de Gruyter Expositions in Mathematics
The Analytical and Topological Theory of Semigroups. K, H. Hofinann. J. D. Lawson. J. S, Pym (Eds,)
2 Combinatorial Homotopy and 4-Dimensional Complexes. H. J. Baues
3 The Stefan Problem. A, M. Meirmanol'
4 Finite Soluble Groups. K. Doerk, TO. Hall'kes
5 The Riemann Zeta-Function. A. A. Karatsuha. S. M. hJrlil1in
6 Contact Geometry and Linear Differential Equations. V. R. Na:aikinskii. V. E. Shata/ov. B. Yu. Sternin
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
10 The Link Invariants of the Chern-Simons Field Theory, E. Guadagnini
11 Global Affine Differential Geometry of Hypersurfaces. A .-lvf. Li, U. ,)'iIl1OII.
G. Zhao
12 Moduli Spaces of Abelian Surfaces: Compactilication. Degenerations. and Theta Functions, K. Hu/ek, C. Kahn. S. H. Weilltrauh
13 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 Decompositions and Integral Lattices. A./. Kostrikin, P. H. Tiep
16 The Adjunction Theory of Complex Projective Varieties. M. C. Be/tramel/i, A. J. Sommese
17 The Restricted 3-Body Problem: Plane Periodic Orbits. A. D. Brullo
18 Unitary Representation Theory of Exponential Lie Groups. H. Leptin, J. Ludwig
Blow-up in Quasilinear Parabolic Equations
by
Alexander P. Mikhailov
Michael Grinfeld
de Gruyter Expositions in Mathematics 19
Editors
O. H. Kegel, Albert-Ludwigs-Universitiit, Freiburg V. P. Maslov, Academy of Sciences, Moscow
••••••••••••••••••••••••••••••••••••••••
S£'rgei P. KurdvulIlol', Alexwuler P. Miklwilo\'
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 blow-up 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.
Chapter I Preliminary facts of the theory of second order quasilinear parabolic equations . . . . . . . . . . . . . . . . . . . . . . . . I
* I Statement of the main problems. Comparison theorems ..... I * 2 Existence, uniqueness, and boundedness of the classical solution. 6
* 3 Generalized solutions of quasi linear degenerate parabolic equations 14
Remarks and comments on the literature . . . . . . . . . . . . . . . . . 35
Chapter II Some quasilinear parabolic equations. Self-similar solutions and their asymptotic stability . ., 38
* I A boundary value problem in a half-space for the heat equation. The concept of asymptotic stability of self-similar solutions. . . . . . . .. 39
* 2 Asymptotic stability of the fundamental solution of the Cauchy problem 47
* 3 Asymptotic stability of self-similar solutions of nonlinear heat equations 53
* 4 Quasilinear heat equation in a bounded domain. . . . . . . . . . . .. 61
* 5 The fast diffusion equation. Boundary value problems in a bounded domain. . . . . . . . . . .. 67
* 6 The Cauchy problem for the fast diffusion equation 68 * 7 Conditi~ms of equivalence of different quasi linear heat equations 74
* 8 A heat equation with a gradient nonlinearity . . . . . . . 84 * 9 The Kolmogorov-Petrovskii-Piskunov problem . . . . . . 87
*10 Self-similar solutions of the semilinear parabolic equation U, = t:.11 + IIlnu , 93
* II A nonlinear heat equation with a source and a sink. . . 99 *12 Localization and total extinction phenomena in media with a sink 101 *13 The structure of altractor of the semi linear parabolic equation with
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Preface to the English editionvi
Die Delllselle Bihlio/llek - Cata/ogillg-ill-l'lIhlica/io/l Data
Lihrary of COllgress Ca/a/ogillg-ill-l'lIhlim/ioll Data
(0 Printed 011 'll:id.hec paper which falls within the guidclinc~ of the ANSI 10 ensure pcrmancm:c ;JIlJ durahihl)
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 non­ stationary 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 blOW-Up 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, pro­ vided 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 blOW-Up, 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 self-similar (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 impor­ tant element of evolution of such systcms is finite time hlow-up hehaviour, which represents a kind of stable intermediate asymptotics of the evolution. Without studying tinite time blOW-Up. the picture of the nonlinear world would be incom­ plete. 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-
Preface to the English edition Michael GrinfelJ Department of Mathcmalil'~
University of Strathclyde :!6 Richmond Street Glasgow Cit IXH. UK
Trallslator
Current address of V. A, Galaktionov' Department of Mathematics Universidad Aulonoma de Madrid 28049 Madrid. Spain
Rezhimy s obostreniem v zadaehakh dITa kvazilincinykh para­ bolicheskikh uravnenii. English
Blow-up 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 3-tl-OI2754-7 I. DifTerential equations, Parabolic. I. Samarskii. 1\. 1\.
(1\leksandr Andreevieh) II. Title. III. Series. QA372.R53413 t995 515'.353-de20
(U Copyright 1995 by Waller de Gruyter & Co., D-107H5 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.
Blow-up 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 3-11-012754-7
NE: Samarskij, Aleksandr A.; Grinfcld, Michael [Ubers.]; Rezimy s obostreniem v zada(;ach dlja kvazilinejnych paraboliceskich uravnenij <engl. >; GT
1991 Mathematics Suhject Classification. 35-02; 35K55. 35K65
KeJ'II'ords: Nonlinear heat equations. combustion. blow-up. asymptotic behaviour. maximum principle, intersection comparison
Title or the Russian original editiol1: Rezhi~lY s obostreniem v zadachakh dlya kvazilinejnykh parabolicheskikh uravnenij. Puhlisher: Nauka, Moscow 1987
With 99 ligures.
AliI/lOr.\'
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
••••••••••••••••••• ~J
130 130 135 142 158 174
Chapter 1Jl Heat localization (inertia) . . . . . . § I The concept of heat localization . § 2 Blowing-up self-similar solutions § 3 Heat "inertia" in media with nonlinear thermal conductivity § 4 Effective heat localization ..... Remarks and comments on the literature
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 Ifl-criticality 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 ...
Chapter IV Nonlinear equation with a source. Blow-up regimes. Localization. Asymptotic behaviour of solutions . § 1 Three types of self-similar blow-up regimes in combustion . § 2 Asymptotic behaviour of unbounded solutions. Qualitative theory of
non-stationary averaging. . . . . . . . . . . . § 3 Conditions for finite time blow-up, 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 self-similar solutions . § 6 Asymptotics of unbounded solutions of LS-regime in a neighbourhood
of the singular point . . . . . . . . § 7 Blow-up regimes, effective localization for semilinear equations with
aSOUITe .
ixContents
Bibliography 506 Index. . . . 535
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 multi-component media 1"467 § 5 Finite diffcrence schemcs for quasi linear parabolic equations admitting
finite time blow-up, . . . . 476 Remarks and comments on the literature 502 Open problems 505
Chapter YI Approximate self-similar 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 self-similar solutions in the degenerate casc 375 § 3 Approximatc self-similar solutions in the non-degenerate case.
Pointwisc estimates of the rate of convergence . . . . . . . . . 386 § 4 Approximate self-similar solutions in the non-degenerate case. Integral
estimates of the rate of convergence. 398 Remarks and comments on the literature 413 Open problems 413
238 257
Contentsviii •••••••••••••••••••••••••••••••••••••••••
Introduction
Second order quasilinear parabolic equations and systems of parabolic quasilinear equations form the basis of mathematical models of diverse phenomena and pro­ cesses 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 filtra­ tion of gases and liquids in porous media, diffusion of neutrons and alpha-particles: 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
•••••••••••••••••••••••••••••••••••••••••
---_......•_------
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 so­ lutions 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 direc­ tion 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 ap­ proach. 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 spatio­ temporal 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 self­ organization 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 spatio-temporal "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 mathe­ matical 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.
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 experi­ ments 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 blOW-Up behaviour (physical terminology). Nonlinear evolution problems that admit unbounded solutions are not solvable globally (in time): solutions grow without bound in finite time inter­ vals. 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 self-focusing of light beams in nonlinear media. non-stationary structures in magnetohydrody­ mImics (the T-Iayer). 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 blow-up 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 non-stationary 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 non-stationary
dissipative processes.
xii Introduction Introduction xiii
In this book we present some mathematical aspects of the theory of blow­ up phenomena in nonlinear continua. The principal models used to analyze the distinguishing properties of blow-up 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 spatio-temporal structure of blow-up 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 blow­ up 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 blOW-Up phenomena.
The present book is devoted to the study of such model problems, but we em­ phasize 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 blow-up phenomena in nonlinear continua: unbounded growth of temper­ ature, 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 blOW-Up phenomena in parabolic problems is by no means ex­ hausted by the range of questions reflected in this book. It will not be an exag­ geration to say that studies of blOW-Up 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 so-called 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 blow-up regimes is accompanied by the appearance in the medium of complex. as a rule discrete, collections of eJ. with diverse spatio-tem­ pond structure. An intrinsic reason for such increase in the complexity of organi­ zation of a nonlinear medium is the localization of dissipative processes.
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 non-equilibrium 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 blOW-Up 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 hlow-up in two specillc problems for parabolic equations with power law nonlinearities. In subse­ quent 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 blow-up 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 blow-up regime. /II (I) = (T - I) 1/": heat localization in this case is graphically illustrated by the simple separable self-similar solution':
xiv Introduction
xv
(I)
•••••••••••••••••.1 •••••••••••••••••••••••
wherc Ls = 21T(IT + I) I /2 I (T is the fundamental length of the S-regime. 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 spatio-tcmporal structure of unbounded solutions is based on a particular "comparison" of the solution of the Cauchy problem with the corresponding self-similar 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 blow­ up 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 non-dccreasing function of I), is a natural consequence of the Strong Maximum
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 non­ localized (II < -I I(T) power law boundary conditions; corresponding self-similar solutions are constructed; analysis of the asymptotic behaviour of non-self-similar 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 S-regime 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 self-similar 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) + uli . I > 0, x ERN, where (T :::: 0, (3 > I are constants. A number of topics are investigated for (T > O. We construct unbounded self-similar solutions. which describe the asymptotic phase of the development of the blOW-Up 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 non-occurrence (I < (3 < if + I)
of localization of unbounded solutions. Localization of the combustion process in the framcwork of this model is illus­
trated by the self-similar solution (S-regime) for (3 = (T+ I. N = I, in the domain (0, To)xR:
where n is a smooth domain in RN , A(u) is a nonlinear second order elliptic
operator with smooth coefficients, there is a well-known comparison principle for sub- and supersolutions. Let II ::: 0 and v :::: 0 be, respectively, a super- and a subsolution of cquation (3), that is.
(5)
(4)
(3)
xviiIntroduction
u, = AUt). (I, x) E G = (0, T) x n,
1/;"1 = L (I') (U("I, I'VI/(") I, ~l/iI'), (t, x) E G, /! = I, 2,
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 con­ structive analysis of nonlinear parabolic equations started precisely from an under­ standing 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 differ­ ent 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
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 of existence, N(r) cannot be strongly decreasing; in any case, if N(O) > °then 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 blOW-Up 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 RN
, 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 .
(2)Ixl < Ls12,
••••••••••••••••••••••••••••••••••••••••••
il 1"1, 111 '. II' 121 ' () ';'J" (L - (p. (/' r) ~ L (p, q, r)) ::: 0, L (p, (/'/0 (p. (il):s. . , r
••••••••••••••••••••••••••••••••••••••••••
ii, = A(I/) =: (k(I/)I/,)" t E (0. n. x> 0;
iI(I.O) = 1/1 (I) ---+x. t ---+ T ; u(O, x) = ilo(.r) ::: 0, .I' > O.
so that the equation
IIA(II,(I, .)) - B(I. 11,(1, ·))11 « IIB(I. 11,(1. ·))11
admits an invariant solution II = II, (I. xl generated by the given boundary condi­ tion: 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
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)
is the construction and analysis of the corresponding self-similar or invariant so­ lutions. 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 self-similar 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,
heat equation of general type (by comparison with self-similar solutions of the equation 11, = (11"1/,)" cT ? 0, which are studied in detail in Ch. Ill.) Using the concept of III-criticality, we derive conditions for non-existence 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 non-stationary processes of nonlinear heat conduction, described, for example, by the boundary
value problem
() lI') R R:-L (p,q.r) ':::0. p.qEj. rE . (ir
Therefore for the comparison 11 121 ::: 111 1) 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):
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 III-criticality of a solution: u;21 ? 11/(11121
) in G, where III is a
smooth function. As applications. we obtain in Ch. Y conditions for localization of boundary
blow-up regimes and its absence in boundary value problems for the nonlinear
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:
(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
From the usual comparison theorem of classical solutions it follows that the in­ equality 11 121 ::: Ull ) will hold in G if Ul21 ::: 11 111 on ilG and for all v E C~;2 (G)nc(G)
xviii
I~-
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 Hamilton-Jacobi type equation:
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 blOW-Up 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 non-stationary solutions
is close to u. for large {. For strongly non-stationary solutions, for example, those exhibiting finite time
blOW-Up (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 AI (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 IVA\. this fact allows us to describe quite subtle effects connected with the development of unbounded solutions.
In addition, in Ch. VII we analyse blOW-Up 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
." ,;\ .. 1"\. \' " .' ,~ ,to....
" ' .: I
Applied Mathematics of the Russian Academy of Sciences, Applied Mathematics Department of the Moscow Physico-Technical Institute, and the Numerical Anal­ ysis 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.
xxiIntroduction
(13)
Introduction
xx •••••••••••••••••••••••••••••••••••••••••
Chapter I
, (I)
(2)II, = 8 (II) = 'V. (k(lI)'V II) + Q (II).
or with nunlincur hcal cqualiollS wilh sOllrce (sink),
Preliminary facts of the theory of second order quasilinear parabolic equations
In the majority of cases we shall deal with quasi linear parabolic equations of the
following type: nonlinear heat eqllations,
In this introductory chapter we present well-known 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 uncompli­
cated 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 par­ ticular (self-similar 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.
•
2 I Preliminary facts of the theory of second order quasilinear parabolic equations
1/(0. x) = I/o(x)::,: O. x ERN; I/o E C(R"'j, sup I/o < 00. (5)
3
(6)
2 The Maximum Principle and comparison theorems
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 non­ negative. This is cnsurcd by the Maximum Principle, which plays a fundamental part in practically all aspects of the theory of nonlincar parabolic equations.
1I0(X) = UI (0. x). x E an.
Theorem 1. LeI 1/( II alld 1/(21 be Iloll-negmil'e classical SOlllliolls (~r eql/atioll (2)
ill (0. T) x n. sl/ch Ih(lI. moreol'l'r.
The Maximum Principle characterizes a kind of "monotonicity" property of solu­ tions 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 ex­ tensively 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 are especially important when the domain n is unbounded). Therefore we move 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.
Wc are looking for a solution in the class of functions bounded uniformly in x E R N
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:
(4)
(I')
II, = A(II):= k(II)/l1l+k'(1I)1V'1I1 2
.
11(0. x) = lIo(x) ::': 0, x En; 110 E c(n). sup 110 < 00; (3)
11(1, x) = 111(1. x) ::': O. IE (0. T), x E an; III E C<lO. T) x an). SUpll1 < 00.
The function 110(.r) in (3) can be considered as the initial temperature perturba­ tion. 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 n for I E lO, T).
Apart from the tirst boundary valuc problem, we shall also consider the Cal/ch,'
problem in (0. T) x RN with the initial condition
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, where T > 0 is a constant and n is a (possibly unbounded) domain in R
N with
a smooth boundary an, the equation under consideration. togethcr with the initial
and boundary conditions
where /l is the Laplace of!eralor.
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 a non-negative and sufficiently smooth function: k(lI) E C2 «0. 00)) n C([O. 00)).
If 1I > 0 is a sufficiently smooth solution, then (I) can be rcwritten in the form
••••••••••••••••••I. •••••••••••••••••••... ••
Theil
4 1 Prcliminary facts of the thcory of second ordcr quasilincar parabolic equations •••••••••••••••••••••••••••••••••••••••••
5
( 16)
( 15)
( 14')
( 13)
1I1 = F(lI, \Ill, 6.lI,I, x),
all/iill = a(lI. t, xl. t E 10, n,.r E an,
all. /an ::: all/an::: alit /all, I E [0, T), x E a!!
§ I Statemcnt of the main problems. Comparison theorcms
where a( II, t, x) is a Sll fliciently smooth function [10 I. 3381·
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
all/all = 112(1. x). I E (0, T). x E au; 112 E C. sup //2 < DC. (14)
allo(x)/an = 112(0. x), x E aU,
(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
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
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:
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 n. Then a new compatibility condition arises:
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 n. The parabolicity condition here has the form
Statements similar to Theorems 1,2 hold also for nonlinear parabolic equations of general form, in particular. for essentially nonlinear (not quasi linear) equations
(9)
( 12)
( 10)
(II)
(8)
allt/al ?: B(lII)' all /al::: B(I/ ) ill (0, T) x n.
II .. ::: II ::: 11( in [0, T) x n.
II. (0, x) < 1I0(X) ::: III (0, x), x E n;
// (I. x)::: 1I1(1, x)::: III (I, x), t E [0. T), X E au.
Theil
and .fil/·Ihermore
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 11121 ::: 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·
Proposition l. LeI Q(O) = 0 alld leI II(X. r) he a classical sollllioll o( Ihe prohlem
(2)-(4). 71lCII 1I ::: 0 ill 10, T) x U.
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
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).
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 on the analysis of the "linear" parabolic equation for the difference z. = 1I
121 - III II
and in essence uses the sign-definiteness of the derivative 6.;: at an extremum point
of the function ;:. As a direct corollary of Theorem I we have the following
1 Conditions for local existence of a classical solution
6 I Preliminary facts of the lheory of second order quasilinear parabolic equalions
§ 2 Existence, uniqueness, and boundedness of the classical solution
(3')
Proposition 2. LeI Ihe jilllCliolls k(II). QUI) be sl(tJicielllly smoolh fiJI" II 0. Q(O) = o. If Ihe cOlldilioll
*2 Existence, uniqueness, amI boundedness of theclassical solution 7
J!(p) :::: aF(p. C/, r, I. x)/ilr:::: p(p).
holds. Ihell Ihere exisls a local classical sollllioll of Ihe bOll/lllary \'{IllIe problem ( I. 2)-( 1.4) .. IIl11reover. if 1/11 ¥= 0 ill n or if II I (0. x) ¥= 0 Oil an. thell 1/(1, x) > 0 ill
n for all admissible f > O.
k(lI) ::: Ell = const > Ofor 1/ > O. (4)
I In this way we refer to formulae from previous sectiolls; in this case it is from * I.
A non-negative 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 non-differentiable, 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 RN . 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.
For equations of the form (I) the uniform parabolicity condition has a particu­
larly simple form.
for arbitrary I E [0. T), x E n, p ::: 0, q, r ERN, where the continuous functions Il(p) and p(p) are strictly positive. Condition (3') means, in particular, that the second order elliptic operator in (3) is non-degenerate 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
if they are IIlliformly parabolic. This means that
(3)
( I)
II f = V'. (k(II)'\7I1) + Q(lI),
N
II, = L a ,J (II. '\711.1. X)II"I, + a(lI, '\711, I. xl. i,J'7!
the existence of a global solution is equivalent to its boundedness in n on an arbitrary interval (0,7'). Namely: a global solution is defincd and bounded in n for all I ::: 0, while an unbounded solution is defined in n on a finite interval (0, Til), such that moreover
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
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.
Questions of existence and uniqueness of classical solutions of boundary value problems for nonlinear heat equations are studicd in detail in the well-known 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,
•••••••••••••••••••••••••••••••••••••••••
2 Condition for global boundedness of solutions
•••••••••••••••••••••••••••••••••••••• .~
••
11(1. x) = n. / > 0, x E an.
III = illl + Q(lI), / > n. x E n.
in a boundeJ Jomain n E RN with a smooth boundary an. Letll(O. x) = 110(.1') ::: n in n, 110 E nIi), 110 'f=. 0, and
Example 2. Let us consider a bounJary value problem for a semi linear equation,
lI(t) -, 00. I -, I'll'
*2 Existence, uniqueness, and boundedness of theclassical solution l)
o
Proposition 3 reflects one aspect of the problem of unboundedness of solu­ tions. 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 bound­ edness 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 RN 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.
furthermore
! "lfI_~=I.
. III Q(.,,)
From this it can be seen that lI(() is defined on a finite time interval (0. To) where
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,
(6)
(5)UI21 ((.X) == M =const:::: max{supllo,SUplll}'
that is, II is globally bounded. Necessity. This follows from the following simple example.
uoCr) == 111 = const > O. x E R N
.
lI(I. x):": lII21(!), IE (0. n. x E n.
where thc constant M satisfies (5), The function lI121(1) is determined from the
equation
.L"·'(fI ~i~) = /.
where, morcover, by (6) lI121(1) is defined for aliI E (0. (0). Then from Theorem I.
by setting III I I == II we obtain that
Proof Sufficiency. Let us use Theorcm I. As u121 (1, x) let us take the spatially
homogeneous solution ,pi (I) of (I):
is l/ Ilecessury ulld sllfficielll cOlldilioll Fir glohal hOllluledlle.l's or UIlY .l'ollllil!l1 of
Ihe prohlem (\.2)-( 1.4).
Proposition 3. In equation (I), /1'1 Q( lI) > 0 jiir II > 0, Theil Ihe cOlldilioll
we see that conditions (1.6) and (1.7) hold, so that u(l. x) ::: M, that is, u is bounded in n 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.
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
8 I Preliminary facls of the theory of second order quasi linear parabolic equations
Then E(O) = Eo and furthermore. as follows from (9). E(I) satisfies the equality
(17)
(16)
H(O) = Ho = I' lIoCr) dx > O. .Ill
aU/ail = O. t > O. x E an,
dH(I)
dl
and integrate by parts. taking (16) into consideration. we have
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 no-flux Neumann boundary condition.
§ 2 Existence. uniqueness. and boundedness of theclassical solution II
with initial perturbation u(O, x) = uo(x) :: 0 in n. Let us show that any non-trivial (II ¢ 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
[.
• lOll Q(7/) - AI 7/ -
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.
Hence under our assumptions we have that E(I) > Eo for all I > O. and conse­
quently
(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
dE(t) ., ' -'J( :: -A,E + Q(f:). I > 0; E(O) = Eo :: 80 .
(\ 5)
(14)
(\ 3)
( 12)
(\ I)
61/1 + AlII = O. x En; I/,(X) = O. x Ean.
dE(t) l' l'-_.- - 6I1(1,X)I/'j{X)dx+ Q(II(1,X))'/II(X)dx. dt -. II . II
I' 611(1, x)l{ll(x)dx = .Ill
To prove this. let LIS introduce the function
Then for any initial functions uo(.d :: 0 sllch that
Let Q( u) - AIII > 0 for all II :: 80 = const > O. and furthermore
Integrating by parts and taking into account (10) and (II), we obtain
the solution of the problem is unbounded and exists till time
(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 C2(R,l is a convex function:
and by '//1 (x) the first eigenfunction. which is known 1283. 3621 to be of constant
sign in n. Let IPI (x) > 0 and
Let us denote by A, = AI (n) > 0 the first (smallest) eigenvalue of the problem
•••••••••••••••••••••••••••••••••••••••••
11'(1) = lI"(I), I > 0; u(O) ::::: o.
11(0, x) == 0, x E !T 11(1, X) ::::: 0, I > 0, x E an.
with the conditions
Hence we obtain the condition
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 so­
lutions, 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 non-unique.
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
*2 Existence, uniqueness, and houndedness of theclassical solution 13
where T ~ 0 is an arbitrary constant. Therefore, due to non-differentiability of the source for II = 0, there appear
from the zero initial condition (19) non-trivial solutions that grow at the same rate on the whole space. Let us note that for a E (0, I) all the functions lJ7 (t) are
classical solutions. since lJ T E C I ([0, (X))).
It is not hard to see that similar non-trivial solutions of the Cauchy problem can be constructed in the case of arbitrary sources Q(II) > 0, 1I > O. if
Solutions of this problem are the functions
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
( IlJj
( IX)
.
where a E (0, I) is a constant. Here Q(II) ::::: II". Q(O) = 0, Q'(O+) ::::: x. Let
~~H(I) ::: (meas n) Q(~~) , I> O. til meas!}
3 lJniqueness conditions for the classical solution
/
I < *' == -- < 00, - , II,,/meas l! Q( T))
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 coeffi­
cient Q, which is a non-negative 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 non­
differentiable for II = 0, Q E C l «0, (0)). The following example shows what this
can lead to.
and therefore lim sup, lI(I, x) ::::: 00, I --, Til::: T I •
Therefore by (8) it follows that the energy H(t) (and therefore 1/(1, x)) is defined and bounded only on a bounded interval (0, T I), where
we obtain from (17) the inequality
Using the Jensen inequality
12 I Preliminary facts of the theory of second order quasilinear paraholic equations
14 I Preliminary facts of the theory of second order quasilinear parabolic equations
(5')
(5)
(3')
(4)
(3)
(2)
[
, 0 r1
so that the function
d ( dr\') dl's-, k(I )-'-' + A-'-'- = 0 d~\ d~ dl; ,
where <1>-1 is the function inverse to <I> (it exists by monotonicity of <I); see (5/)). Let us extend Is into the domain {~ > OJ identically by zero; it follows from (3')
1 Examples of generalized solutions (finite speed of propagation of perturbations, localization of boundary blOW-Up regimes and in media
with sinks)
dis . k(rs)- + AI \' = c."dg , .
Selling C = 0 (what this corresponds to will be made clear in the following), we obtain the equality
or, which is the same,
where A > 0 is the speed of motion of the thermal wave. Substituting the expres­ sion (3) into (2). we obtain for Is(~) ::: 0 the equation
and let us construct its particular self-similar solution of travelling wave type:
Example 6. (tinite speed of propagation of perturbations) Let us consider equation (1.1) in the one-dimensional case:
(I)k(O) = O.
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,
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 sufficiently smooth: k E C"«O. 00)) n C([O. 00)), Q E C I
([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
where lJT(t) are the functions defined in (21), To conclude, let us observe that non-uniqueness 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 RN , 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.
§ 3 Generalized solutions of quasilinear degenerate 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 non-differentiable 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
~ i.I .I. I. II. ••••••••••••••••••••••••••••••••••·4;
•••
~
rUs (t, x)
Fig. 2. Travelling wave in the case of inlinite speed of propagation of perturbations
o --_:::=~---====--:=====.­.1.'
.0 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.
OL-_--' .l.-__-l..__....1.--:;;.X
§ 3 Generalized solutions of quasi linear degenerateparabolic equations 17
Fig. 1. Travelling wave in the case of linite speed of propagation of perturbations
Condition (5) is necessary and sufficient for the existence of a compactly sup­ ported travelling wave solution. If it is violated, that is if
(6)tI,\(t, x) = «>"[,1(,11 -- x)j·l. I> 0, x E R.
W(I. x) = --tl:~(lIsl\ = (T'/" ,11"1 '1/"1(,11 - xl j 1'/".
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.
Let us check again that the heat flux is continuous at the points xf(l) = At.
Indeed.
k(tI) == II". IT = const > O.
Then <»(11) = II"/(T. «> '(tI) = «(TII)'/", To = 00 and the travelling wave solution has an especially simple form
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
liS (I , x) == O. x> AI, IE (0. To).
IIs(I,x)=llrA(AI~X)II'/", I >(). x>O.
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):
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
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 self-similar solution:
Example 7. (localization in a boundary blOW-Up regime) Let
(14)
( 13)
( 12)
I' 11(1. x) dx = Eo = const > D .IRS
II, = '\7. (11"'\711). I> D. x ERN.
(this is a solution of the instantaneous point source type). We shall look for it in the self-similar ansat7.
having constant energy
Us (t.x)
Example 8. Let us find a solution of the Cauchy problem for an equation with a
power law nonlinearity
Fig. 3. Travelling wave self-similar solution (10) localized in the domain (0. 1/A)
I"" k(1/) -- d1/ < 00.
, I 1/
This follows immediately from the representation (6) of the corresponding travel­ ling wave solution generated by the blOW-Up regime.
Let us consider an example of a generalized solution of the heat equation in
the multidimensional case.
It is clear thal the same comparison argument and thus the same result on local­ ization of arbitrary boundary blow-up regimes, also holds in the case of coefficients
k that satisfy (II)
, _ { - Inll - 1.( At - x) 1, 11.1'(1, .\) - D.
k(lI) = lIexpl-1I1. II :::. D.
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 = 0, the solution
is bounded from above uniformly in I:
and it grows without bound due to the boundary blow-up 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 blOW-Up 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" self-similar 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(To. x) :::
IIs(To• x - I/A) < 00 for x > I/A.
which is dehned for a bounded time interval [0, To), where To boundary condition at x = 0 corresponding to (10) has the form
Then it is not hard to see that the solution (6) has the following form:
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 blOW-Up, The study of this interesting phenomenon in various problems occupies a substantial
part of the present book.
18 I Preliminary facts of the theory of second order 4uasilinear parabolic equations •••••••••••••••'.•••••••••••••••••••••••••
holds. Equation (19) is equivalent to the equation
20 I Preliminary facts of the theory of second order quasi linear parabolic equations
(}iT(t'(O) = 0
[ J I lITo _ If' 2_ 2
(1]) - 2(NlT + 2)(1]0 1])t
[ (
ll'( x -( '. - '-----.d. ) - 2(NIT + 2) 1]0 (2/INIT+21 + .
_ { .. NI1[2(NlT + 2)J I!IT l'(N/2 + I + l/lT) L' }IT!INIT+21 1]0(l:0) = 7T . I' 'eOIT (i/IT + I)
Example 9. (localization of heat in media with absorption) Let us consider the
equation
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
Introducing the new independent (time) variable
where II is a new unknown function. Then the equation for II takes the form
exp(ylT/}V, = 'V. (dr'Vv). 1 > O. x E R N
I T = T(t) == -II - exp(-ylTI}I. T E 10. 1/(Y1T)).
YlT
For any ( > 0 it has compact support in x. while as ( --, 0+. it goes to a rS-function: lls(1. x) -> 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.
~ 3 Generalized solutions of quasi linear degenerateparabolic equations 21
11(1. x) = exp(-yl}v(l. X),
Thus the required self-similar solution with constant energy has the form
Here 1]0 is a constant, which is determined from the condition (18):
Its solutions have the form
( 17)
(16)
(15)crr,,-I{! - (3(,,-1 ~ ilfJ. (: = ("IIT+II "1{3'V..,. ((}IT'Vc. 0). L 'j>: S, "<
I (""
tT I ~iJO N v 'Vf . ({) 'VfO) + --- L -, {;, +---0 = 0, {; E R' ,
. N1T+2 I d{;; NlT+2
I' OWd{; = Eo. (18) .InN
Let the function 0 be radiallr S\'IIIII1('lri1', that is, let it depend only on one COOl"
dinate: 0 = H( 1]), 1] = Igi ;;: O. Then equation (17) takes the form
_1_( N .IOITO')' .j- __1-._ 0'1] + _N__O= O. 1] > 0; (19) 1]N .. I1] NlT+2 N1T+2
moreover, by symmetry we havc to require that the condition
(r/" I (}iTO')' + --'-(01],v), = O. 1] > O. N1T+ 2
I' 11(1. x) dx == I' ("0 (~) dx == t',!Nf3 I' {}(O d{; .IR' .IRS (I' .IR'
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 I (tiT()' + __I_{}1]N = O. 1] > 0; ()iTO'(O) = O. Nlr + 2
as well as the condition
Then it follows from (15) that the function (} ;;: 0 satisfies the following quasi­
linear elliptic equation:
(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. the desired solution has the form
From here we have the necessity of the equality a-I = a(l' + 1) - 2(3; then the terms involving time can be cancelled. Furthermore, using the identity
where U', (3 are constants, and where O(g) :::: 0 is a continuous function. Substituting (14) into (12), we obtain the following equation:
liT = 'V. (v"'Vv),
(25) u, == 'V. (k(u)'Vu), , > 0, x ERN.
§ 3 Generalized solutions of quasi linear degenerateparabolic equations 23
l7 l ( a( .) l11-;- _ k(II)'V1I . 'V j + Q(II)! dx d' + 1Io(x)!(0, x) dx ::= O. (24) .0 . {I a, .!I
for which we consider the Cauchy problem with an initial function of compact
support
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 re­ quirement 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 n: 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
degenerate in the domain under consideration. Generalized solutions of quasi linear degenerate parabolic equations were stud-
ied 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 gen­ eralized 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 informa­ tion 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
(I( 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 an and for r == T. we have the equality
(23)
where g(1) == I + T(I). This solution has the degeneracy surface
l I - exp( -yeT' 11 1
/{N<T\ 2)
IXf(l)1 = TID 1+ ' r 2: 0,yeT
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 == lim IXJ(I)I = TID 1+-. < 00, '.'C>.) yeT
= exp( _y' llg(l) 1- N/{N" I 21 r (T l2(N(T + 2)
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.
11(1, x) =
Definition. A non-negative continuous bounded function u(T, x). which satisfies the boundary conditions (1.3), (1.4) will be called a gCllerali,.ed (weak) sollllioll
The examples we considered in subseL:tion I allow us to demonstrate many of the properties of generalized solutions of quasi linear degenerate parabol