm(/l \I'(/\'{':
To,
(21 ) 2 E\pollell{ia! bOlllldary regime
A different asymptotically stable selfsimilar solution of equation ( I I) exists in the case 111 (t) = 1'1 for ( > O. Here liS has the form 1I.\(t,
The function
Is ::: 'tT
.,
Us!s)
x)
= c/fs(Tf) , Tf = xl expllTtI2}.
( 19)
0 satisfies the boundary value problem I
(T
_/.
+ 2.1sTf .Is = 0, Tf
.
> 0, '!s{O)
=
I, fs(oo)
= o.
(19')
solvability of which is proved as in the analogous problem for power law regimes. The nature of the motion of the thermal wave in this case is more or less the same as in Figure 6, the difference being that due to the more vigorous exponential boundary heating, the halfwidth of the wave grows with time faster than any power: x;~f(t)=Tf"fexpllTtI2},{ >0 (/,(1].. /)= 1/2). Due to asymptotic stability of the selfsimilar solution (19) this estimate holds for large ( for a large class of nonselfsimilar solutions. The same is true about the law of motion of the front point of the thermal wave:
where Xo = 12(lT + 2)llTjl/2. The position of the front point in (21), xJ(t) == Xu, is constant during all the time of existence of the solution I E (0, To) and heat from the localization domain x E (0. xo) does not penetrate into the surrounding cold space, even though everywhere in the domain (0, xo) the temperature grows without bound as { + To· A schematic drawing of such a heating process (heat localization in the Sregime) is to be seen in Figure 7, which shows the essential difference between the influence on a nonlinear medium of a boundary blOWUp regime (20) and of ordinary regimes (see Figure 6). The depth of penetration of the localized wave is. just like the position of the front point. independent of time; from (2 I) it follows that x;\r(f) == xo( I  2 lT / 2 ). 0 < I < To· The selfsimilar solution (21) is asymptotically stable. In Ch. V we shall show that the heat localization of boundary heating regimes which exhibit finite time blowup occurs also in arbitrary nonlinear media described by general heat equations of parabolic type. It is important to note that not every boundary blOWUp regime guarantees heat localization. For example, if we take a different power type regime: 111(1)
= (1'0 t)",
0 < {< To.
(22)
~
II Some quasi linear parabolic equations
60
61
4 Quasilinear heat equation in a bounded domain
where the constant ~ .. r E R+ is slll:h that HS(~"I) == 1/2. For II E (I/lT,O) we have that x;~r(l) ~, 0 as I ~ 1'(;, so that the halfwidth (in a certain sense the depth of penetration) of the thermal wave decreases during the heating process down to zero. A detailed analysis of the localization phenomenon in boundary val4F problems for heat equations is presented in Ch. III (for equation (11) for IT::: 0) and in Ch. V (for arbitrary nonlinear heat equations). Equation (II) has a number of other interesting selfsimilar solutions (see Comments). Let us present, for example, an interesting invariant solution, which espccially clearly demonstrates localization of a thermal wave front under the action of the S boundary blowup regime. It is not hard to check that equation (II) has the following exact generalized solution: ,
11,(1. x) =(1'01)
Fig. 7. Evolution as
1 ,
T(l' of the localized selfsimilar solution
(21) (S
blowup regime)
(23) where Os ) () satislies an ordinary differential equation. For II < 1/ (T the function Os«(J has compact support, ~J :::0 meas supp Os <. 00 (see Ch. Ill). Then it follows from (23) that the front point of the thermal wave moves according to
and x}(1) , 00 as ( , T(l' Evolution of the thermal wave in this case is not substantially different from that of Figure 6; however, the heating of the whole space {x > OJ to infinitely for high temperature takes only a finite amount of time (lIs(l, x) > 00 as > all x ::: 0) _ For II <  1/ IT the boundary regime (22) is called the HS ii/OWlip
To
regime. On the other hand, if II E (1/ IT, 0), then it is the LS h!(}ll'II{I regime, which leads to heating localization. Furthermore, from the spatiotemporal structurc of the selfsimilar solution (23), unbounded growth of temperature as I > To occurs only at the point x :::: 0; everywhere in the space {x > OJ it is bounded from above uniformly in t E (0, To). This is indicated, in particular, by the law of motion of the halfwidth of the thermal wave:
(24)
1
where Xo :.:: [2(IT + 2)/ITI I12 . It corresponds to the initial function 11.(0, and a boundary regime which is close to a power type one: 1I,(I.O)=(To I)I/fT
where II <. I /IT (in (20) II :.:: I /IT). then there is no localization. To the regime (22) corrcsponds the selfsimilar solution
, ] Iff!
1/" [ (lx/XO)2_(II/T o)/I"I1
x)
""]1/" [ I(II/To)/I"II
==
0
(25)
and obviously 11,(1,0)
= (To 
I) 1/"( 1+ o(
1)) as
I + To,
so that this is indeed a boundary blowup Srcgimc and the solution (24) grows without bound in the localization domain x E 10, xo). However, the front of the thermal wave. which corresponds to (24), is not (unlike (21) immobile. It moves according to xj(l):.:: xo[1  (I _1/T o )I/(,,,2'!. (E 10, To), the wave is localized and xj(l) > Xo as ( , T(). By comparing (21) and (24) it is easy to see that close to the blOWUp time I = To, the solution II, (I, x) is close to the selfsimilar solution (21). In Ch. IV we shall show that this selfsimilar solution is asymptotically stable not only with respect to small perturbations of the boundary function. as in (25), but also to perturbations of the nonlinear operator of the equation. that is, of the thermal conductivity coeflicienl.
§ 4 Quasilinear heat equation in a bounded domain In this section we consider other problems for the nonlinear heat equation in the multidimcnsional case: II, :.:: j.1I,,1 I, IT
= (onst > O.
(I)
•• •• •• •• •• •• •• •• •• •• •
•• •
•• •• •• •• •• .! •• •• ••
•• •• •• •• •• ••• •
•• •• •• ••• •e•
•• •• ••• •• ••
•e, ••
§ 4
II Some quasi Iinear pambol ie equations
62
Let n be a bounded domain in R N with a sufficiently smooth boundary that in n an initial heat perturbation is given, 11(0, x)
= lIoCr)
:: 0, x E
n;
II;~ I' E C(O) n
an.
Assume
Proposition 8. Lei Ihe ilzitiallimctioll
(2)
where 0 < T I < T c <
00
1 The boundary value problem with Dirichlet conditions
11(1, x)
= 0,
I > 0, x E
an of the domain
Proof: Validity of (T c + I)
U:
au,
(3)
which corresponds to outflow of heat from the boundary (processes with the adiabatic "isolation" condition on the boundary are considered in subsection 2.) Clearly 1/(1, x) + () as I + 00 everywhere in n, as heat is taken away through the boundary. How docs the evolution of the initial perturbation proceed? At what rate docs the extinction process occur? These questions can be answered by analyzing the selfsimilar solution admitted by equation (I): 11.1'(1, x)
= (T + I)
I/"Is(.r), I > 0, x E
:0.
(4)
Here T > 0 is an arbitrary constant. Substituting this expression into ( I ) and taking into consideration the boundary condition, we obtain for Is :: 0 the following elliptic problem: (::"f~11
I + Is = 0, x IT
E U;
I(x)
= D,
x
E
an.
(5)
For any IT > 0 it has a unique solution, strictly positive in n (existence of the solution can be established. for example, by constructing sub and supersolutions of the problem; sec 17, 211) It turns out that (4) is stable with respect to arbitrary bounded perturbations of the initial function 1I0(X). that is. for 1+ 00 the expression (4) correctly describes the evolution of any heat perturbation. Without considering the details of this. let us restrict ourselves to proving a simple assertion. To describe the asymptotics of the solution. let us introduce. as usual, a similarity representation of the solution of the problem (I )(3) by the expression I(I,x)
= (I
1+ 00.
(6)
+1)1/"II(1,X). I> O.X E U.
Stability of the selfsimilar solution (4) will mean that I(I,
x) '*
(2) be slIch that
Is(.r) in
n
as
Hence
(8)
on
En,
(7)
are COllstlilltS. Theil
III(I.·) 
Let zero temperature be maintained on the boundary
1I0(X) ill
T~I/" Is(x) :::: lIoCr) :::: T I"/" Is(x). x
HI (il).
63
Quasilinear beat equation in a bounded domain
IS(')lIcllll
= 0(1
I) > 0, 1+
x.
(8)
follows from the comparison theorem. Indeed. by (7)
I/"Is(x)::::
11(1,
x) ::: (T,
+ I) I/,,/s(x) in
R
cx
n.
follows immediately.
(9) 0
Therefore if conditions (7) hold, the amplitude of the heat perturbation decreases at the rate sup 1/ (I, x ) .\(1I
= ( sup j .S (X )) I ····1/". + 0 ( t .. 1/,,) .
I > 00,
.,dl
and furthermore the maximal value of the temperature is attained at an extremum point of h·(.r). Thus in the framework of Proposition 8. the evolution of the heat conduction process for large times is entirely determined (in terms of the function /s(x)) by the spatial structure of the domain and by the exponent (J' in the thermal conductivity coefflcient k(lI) = (IT + 1)1/". The proof of convcrgence in the case of arbitrary I/o to follows in essence along the same lincs. We havc to show that aftcr a finite time 10 > 0 the tcmperature distribution u(lo, x) will satisfy (7), whence the estimate (8) will follow. Let us clarify this assertion (the arguments bclow illustrate an application of criticality conditions for solutions of parabolic equations, which will be L1sed systematically in Ch. V). Let the initial function 110 E Cdl). I/o t 0, be sufficiently small and have compact support in n: supp 110 C n. Then the lower bound of (7) does not hold for any T, :> 0, since Is(x) > 0 in n. Let us show that we still have stability of the selfsimilar solution in the sense of (8). The equation for the similarity representation (6) has the form
n
~£ = (::,,1'. rr11 + ~
I u·
r. T
:>
O. x E
n; .t'= 0, T
> 0,
x E iiU,
(10)
where we havc introduced the new "timc" T ;::; In( I + t): R r .> R+. Since I\(x) satisfies the problem (5), the cquality (8) has the interpretation that as T + x. the solution of (10) stabilizes to its stationary solution. which, as we have mentioned already. is unique. For simplicity, let 0 E nand 0 E supp uo. Let us consider the family of stationary solutions v = v(r). r = lxi, of equation (10): I
(r rN I
N
I (l'"j
I
)')'
I + v = O. u
(II)
*4 Quasilincar hcal cquation in a boundcd domain
II Somc quasilincar parabolic cquations
64
65
which satisfy for I' = 0 the condition v' (0) == 0 (condition of symmetry with respect to the point I' == 0) and v(O) == Vo == const > O. The solution of this Cauchy problem for the ordinary differential equation (II) exists and is strictly positive in some ball B," == {I' < 1'01, where 1'0 == ro(vo) < 00. such that v(ro) == D. Here ro(vo) + 0 as Vo ...... 0 (see § 3, Ch. IV). Let us choose VI) so small that B," C n. Then we claim that the solution of equation (I D) with the initial function
Due to diffusion all inhomogeneities of the initial perturbation will be smoothed out with time, and as a result as I + 00 the temperature field must stabilize to a spatially homogeneous state. Its magnitude is uniquely determined from (14), and therefore we can expect that
( 12)
Without giving the detailed proof of ( 15), let us make some clarifications. using only two standard identities satisfied by the solution of the problem (I), (2), (13). The first of these is obtained by taking the scalar product in L"(n) of equation (1) with II cr + I and integrating by parts:
is critical: ilI/il7 ?:
0 in R, x
n nIx E n I f(7.
x)
> OJ.
This is a direct consequence of the Maximum Principle (see Ch. V). Therefore the function f(7. x) does not decrease in 7 everywhere in n and, if Vo is small. is bounded from above by the stationary solution fs(.r). Therefore at each point x E n there exists the limit f(7. x) ~ f,(x), T + 00. Then, by thc usual Lyapunov argumcnts (see § 5, Ch. IV), wc can provc that thc limit function j',(x) has to coincide with the unique solution of the stationary problem (5). As far as arbitrary, sufficiently small initial perturbations of /(0. x) are concerned, note that under each of these we can "place" the indicated critical solution. which, by the comparison theorem, by stabilizing to the stationary solution, will force stabilization to it of any other solution lying between itself and the stationary solution. Thus the selfsimilar solution provides us with information concerning the behaviour for large times of a wide variety of solutions of the problem for more or less arbitrary initial perturbations. Let us emphasize that the asymptotic spatio'temporal structure of solutions of the problem (I )(3) depends in an essential way on the geometry of the domain n. A slightly different situation arises in another boundary value problem for equation (1).
2 The boundary value problem with the Neumann condition
Let now the no heat flux condition
all" I 1/iJll == D.
I > D. x E
an.
( 13)
be imposed on the boundary. Here iJ/iJll denotes the derivative in the direction of the outer normal to an. It is not hard to foretell the asymptotic properties of the solution, based on physical intuition concerning the behaviour of diffusion processes. By the adiabatic condition ( 13 l. the total heal energy in n is conserved:
1/(1. x) + _ _1_
I
.Ill
11(1,
x) dx
== I'
III
110(.1')
dx
= Eo.
( 14)
d
crt "
+~) ( , 11 11 (1)11/"'''1111 IT
.:..
(/
,
1I0(.\") dx
==
'V
IIII"
== Ii"".
r t
(15)
I , 00.
I'
(1)11'.'llll' I?:
()
(16)
.
The second one is derived by multiplying the equation by (11"11), and then integrating the resulting equality in I (see § 2. Ch. VII). As a result we have
+ I), + 2)
4(lT
/./ Ii (1/ 1+"/2'," /. 1 <7 crl·1 (I) 112I." III I == ) I (,1 ) II L' Il! I l ,I .+::; 11 Y 1/
(IT
.0
.:..
( 17)
== 2'I 11<7YII ocr I 1 II"12 Ill)' Passing in (17) to the limit as so that the limit
I ..... 00.
we see that the tirst integral converges. (18)
exists. Comparing (18) with the equality (16), we obtain ao == 0; otherwise the > () is'negative for larne I function Iltlll,n", 1:" '1111 . '" . The condition ao == D in (18) means that 1/,,11 (I, x) converges to a spatially homogeneous state almost everywhere (in fact, by sufficient regularity or the generalized solution, everywhere in n). Then the energy conservation law (14) guarantees stabilization (15). lt is not hard to derive an estimate of the rate of stahilization to the average value of the temperature. Proposition 9. We h(/\'e Ihe cslimalc 1111(1.') 
where
I'
f
meas! 1 .Ill
domaill
fi,",lii.!llll
K > 0,
n.
/I
== I'
III
(11(1. x) ..· [i",,)' dx ::: Ke /'/ . O. 1  ' 00.
~ 5 The fast uiffusion equation. Bounuary value problems in a bounueu uomain
II Some quasi linear parabolic equations
66
§ 5 The fast diffusion equation. Boundary value problems in a
Prol!j: Let us take the scalar product in L2 (n) of the equation (I) with u. Then,
bounded domain
after integrating by parts, we obtain I d, 2,llu(t)1I7."lll ( I
= (tr+
/.
I)
,
In this section we shall consider properties of solutions of quasilinear parabolic equations of nonlinear heat transfer with coeftlcient k(u) > 0 which grows unboundedly as u > O. These are the socalled fasl d!rtilsion equalions. These include the equation with the power type nonlinearity
(20)
u"(I,x)I'i7u(t.xWdx.
. 11
1/ to fi,II' > 0 as I > 00, there exists I. ~ 0, such that for all I > I. we have the inequality u(l, x) ~ fi,"/2 in n. Then. estimating from above the righthand side of (20). we obtain
In view of stabilization of
I d , ;;, 1II/(I)lli'il!l:::: (rT+ _ I I
Setting
II 
11(/1'
= w, we
I)
(Ti'" ~)
substitute in (21)
r
.Ill
I
,r
. II
.0
11=
'
fi,", +
1/1.
By
u, =
(21 )
l'i7u(t.x)ldx. I> I•.
(22)
Then, since 'i7u =: 'i7w and d 1 d /. 1 .) , lIu(r)lIi.'illl =: I (ur + 11;11' + 2Ti(/!,w) dx = I
I
I I . II
d{l
=1 ( I
I'
wdx+ 1
. II
.
l!
fi~t'dx+2fi",.
I } =:1I11'(1)1Ii,:il!I' d wdx
1
dl
II
we derive from (21) thc estimate
d
1
dIll w(t) II i.'il!1 :::: 
2(IT
+ I)
(11(/1')"
2
(23)
1
II 'i7w(l) Ili.'llll·
Using the wellknown inequality [3621
= 111 
III < I is a constant (if, as usual we set I E (1.0)). The heat conductivity k(lI)
d
1
dl II w (r)IIi!(l!1 :::: 2(IT + I)
which coincides with the estimate = 2(IT + I )A I (n)(fi,",/2)" > O.
v
= lIoC\)
> O.X E
n:
()9),
if we set K
= II w(t.) II i.'ll!1
<
00
and 0
'
I. then in this case grows without bound
(2)
I/o E C(O),
(3)
In this problem we have lolal eXlinclioll i/l fillile lillie. This is relatively simple to prove by constructing the selfsimilar solution
= 1(1'0 1)+ \1/1,·/11) fls(X),
To
= const
> O.
(4)
The function (4) is such that liS == 0 for all I ? To. Let us note that for the derivative alls/al has no jumps at I == To, so that liS is a classical solution. Substituting the expression (4) into the equation (I) and taking into account the boundary conditions, we obtain for the function PS > 0 the elliptic problem: I (5) !:lp:~! +   P s = 0, x E n; Ps 0, x E an.
III E (0, I)
II w(t)lli'I!I):::: II w(t.)lIi',!li exp{2(IT+ 1);\,(111/1./2)"1\. I> I•.
111
u(t.x) =0.1 > 0, x E an.
IIs(l. x)
Hence
= IT +
> O. The name "fast diffusion" is related to the fact that since the heat conductivity is unbounded in the unperturbed (zero temperature) background, heat propagates from wann regions into cold ones much faster than, say in the case of constant (111 = I in (I)) heat conductivity. and even more so than for III > I. where we have finite speed of propagation of perturbations. This superhigh speed of "dissolutiqn" of heat implies a number of interesting properties of the process. We shall descrfbe these in some detail, using mainly the technique of constructing various selfsimilar solutions of equation (I). As we have not encountered such equations before. let us make the preliminary observations that for III E (0, I) solutions of boundary value problems and of the Cauchy problem exist. are unique and satisfy the Maximum Principle: in particular, comparison theorems hold. Here. wherever this docs not contradict the boundary conditions, the solution can be taken locally to be strictly positive and therefore classical (see the Comments). Let us consider for (I) the boundary value problem in a bounded domain n (an is its smooth boundary) with the conditions
11(0, x)
(fi,", , 2 )" Alllw(t)lIi'il!I' I > I•.
III
= 11111
1/
II 'i7w 11;'il!1 ~ Alllwlli:'lll' which holds for all functions w E H'(n), aw/illl = 0 on an, which satisfy the condition (22) (here AI = AI (n) > 0 is the first eigenvalue of the problem ;),,'1' + AlII = 0, x E n, at/,/all = 0 on an). we obtain from (23) that
(I)
!:llllll,
where 0 < IT
as
(14)
w(t,xldx=:O.
67
I 
11/
=
~ 6
II Some quasi linear parabolic equations
68 Setting p~~'
= lOS.
we arrive at the equation . I y. aw, + ·_·_w· = 0, I' = mI . Im S
(5')
> I,
with the same boundary condition Ws = 0 on iH L The function Ws does not exist for all I' :::: 11m; if N ::: 3 and I' ). (N +2)/(N2), then the equation (5') has solutions that are strictly positive in R N , while for n a ball of arbitrary radius, there is no solution with the condition li'Slilll = () (see § 3, Ch. IV). On the other hand, if I < I' < (N + 2)/(N . 2),. the required similarity function can always be found. However. for our ends it is not essential for the problem (5) to be solvable. We shall use the selfsimilar solution (4) only to tind majorizing upper bounds for the solution of the problem (I )(3). Proposition 10. LeI 0 < III < I. Then for allY illilial/imction 110 ill Ihe prohlem (l )(3) Ihere is complele eXlinclion ill .finile lime: Ihere exisIs To > 0 sllch Ihal u(t, x) == 0 in Fir al/ I ::: To.
It is assumed that IlnC,) 4 0 as Ixl 4 00. Naturally. if this condition is not . . . met, and for example IInC,) ::: i5 > 0 everywhere In RN . then by tIle comparison theorem 1I(t. x) ::: 8 in R N for all I > 0, that is, in principle there can be no total extinction.
1 Conditions for total extinction in finite time Let us consider the selfsimilar solution, which describes the total extinction process in the Cauchy problem. We can derive a whole family of such solutions: liS (I, X)
n
n
I" ps (s1),:?;' = X. /[(l' II
m IIII' . I)I 111'"l

(3)
.\
uli's 
1+II(mI)" 2
1//11.
vli's'?;
+ IIW SI/m = () , .,J...··ERN.
(4)
For our ends, it suffices to consider radially symmetric solutions, which depend on one variable, T/ = I~l All these satisfy a boundary value problem for an ordinary differential equation,
l
,
n.
() ::: 1I(t. x) ::.: [( 'I'll p
= [ ( l ' 11' I)"
where 'I'll > 0 and /I > I are constants, Substitution of (3) into (I) gives the following elliptic equation for lOs = p~' > 0:
n
Proof If 111 E ((N  2)/(N + 2), 1), N ::: 3, or m E (0. 1), N < 3, let us take an arbitrary bounded domain n ', such that c n' and let us denote by Ps(.,) the solution of the equation (5). which is positive in n' and satisfies ps :::: 0 on nn'. Then, since c n we have Ps > 0 on an and therefore we can always find To > 0, such that uo(x) ::.: Ti/ I I· III} Ps(x), x E By the comparison theorem we have
69
The Cauchy problem for the fast diffusion equation
I) 1 J1/11 /II} ps(.r) , x E
_I__ (T/v I w~\)'
T/N
_
I
.!_::!=.~~~II
 I) (W.V/II)IT/
+ IIW~//II = O.
T/ > O.
n.
lJ)~\(O) = 0, 11'.1'(00) = O.
n
and therefore u == 0 in if I > 'I'll. If on the other hand m E (0, (N  2)/(N + 2)j, N ::: 3 (I' = I/m ::: (N + 2)/(N  2) and the boundary value problem (5') can be insolvable), we take as ps(x) the solution of equation (5) which is strictly positive in R N Then ps > 0 on an and the same argument applies. [J
(5)
I
(6)
This problem (in fact. just as (4)) is solvable not for all m E (0, I), II > I. Lemma l. LeI N ::: 3, 0 <
Proof Let us consider the Cauchy problem for (5) in R, with the conditions
§ 6 The Cauchy problem for the fast diffusion equation w(O)
Let us see, whether it is possible to have tolill exlil/Cliol/ il/.!inile lime in the Cauchy problem for the fast diffusion equation II,
=
~II/IJ ,I > O.
11(0, x) :::: uo(x) >
X E
RN :
0, x ERN;
11/ E
(0. I),
SUpllO
< 00.
(I)
(2)
The situation here is more complex than for a boundary value problem in a bounded domain; however, it can also be analyzed using selfsimilar solutions.
= fL. w'(O) = 0,
(8)
where fL > 0 is an arbitrary constant. Let us prove that every solution of this problem delines. under the above assumptions, a required function Ws· Local solvability of the problem (5), (X) for small T/ > () is established by considering the equivalent integral equation. ·\)bviuusly. in this case 1/ classical sulution 111 R I x R''I.
*6 The Cauchy problem for the fast diffusion equation
II Some lJuasilinear parabolic equations
70
Let us show that this local solution can be extended to the whole positive semiaxis T/ E R t  and satisfies the second of conditions (6). First let us note that the solution is monotone decreasing in T/, since assuming that at some point T/III > 0 the function W has a minimum (W(T/III) > 0, W'(T/III) = 0) leads to a contradiction; this follows from the form of the equation. Assume the contrary, that is. that the function u' vanishes at some point T/ = T}. > O. so that w(T/) > 0 on (0, T/.) and w(T/.) = O. Clearly. w'(T/.) :s o. Integrating equation (5) with the weight function T/NI over the interval (0. T/.). we obtain the equality
It is convenient to formulate an optimal condition on the initial perturbation which ensures total extinction. employing a solution of equation (4) of special form. It is not hard to check that for 0 < 111 < (N  2)+/N there exists the solution lIo(x).
Here P.\(O + ex; as ~ + O. which. as will be seen below. is not essential. This function corresponds to a solution. which becomes extinguished everywhere (apart from the point x = 0).
(9)
ui·(I. x)
where we have introduced the notation
e(N. erN,
Ill.
= ICT o _
Ill. n) = ~ [n (N ~ Ill) _ 2
71
1)j l/ll"1II1
l
= [2111N (N  2 _
/ 111)]1 11
N
IIll
""Ixl
:~/IIIlil.
x
(II)
Using (II) as the majorizing solution in the comparison theorem. we obtain
if strict inequality in (7) holds. Therefore the equality in (9) is impossible, since its lefthand side is strictly negative. Thus. w(T/) cannot vanish. From equation (5) it follows then that w(T/) + 0 as T/ ~ 00, that is. W satisfies the boundary conditions (6). If, on the other hand. we have in (7) the equality /I = 1(N  2) /N  III I. then the problem (5). (6) has solutions of the form
Proposition 11. LeI N ::: 3. 0 < he sllch Ihat
111
< (N  2l/N. and leI Ihe inilialfilllclion IIO(X)
(12)
r
where rJ~ > 0 is an arbitrary constant.
111 '"(1"
111/2 . W II'" (T/I'j) s
Then Ihere exisls To > 0 stich Ihal
11(1,
x) == 0 in R N fill' all
I :::
To·
(10)
Corollary. For 0 < III < (N  2)+/N. i/l general. Ihere is no COll.1'elWllion (~f energy: if lIO ELI (R N land CIIndilion (12) holds, Ihen
o
(13)
The family of selfsimilar solutions (3) makes it possible to ontain. using the comparison theorem. a condition on lIO(.t) , which is sufficient for total extinction in finite time. For example, if tlo(.t) :s tls((). x) in R N • then 11(1. x) :s tls(l. x) in R, x R N , and therefore lI(I. x) == 0 for all I ::: To. Selfsimilar solutions (3) provide us with the following law of motion of the halfwidth of the heat extinction wave: T/,'! 1(7' 0  I),
RN\lOI.
I];
n) < 0
. ( I )1 I'\('1
E
= '2I 10\.11111 ( () l.
where, moreover. I + n(1I1  I) < 0 for all 1/ satisfying (7). Therefore IX<,((I)! > 00 as ( ~ Til' which agrees well with the property of fast diffusion processes mcntioned above: with ever increasing speed heat (lows out of the region into infinitcly distant regions. where the thermal conductivity coefficient is infinitely large.
Ihal is
1111(1.
·llll.'IR"'# lI t1 o()III.'IR'I·
:s
P/'O(I{ of Proposition II. By condition (12) there exists To > 0, such that /Jo(.t) lIs(O, x), x E RN\\OI. Therefore from the comparison theorem we obtain that 11(1. x) lIi(l, xl, I > D. x E RN\IOI and therefore 11(1. x) == 0 in RN\lOj for
= To.
:s
It remains to show that total extinction also occurs at the point x = O. For that it suffices to notice that the function liS (I . x  xo) where Xo :f 0 is an arbitrary point of R N. is also a solution of equation (I) in R) x IRN\(x = xo)}. and then compare 11(1. xl with this solution using similar arguments. 0
I
Example 2. Let us set
qldll)
= minlkll, 1I'1I}.1I::: 0;
k
= 1.2 .....
~ fJ
II Some quasi linear parabolic equations
72
It is clear that the functions cjJdll) are continuous for II ::: 0 and q)dll) k 4 00 for any II ::: O. Then the solution of the Cauchy problem for (lldl
= b..cjJdlld,
I >
4 11
111
as
N
0, x E R ,
with the initial condition (2) exists and is unique for any k. Since cjJ~(II) is not singular at II = 0, generalized solutions conserve energy:
Jr. R,
II d I,
x)
dx
=
r. JR'
I/o (.r)
d x,
I :::
0: k
2 Conditions for existence of a strictly positive solution Let us show first that for III <: (N .. 2)/N, N ::: 3, not every initial perturbation uo(x), such that lIoCr) + 0 as 1.11 ..+ 00, ensures total extinction in flnite time. This is established by constructing other selfsimilar solutions of equation (I) in R+ x R N , which do not have that property:
= explldT + I)lg~(~:),
g = 1.11/ exp{a( I . 1Il)(T + 1)/21,
T
~6 > 0 is an arbitrary constant.
Therefore for 0 < III ::: (N  2)+IN there exists a solution of the Cauchy problem that becomes totally extinguished (Proposition II) and a solution that does not. It is of interest to compare for which initial perturbations I/o one or the other mode of evolution will occur. Determining the asymptotic behaviour of the problem (15), (16) as ~ + 00, we obtain the following
Proposition 12. LeI N ::: 3. 0 < III <: (N  2) IN alld leI Ihe illilial./illlclioll 110( x) he slIch Ihal fiJI" al/ sl/fficielllly large Ixl
= I, 2, .
(the fact that q)~(II) has a jump discontinuity at II = k 1/111111 is not important, since, for example, we could smooth (!JA in a neighbourhood of the point of discontinuity of the derivative). Therefore under the conditions of Proposition II (see N the Corollary) the sequence lid!. x) cannot converge in the norm of L I (R ) to lI(I, x), the solution of the original problem (I), (2), which corresponds to k = 00.
us(l, x)
73
The Cauchy problem for the fast diffusion equation
(14)
= const ::: O.
N
*where It > D. Then liS ,. 0 in R as ( ....+ 00 and 11.\' > 0 everywhere. The function g.\ > 0 satisfies the ordinary differential equation
1I0Cr) :::
Theil 11(1,
x) >
0
ill
= const
Klxl C/11'I1I'\In 1.1111/11 "", K
R N fiJI' al/
I >
(18)
> O.
O.
°
Proof If (18) holds, we can always pick I t > 0 and T::: in (14), such that 110(.1) ::: lIS(O, x) in RN , and then lI(I, x) ::: 11.1'(1, x) in R I xR N , which ensures strict positivity of the solution (so that there is no total extinction). This same inequality allows us to estimate the rate of decay of the amplitude of the temperature prolile: it can be at most exponential. 0 Let us note that the "boundaries" of the sets ( 12) and ( 18) in the space of initial functions (in the Ilrst set we have total extinction, which is absent in the second one) arc very close and differ only by a slowly increasing logarithmic factor. Let us show now that the restriction 11/ E (0, (N  2), IN) is essential for total extinction in finite time to occur. Below we provide examples of positive selfsimilar solutions, which exist for 111 > (N .. 2) I IN and conserve energy. Let us seek these solutions in the form ( (9)
I
.N.1
gNI(g
.111/'
a(IIIl)
,
(g\)) +~(g,)g+ltgs=D,~>O, gs(O)
= D, g\(oo) = D.
(15) N
(16)
Exactly as in Lemma I in subsection I, we show that this problem has nontrivial solutions if III ::: (N  2)1 IN, thaI is, also in cases when total extinction in finite time is possible. However, (14) are strictly positive in R N for alII> O. In particular. if III = (N  2llN (the "critical" case), the problcm (15), (16) can be integrated explicitly and the selfsimilar solutions have a simple form: IIs(l, x)
=
/,.erl
= [
:r;~;,!12~ (~xP{;H~.7,~)11 + ~0)]
1/11
Hli
> D, I >
O. x ERN, ( 17)
where I < 0, T > 0 are constants. Here 1/.\ :> 0 in R for all I > O. Substituting (19) into equation (I), wc obtain for the function !/Is = fl~~'(TJl the equation
> 0
I + 1(11/  I) + ~ .._(W 1/1/1') T]
(20)
I
TJN ... I (TJ
N
I
"
w~)
Ws(O)
= 0, w\(oo)
1/1/1
lwI
= O.
= 0, TJ
> 0,
(21)
It is not hard to show that if the condition (N  2),./N < 111 < I is satislied for any I ::: [(N  2)1N  11/1 I < 0, there exists an inllnite number of functions ws( TJ) > 0, which satisfy (20), (21) (see the proof of Lemma I). In the particular case 1= [11/  (N  2)INI 1 there exists a selfsimilar solution (19). which can
be written down explicitly: IIS(t.X)=(T+t)N I1 2.fNIIII.III{
(IIll)
2/1ll/llN  (N  2)J x
X [712 + "0
1/11·
2
(1'
Ixl ·IN + t)2/[IIIN
211
]}
(T/~ = const > 0). It exists for all (N  2) ,.IN < which is conserved:
r
.lit"
11.1'(1. x)dx
= /.' . It'
lI,dO . .I)d.l
(22)
Ill)
I >
III
==
O.
X E
R
N
< I and has tinite energy.
IIPslIl.'IR"I'
The selfsimilar solution (22) is the analogue of a solution of instantaneous point energy source type, which was considered in Example 8 of Ch. I for the case III > I (that is, IT = III  I > 0). Thus. for III > (N  2)+/N there are solutions with conserved finite energy. In other words, in this case there is no absorption of heat in infinitely distant regions, which happens when 0 < III < (N  2)I.jN (Proposition II). Furthermore. using the selfsimilar solutions (19) and the method of proof of Proposition 4 of Ch. L it is not hard to show that in this case there is no finite time extinction and energy is conserved (sec Comments).
*7 Conditions of equivalence of different quasilinear heat equations
7 Conditions of equivalence of different quasilinearheat equations
The examples of really nontrivial and "nonobvious" strict equivalence are relatively few. For that reason their role in the systematic study of properties of solutions of quasi linear parabolic equations is, in general, not an important one. Nonetheless, this approach sometimes affords a considerable simplification of the problem. We consider below certain simple transformations that establish equivalence of different equations. We will not analyse in detail the structure of such transformations or discuss their constlUctive aspect: how simple is it to reconstruct a solution of one equation using a solution of the other? From the practical point of view this last question is very important: frequently it is easier to solve numerically the equation itself than to implement numerically the equivalence transformation. In most cases we shall deal with an equation. without posing a specific boundary value problem for it. and for that reason we will pay no attention to the behaviour of its coefficients. These have to be taken into account in the formulation of boundary value problems.
1 Simplest examples We have already encountered an equation which can be reduced by an equivalence transformation into a simpler one. This is a quasilinear equation with a linear sink: IIi = CUl tI + I  II, which can be transformed by a change of variables II = e'' v. tll I ('lTldl = dT into an equation without a sink: Vr = uv  . Using these elementary transformations we can establish localization of heat perturbations in nonlinear media with volumetric absorption. Let us consider another simple example. Example 3. The semilinear parabolic equation E"(II) = UII +,1\1111E'(u)' 1
Above, using a range of examples, we demonstrated asymptotic equivalence of solutions of nonlinear parabolic equations cOlTesponding to different boundary data, as well as equivalence as I , 00 of solutions of different parabolic equations obtained by perturbing nonlinear operators. The idea of this asymptotic equivalence (asymptotic stability of approximate selfsimilar solutions) will be widely used in the sequel. Here we consider the question of equivalence of equations, understood in a strict sense. Are there different quasilinear equations that can be reduced to each other by a certain transformation? In other words, is it possible to transform a nonlinear heat equation with a source ora sink into a simpler equation, one with better understood properties? In the general setting this problem is studied in the framework of the theory of transformation groups and is known as the Backlund problem (its precise formulation and constructive methods of solution are to be found in 1221\).
75
II f
(I)
where E : R+ > R +, E E C 2 , is an arbitrary monotone function, can be reduced by the change of variable v = E(II) to the linear heat equation lIt
= uv.
(2)
Let us consider more complicated transformations. Example 4. Let u with a source:
= 1I(t. r), u,
r
= Ixl.
be a solution of a nonlinear heat equation
I N. I t I ) tt,l = ;:NI.(r II UI'I'+u .
(3)
.
where IT # .. I. Examples quoted earlier show that solutions of equations with a source have properties that are significantly different from those of solutions of
~ 7 Conditions of equivalence of different quasilinearheat equations
II Some quasilinear parabolic equations
76
nonlinear heat equations. Let us try to get rid of the source in the righthand side of (3), so that only a diffusion operator remains there. To that end let us use the transformation y
= 4)(1'),
u"+1 (t, 1')
= r/J(r)v(t, y).
77
Proposition 13. III the equation (9) the transfiJl'll1atioll
Then it is easily verified that the function v :: () satisfies the equation
I
+;:N:I II' NI/'A' ",/) +(1' N
1//' , IIIJ)!1',+
[I
(4)
r
 (rN.III/)'
1
N
= (v
v,
To get rid of lower order terms in (4), we set I
remol'l's the source ill the riMhtlwlld side, IVhile the jilllctioll
" +(IT+I)rll] v. rN .. I·(r N _ I III)
l'
satisfies the equatioll (II)
.1/\\),.
For equations of general form
+ (IT + 1)111 = 0,
(5)
1/4/ + (rN.IIII
r N 11
II,
= (j,¢(u) + Q(II)
( 12)
(6)
there also exist transformations that remove the source term Q(u); however, here the resulting equivalent equation is no longer autonomous.
(7)
Example 5. Let us set in (12) new equation
If the function ifl satisfying (5) is known. then
I
=.
dr r Nl rfi2(r) .
In particular, if
ll)(r)
=
Let us choose the
f
In
exp{±!lT+ III'r},
= T ..
21lT
v,
I ___;; cxp(T21IT + 11 1/.
+
11 1/21'1.
III I1"HI/I" 1.11(1')
Selling
Vi /ltrlll
1/1"'111
= U,
) ,
== ;JIN~~II(lT
+ I) 'V",
Y
A, = ,/,(1').
we obtain the onedimensional equation without a source, U,
=
III 11,,·14 1/1,,'·11 (1') I
v(t,
xl we obtain the
that is,
dT/
  = (+ ("(v). Q(T/)
(U"U,),.
I
_
= :.11
v
( 13)
I:~.(f. v)
For example, in the casc of an cquation with powcr law cocfficicnts
If conditions (5), (6) hold. then thc equation for the ncw function v has the form (V
=
where ("(v) is an arbitrary function. After this change of variables we obtain for a paraholic equation, whose cocfficients depend on thc variable f:
I'
111(1')
"If""
,
=3
I
v
+ E>', = 11
+ (IT + I )K = O.
+ I < 0, then for N
(T
Then for
E;
The system (5), (6) can be solved explicitly, for example, in the case N = 1 (concerning this see below), as well as for N = 3, when by the change of variable 111(1') == K(r)/r equation (5) reduces to K"
= E(t, v).
II
(~)
It has a particularly simple form in the casc N = I. IT = 4/3. Then the system (5), (6) is easily solved. As a result we obtain the following
/I,
= t:.1I" + I + lI!i,
IT ::
o. f3
(14)
> I,
which will bc studied from different points of view in subsequcnt chapters, the transformation E has the form E(f, v)
= ICB
0_
I)(l'(v) 
1)1
I/I!i
II
It is convenient to choosc the function I'(v) so that for t = 0 the transformation is the identity, E(O. v) == v. This gives us I"(v) == v l /3 /(f3  I), so that finally we have I !i 1/1!i· II E(f,II)=IIl'
*7 Conditions of equivalence of different quasilinearheat equations
II Somc quasilincar parabolic cquations
78
Let us derive from (16) an equation for the function 1/1. It is easy to check that
Equation (13), which is equivalent to (12), then has the form
i~¢ = _ i~~ (i~I~)  I , ~¢ = (~1~) ill
Transformations of this kind turn out to be quite useful in the study of semilinear parabolic equations and will be employed in 7, eh. IV.
*
2 The "linear" equation
1I,
=
79
2 (1I I1 x )x
dl
(Ix
and therefore
Since
dx
I ,
~2 ~ = _ '.12y;2 (~f/~) :1 dx
dx
dx
iJx
I{il~al _~2_~} =0. ( ~I~) at dx·
(1//, )',1 i
0 (whicb is equivalent to the condition 11
i
0), 1/1(1, X) satisfies
the linear heat equation We move on now to more complicated equivalence transformations. Let us show that the nonlinear heat equation with coefficient k(lI) = 11,2 is equivalent to the linear equation. Let 1I(t, x) be a solution of the equation (15) such that 11(1, x) is a sufflciently smooth function which is not zero in the domain under consideration. Let us fix a point (to, xo). Integrating (IS) in x we obtain the equality
(19)
It is not hard to effect tbe inverse transformation and to show lbat a solution of equation (19) transforms into a solution of the original equation (15).
Example 6. Let us consider the fundamental solution of the heat equation (19): _ t/J(I, x)
{I
2
 41
}
¢(t,x)
= 141In(xt I I')1 I I'.
lI
However, 11 = ¢, is the solution of equation (15), that is, lhe fundamental solution (20) transforms into the following solution of the "linear" equation (15):
or, equivalently,
*{/" I
11I/2 exp
Equalities (171. (18) define the required function
iI /" 1I(t, y)dy = u 2(1, x)u\(t, x)  1/'2(t, xo)u,(I, xo), iJl"
=
1I(t, y)dy
+ l'
'\0
11 2(T, XO)U,(T,
xo)(IT}
= U· 2 I1,.
.11(1
11(1, x)
tin = ~
[ In ( XI I /2 )] 1/2
Denoting the expression in braces by
It makes sense for XI I/2 E (0, I). The equation with the coefficient
(from which it follows that el), function ep:
==
11),
we obtain a new parabolic equation for the ,.7
u
'
II 2
has other interesting properties.
Example 7. Let us consider the "multidimensional" equation
(16) It is easy to check that the same transformations
Let us introduce the new independent variables
= el)(I,
=
(21 )
iJep
:) = (¢ ,) c/J. (t ,I'
k(lI)
x),
1= t.
( 17)
1I(t.r)=r l NePr(t.r). I=t, i'=ep(t,r); r=III(I,f).
reduce (21) to the form
Solving the first equality with respect to x, we obtain ( 18)
(22)
*7 Conditions of equivalence of different quasilinearheal equations
II Some quasi linear parabolic equations
80
= 2 we
For N ::::: 1 we obtain a linear equation; for N
81
However, from equation (25) we have that
have the equation
[
' 11,(1, y) dy
= k(II)U, 
k(u)u,IIl,',,) ,
• ·\0
n
= In III this equation reduces to a onedimensional By a change of variable rt(1, equation with exponential nonlinearity:
and therefore (28) means that x,
iii
= ((,cl' rtf) ,
If, on the other hand, N > 3, then, setting equation where
)IN
= 2(N 
= k(u)u./u,
Then from (27) we obtain
t/!"'N we obtain from
(i
(22) the
_ IIi
FurthemlOre. since
X,
= Ilu,
u,
k(lI) ,
(29)
=  ' u0 + lII~, 11"
the other derivatives are easily computed:
1)/(2  N) < O.
(30) 3 Equivalence conditions for equations of general form (31 )
Below, using the same transformations, we show that to each heat equation corresponds an equivalent heat equation with a different heat conductivity coefficient.
For a power coefficient k(lI) = II", the equivalent equation has the coefficient K (II) ::::: 11'1"+"). which means that equations with coefficients k I (II) :::::
are ellllivah'IlI, illlO
(j
SOllllioll iI(i, .\.) of Ihe eqlll/lioll
rt r :::::
(_I'uk (~) iI ) ,,·1
,.
II"', k"(II) ::::: u'"
II
Let us compute the derivatives that enter equation (26). From (24) it follows that II, + II,X, (27) , (il(i, 0\)), == ( __ I ) = II'
,
(32')
\
Proof
11(1, x)
are equivalent if
(26)
,\
If IT, ::::: 0, then according to (32') IT" = ,2 and wc obtain the known result on the equivalence of the equation with k(lI) = II " and thc linear heat equation. Proposition 14 opens new possibilities for constructing particular solutions of certain equations.
has a wide variety of symmetries. For example, it is invariant with respect to the transformations ( + (/ a, x , x, II ;.  In a + II, that is,  In (1' + 1I(t / a, x) is a solution of the equation if it is also satisfied by 1I(t, x). Setting here Cl' = (, ( < 0, and then making the change of variables ( ;. (  To, To = const > 0, we see that (33) has the selfsimilar solution IIS{t, x)
= In(T o 
r)
+ O,,(x),
0 <
(34)
I < To.
Substituting (34) into (33) provides us with the following equation for the function 8 s (x):
Conditions of equivalence of different quasilinearhcal equations
Let us note that the same method can be used to construct an unusual exact solution of the superslow diffusion equation (37)
Its name reflects the fact that the corresponding thermal conductivity coefficient = 1I "e ti" changes for low temperatures 1I > 0 more slowly than any power. Therefore equation (37) can be formally considered as the limit as (T + 00 of the nonlinear heat equation 1I , = (lifT lI,) \' some properties of whose solutions were described in 4. If we consider for (37) the Cauchy problem with a continuous nonnegative compactly supported initial function lI(O, x) = 1I0(X) in R, then the generalized solution lI{t, x) will satisfy the conservation law k(lI)
*
I:
that is, Os(x)
= In(x" /2 + hx + C),
(35)
where 17, C are arbitrary constants. The above equivalence of (33) to the equation (sec (32))
83
1I(t, x) dx
== Eo =
I:
1I0(X) dx
for all ( > 0
*
(see 3 in eh. I). The exact solution given below satisfies the conservation law. Formally it is implicitly given by II. (t, x) =  1/ In( v, (t. x; c)), (36)
allows us to construct a particular solution of the latter equation. For example, let C = in (35). Then, as follows from (23), (24), a solution of equation (36) will be a function flU, X), which is implicitly defined from the equalities
17 =
°
",_,.
{
'lr{t,X) 1 
In (lot) I~ n
1I{t,X)=
__
]}
Ixl
I
=
'I'U'" [,
J
[
In (To 
v" ] n I,?
.0
=.c In ( + c In .t ., (t.,C). .

To conclude, let us state equivalence conditions for more general quasi linear parabolic equations.
Proposition IS.
7111' eqllmio/ls
1I ,
are eqllivale/l(.
= (2+ In(2t))1J.! + (e  w) In(e  w) 
= (k(lI, 11,)11,)"
i , = (!,k (~,  ~~) ii,), ' U
II
II
,
T/"(/Il.~/(JI"I/l(/fio/l (23), (24) (akes a solwio/l II
eqllatio/l info a SOllilioll ()f (he second 0111'.
10
= w(t. x; c)
(c
+ w) In(c + w).
is determined
(38)
It is not hard to check that for I ?:. c" /2 equation (38) is uniquely solvable with respect to the function w(t, x;c) E \0, C) in terms of x E [0, x,(t;e)), where
[
where the function IlIa, :0 is such that X
where C > 0 is an arbitrary constant and the function from the equation
::j=. 0 of (he firsl
( ') e
, 2c~
.
Setting 11,U. x; c) = 0 for Ixl ?:. x.(I; c), we obtain a generalized solution with compact support of equation (37), 11.(1, x), which has continuous thermal flux at the front points of the solution x = ±x,(t; e). It is easy to see that the conservation law
I:
1I.(t, x)dx
== Eo = 2c for all (2
c"/2
is satisfied. It is interesting that at time 10 = c:. /2 the solution 1I. (x, 10) behaves close to the point .\' = 0 as the unbounded singular function IxI 1/3, which is integrable, but not a delta function.
~ 8A
II Somc quasilincar parabolic cquations
84
§ 8 A heat equation with a gradient nonlinearity In this section we consider the properties of generalized solutions of quasilinear parabolic equations, which describe diffusion of heat in a medium, heat conductivity of which depends not on the temperature. but rather on its spatial derivative (gradient), Typical examples of such equations are: III
= (lu,I"u,),.
hcat cquation with a gradicnt nonlincarity
At points of the front of the solution, IV liS I = 0, I1l1s(l, x) ~ (IXfllxl)IIITI/" as Ixl lXII, Therefore if (T < I, then I1l1s(l. xf) = 0 ((3) is a classical solution for x # 0); if (T = I, then I1l1s(l. x f) # 0, while if (T > I then I1l1s = 00 for Ixl = IX/I. In the two last cases I1l1s has on the front surface a discontinuity of the first or second kind. respectively. Let us note in particular that at all points of degeneracy the heat flux
where (T > 0 is a constanl. These equations arc parabolic and degenerate; the thermal conductivity coeftlcient k = k(i'llll) = l'llll" ::: 0 vanishes wherever VII = 0, in particular, at points of positive extremum of the function II ;::;: lI(I. x) ::: 0, or, for example, at the points of the front of a thermal wave which propagates with a tinite speed. Therefore, in general. solutions of the equations (I) and (2) arc generalized ones. Example 9. The Cauchy problem for (2) in R j x R N has the solution
==
W(I. x)
(I)
in the one space dimension. while in the multidimensional case we have the equation (2) III;::;: v· (I'llll"'lll).
•
85
k'lll
= I'llll"'lil
is continuous. This is an important property of the generalized solution. which is taken into account when one introduces the integral identity which is equivalent to (2). The generalized solutions satisfy the Maximum Principle; comparison theorems with respect to boundary data hold for these solutions. Equation (2) describes processes with a finite speed of propagation of heat perturbations over any constant temperature background. For example. the function
=
11(1. x)
I
+lIs(l.X).( >
O.x
ERN.
is a solution with a tinite front on a (temperature unity) background, Equation (I) has a power nonlinearity. Therefore it is not difficult to construct selfsimilar solutions for it in the halfspace Ix > 01 with a regime prescribed on the boundary x ;::;: 0 (see 3). For example, if 11(1. 0) ::= (I + I)"'. 111 > O. thcn the corresponding solution has thc form
*
11\(1. x)
If on the other hand
where A".N
=
(;;:~~2 Y"+
11/"
{;;:(N +1 I~+ 2
f/"
= (I
+I)'"fs(~).{; = x/(I
11(1.0) ;::;: (".
IIs(l. x)
+1)11'1'""1/1"'"1.
then
= ("fs(O.
~
=::
x/exp
{~(}. +2 IT
l' ::: 0, (/ > 0 are arbitrary constants. This is a selfsimilar solution of an instanta
neous point energy source type. It is determined exactly as the analogous solution for the equation with k(u) = II" which has constant energy: .
./.I~N
IIs(l. x)dx
==
r.
lIs(O. x)dx. ( >
0,
These selfsimilar solutions are asymptotically stable in the sense indicated
*
above (see l, 2). We shall consider more closely solutions evolving in a blOWUp regime. which demonstrate the h('(1( /o('(I/i;:a(;ol1 phenomenon.
.fRo"
The solution (3) has compact support at each moment of time: liS (I . x) ::= 0 for alllxl ~ Ix;(I)1 = a(T+t)'/I"INIIH1 1, Its degeneracy points are x = 0 (a positive maximum point) and the front surface (Ixl = IXr(l)ll. From (3) it follows that at x::= 0 the second derivative ~(IIS) does not exist, but that the product I'VII,\I"~I/s is finite, so that for x = 0 the derivative II, is defined, since t2) is equivalent to the equation III = l'VIII"[l1u + IT'll'll/I· ('lu/I'lul)l·
Example 10. In a boundary value problem for equation (I) in the domain (0. To) x R I • let lI(r.O)::=
1+ (T u  1)".0
<: 1
< T u:
II
< O.
(4)
The corresponding selfsimilar solution has the form
From this section we begin to analyse specific selfsimilar solutions of quasi linear parabolic equations with an additional term Q(u) (either a source or a sink) in the righthand side, Some examples of such equations were given in Ch. I. First we consider selfsimilar solutions of travelling wave type in active media with a source, This problem was studied first. and in an exhaustive manner, in the wellknown paper [255 [, It generated a whole range of papers (see Comments). which is the reason this problem is named after the authors of [2551,
1 Statement of the problem
We consider the diffusion process
o·x'ox
II,
Fig.
H.
Evolution as
Ti,'
1 ~
, rr "
in a medium with a source of a particular form:
1+ nlT
,
=
I, Os(oo)
> 0,
(6)
= 0.
= IjlT
(the S blowup regime). equation (6) is easily integrated. The corresponding selfsimilar solution IIs(l. x)
= 1+ (7'11 
. _
I)
I/rr
[(I  xjxo)+IIIr+]l/rr,
IT+2 [2lT(lT+ 1)]l/lrrt21
,10'IT
(IT
(5')
The bchaviour of the function tions on Q( II) it follows that
0,
u E (0, I);
(1',
II E
(0,
Q(u) :::
u E
lYl/,
(this is essential in the following). example, by the source
(3)
10, I J
All the above conditions are satisfied, for
ill. 0 :::
Q(II) = au( I 
II :::
I.
/Q :etli /
/ / /
/
/ / /. /,
o
w
(2)
II·
is shown in Figure 9, From the stated restric
Q(II)
+ 2)
represents a thenn.1! wave with a fixed front point, localized in the domain () < x < Xo during all the period of action of the boundary blOWUp regime, Heat does not leave the localization domain, and for x > Xo the homogeneous temperature background remains the same (Figure 8). The spatiotemporal structure of the selfsimilar solution (5) indicates that if n < I j IT, the inlluence of the blowup rcgime will not be localized and xf(1) ~ S' h'llell1tlecaseIlE(ljlT,O) . I (7 '0' 1)llllIrrl/llrlOI  +ooasl+ 7' 0 ( H .reglme),w we do have localization, such that, moreover, temperature grows without bound only at the point x = 0 (LSregime). This classification coincides with the one given in 3 for the thermal conductivity coefflcient k = II rr ,
*
= Q(I) = 0; Q(u) > Q'(O) = > 0; Q'(II) <
Q(O)
(I'
+ 'J O\~ + nOs = 0, ~ IT _ OslO)
In the particular case n
(I)
> 0, x E R.
1
of the localized S hlowup regime (5')
where the function Os ::': () is a gencralized solution of the boundary value problem (\0d Os) 
= II" + Q(u),
•_ _
._~.H~~_~_·"_~·
~
I
Fig. 9.
Ii
(4)
~ '!
II Some quasi linear parabolic equations
88
The KolmogorovPetrovskii.Piskunov problem
The solution Os :::: 0 corresponding to a given A ~ Ao is unique up to a shifl. This fact is important: if Os is a solution, so will be Os({j + f'), {j' :::.: consl. The natural question that arises is: what speed is selected for an initial perturbation of the "mesa"like form (6)'1 In 12551 the authors prove the following fundamentally important result: in the problem (I). (6). for large I the wave moves at the speed A :::.: Ao. that is, the minimal possible speed. For other noncompactly supported uo(x) the wave may move as I '> 00 with a speed A > An· If we denote. as usual, by xl' J (I) the depth of penetration of the thermal wave (l/(I, xcJ(I)):::.: 1/2), then
lI(f,x)
I 
Formation of a thermal wave in the problem (I), (6), The initial function is indicated by a thicker line For equation (I) we consider the Cauchy problem with the initial condition
= 1I0(X)
::::
0, lIo(.n
:s
I, x
E
R,
(5)
This problem is wellposed, Though we did not define the function QUI) for li < 0 and u > I, this is not important, since from (5) and from the comparison theorem il follows that 0 :s u(l, x) :s I, Indeed, Uf, == I and U _ == are solutions of equation (I), and by (5), LI :s 110(.1) :s 11+; lherefore LI :s u(l, x) :s Uf in R+ x R Let us consider now an initial perturbation of a simple form (see Figure 10):