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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 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. 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 Chern-Simons 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 3-Body 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, Albert-Ludwigs-Universitiit, 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 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.
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. Self-similar solutions and their asymptotic stability . ., I A boundary value problem in a half-space for the heat equation. The concept of asymptotic stability of self-similar solutions. . . . . . . .. 2 Asymptotic stability of the fundamental solution of the Cauchy problem 3 Asymptotic stability of self-similar 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 Kolmogorov-Petrovskii-Piskunov problem . . . . . . 10 Self-similar 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
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durahihl)
Lihrary of COllgress Ca/a/ogillg-ill-l'lIhlim/ioll Data
Rezhimy s obostreniem v zadaehakh dITa kvazilincinykh parabolicheskikh 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 94-28057 515'.353-de20 ell'
Die Delllselle Bihlio/llek - Cata/ogillg-ill-l'lIhlica/io/l Data
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
(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.
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 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, 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 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 important 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 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 Blowing-up self-similar 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. 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
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 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 . . .
316 316 324 331 335 348 353 365 371
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. 386 Pointwisc estimates of the rate of convergence . . . . . . . . . § 4 Approximate self-similar solutions in the non-degenerate 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 multi-component media 1"467 § 5 Finite diffcrence schemcs for quasi linear parabolic equations admitting 476 finite time blow-up, . . . . 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 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 of quasi linear parabolic equations, concerned with the spatio-temporal 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 self-similar (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 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 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 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 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 self-focusing of light beams in nonlinear media. non-stationary structures in magnetohydrodymImics (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.
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 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 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 blOW-Up 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 blow-up 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 blOW-Up 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 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-tempond 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 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 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 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':
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 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) + 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 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 illustrated by the self-similar solution (S-regime) 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 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 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 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. 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 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 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 non-dccreasing 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 III-criticality 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 blow-up regimes and its absence in boundary value problems for the nonlinear
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 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 self-similar 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 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, 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 Hamilton-Jacobi 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 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 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 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
Applied Mathematics of the Russian Academy of Sciences, Applied Mathematics Department of the Moscow Physico-Technical 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 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 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 (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.
§ 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 non-negative 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 Iloll-negmil'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 sign-definiteness 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 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,
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 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
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 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 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 no-flux 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 non-trivial
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 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 lJ 7 (t) are classical solutions. since lJ 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 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 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 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 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 (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 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
3 Generalized solutions of quasilincar dcgencratcparabolic equations
1 Examples of generalized solutions (finite speed of propagation of perturbations, localization of boundary blOW-Up regimes and in media with sinks) Example 6. (tinite speed of propagation of perturbations) Let us consider equation (1.1) in the one-dimensional case:
= (k(II)U,),
ul
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 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 self-similar 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 self-similar 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 blOW-Up, The study of this interesting phenomenon in various problems occupies a substantial part of the present book. Example 7. (localization in a boundary blOW-Up regime) Let k(lI)
= lIexpl-1I1.
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 self-similar 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 blow-up regimes, also holds in the case of coefficients
1/1..
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(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 blOW-Up 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 self-similar 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 a-I = 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 self-similar 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 rS-function: 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 non-degenerate (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 non-negative 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 nOIl-ne!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 self-similar 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 i-th 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 half-neighbourhood 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 self-similar 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 1101l-lIegati,'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 non-differentiable 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 non-trivial 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 one-dimensional 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 non-differentiable 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 non-trivial 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 non-trivial 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 right-hand 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 self-similar 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 blOW-Up: 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 blOW-Up combustion process in the S-rcgimc (sclfsimilar solution (50))
grows without bound as I -> Til' The half-width 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 non-self-similar 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 blOW-Up 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 self-similar solution (50). which we considered in Example 13. Below (see § 5. eh. IV) we shall show that precisely this self-similar solution describes the asymptotics of a wide variety of unbounded solutions close to the blOW-Up time. Finally, we observe that the above solution II •• which is not self-similar. 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 two-dimensional 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 non-uniqueness 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 non-unique 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 non-trivial 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±JA1-4)I/IT ( (T
2
> O.
Thcrefore the required self-similar 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 self-similar 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
self-similar solution (21) of § 3 was established for N = I in 1234] and by a different method in 11871. The proof of stability in the multi-dimensional 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 one-dimensional 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 Lie-Backlund 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 finite-dimensional 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 half-space for the heat equation
Chapter II
Some quasilinear parabolic equations. Self-similar 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 self-similar (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 self-similar 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. Self-similar (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 self-similar solutions is to be determined from the conditions of invariance of an equation with respect to certain transformations. In the general case families of self-similar 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 Lie-Backlund 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 half-space for the heat equation. The concept of asymptotic stability of self-similar 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 half-space for the heat equation
II Some quasilinear parabolic equations
40
motion of the wave of heating, find how its depth of penetration (half-width) 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 self-similar 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 self-similar solution
The self-similar solution (6) constructed above has a simple spatio-temporal structure. From the form of the solution it is easy to determine the dependence of the depth of penetration (half-width) 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 self-similar 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 (non-similarity) 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 self-similar form, 11,(1)
~ I A boundary value problem in a half-space 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 non-self-similar thermal wave.
3 Asymptotic stability of the self-similar 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 spatio-temporal 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 self-similar 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 self-similar 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 self-similar 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 self-similar 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 self-similar 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 ,ll-u\(t)/IIIIII]--O,l->OO.
(21)
.
110(1,') - Os(·)lIcIR,
I
==
sUp~,(J 18(1,~) - ()\(t;)1
= ( 18)
= 0«1
+ 1)/1')
-, 0, 1--+00.
Pro!}/: It follows from the Maximum Principle. Let us sci satisfies the equation ;:,=::",I>O,X>O. and furthermore ::(1.0) son theorem we obtain
43
= 0,
I > 0, and sup, ,(J 1::(0, xli <
1::(1, x)1
:s:
M
= sup 1:.:(0, Xli, ,.(J
00.
= // -
//.1"
If (20) holds, we have the same exact estimate (19) for the depth of penetration of the wave. This result gives us an explicit form of the evolution of the heating process for arbitrary initial perturbations and for boundary regimes asymptotically close in the sense of (20) to a power law dependence.
Then -
4 Asymptotic stability of the self-similar solution with respect to small perturbations of the equation From the compariLet us show now lhat spatio-temporal structure of the self-similar solution is preserved for large I > 0 in the case of a "small nonlinear perturbation" of the original
I > O.
parabolic heat equation.
* I A boundary value problem in a half-space for the heat equation
II Some quasilinear parabolic equations
44
Suppose a sufficiently smooth thermal conductivity coefficient is not constant: k == k(u) > 0 for II ) O. However it is close to a constant for large temperatures: k(u) -,
I. II
(22)
-> 00.
Let us consider the same process of heating, but now in a nonlinear heat conducting medium: (23) II, :::: (k(u)II,)" I> O. x> O. where 11(1, x) satisfies the boundary conditions (2), (3). For convenience, let us introduce the function
45
the requirement that Uo E L 2 (R t ); by (12) the self-similar solution usU, .) is in L"(R+) for allm > O. We restrict ourselves to a formal analysis of equation (25). Let us take the scalar product of equation (25) with win 1}(R t ), and, having convinced ourselves that this product makes sense, let us integrate the right-hand side by parts: this is allowed in view of uniform boundedness of the derivatives u, and (11.\), 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 lIOn-increasing 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 "(R1-l norm implies, in particular, pointwise convergence almost everywhere. Proot: The function
).
*
If condition (22) holds, the right-hand 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 (non-monotone 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 self-similar 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 self-similar 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 self-similar 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(expl-lxI 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 well-known self-similar (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 self-similar 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 self-similar 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 spatio-temporal structure of the solution (5): (9)
(substitution of the solution
into
(5)
(9)
49
Here we use the self-similar 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 self-similar 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!f-simi/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 right-hand 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 self-similar 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 self-similar 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 h-I(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 right-hand 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 right-hand side is nonpositive, cstimating the second one lIsing the Cauchy-Schwarz 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 self-similar 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 self-similar 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 right-hand using the equality
§ 3 Asymptotic stability of self-similar solutions of nonlinear
heat equations we obtain
Let us consider flrst the example of a sclf-similar 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 self-similar 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 self-similar 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 self-similar solutions of nonlinear heat equations
II Some quasi linear parabolic equations
54
For arbitrary k(u) equation (I) admits a self-similar 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 self-similar 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 self-similar 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 self-similar solution Us. Therefore the law of motion of the half-width of the self-similar 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 self-similar 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 self-similar 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 self-similar 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 spatio-temporal 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 self-similar solution is just the travelling wave considered in Example 6 of eh. I. The depth of penetration of the thermal wave described by the self-similar solution (14) has the following dependence on time:
(13)
where thc boundary regime is strongly non-stationary: 1I1 (I) grows without bound with l. Some examplcs of generalized self-similar 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 non-dimensionalized 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 self-similar heating process
*J Asymptotic stability of self-similar 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 self-similar 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 half-width 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 self-similar solution.
lIo(x)
c
Fig. 6. Evolution of the self-similar 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 half-width of the thermal wave is shown by the dashed line. As in I, this self-similar 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 half-width 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 blOW-Up: (20) 1I,(f)::::;: (To - t) l/lT, 0 < { < To. where 0 < To < oc is a constant (blow-up time), The boundary function in (~O) becomes infinite in tinite time: III (t) --+ oc as I --+ To this regime corresponds a self-similar solution of ( II) of an unusual form, a stalldillg {herm(/l \I'(/\'{':
To,
(21 ) 2 E\pollell{ia! bOlllldary regime
A different asymptotically stable self-similar 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 half-width 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 self-similar solution (19) this estimate holds for large ( for a large class of non-self-similar 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 blOW-Up 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 self-similar solution (21) is asymptotically stable. In Ch. V we shall show that the heat localization of boundary heating regimes which exhibit finite time blow-up occurs also in arbitrary nonlinear media described by general heat equations of parabolic type. It is important to note that not every boundary blOW-Up 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 half-width (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 self-similar 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 blow-up regime. It is not hard to check that equation (II) has the following exact generalized solution: ,
11,(1. x) =(1'0-1)
Fig. 7. Evolution as
1 --,
T(l' of the localized self-similar solution
(21) (S
blow-up 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/OW-lip
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 spatio-temporal structurc of the self-similar 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 half-width 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 self-similar solution
, ] Iff!
1/" [ (l-x/XO)2_(I-I/T o)-/I"I-1
x)
""]1/" [ I-(I--I/To)-/I"I-I
==
0
(25)
and obviously 11,(1,0)
= (To -
I) 1/"( 1+ o(
1)) as
I --+ To,
so that this is indeed a boundary blow-up S-rcgimc 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 blOW-Up time I = To, the solution II, (I, x) is close to the self-similar solution (21). In Ch. IV we shall show that this self-similar 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 multi-dimcnsional 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 self-similar 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 self-similar 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 self-similar 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 self-similar 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
-II-II"
== 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/,,1-1 (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 so-called 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 right-hand 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)l-dx. 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'
w-dx+ 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 well-known 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 self-similar 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 super-high 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 self-similar 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 . I-m 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 self-similar 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 self-similar 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(m-I)" 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)~\(O) = 0, 11'.1'(00) = O.
n
and therefore u == 0 in if I >- 'I'll. If on the other hand m E (0, (N - 2)/(N + 2)j, N ::: 3 (I' = I/m ::: (N + 2)/(N - 2) and the boundary value problem (5') can be insolvable), we take as ps(x) the solution of equation (5) which is strictly positive in R N Then ps > 0 on an and the same argument applies. [J
(5)
I
(6)
This problem (in fact. just as (4)) is solvable not for all m E (0, I), II > I. Lemma l. LeI N ::: 3, 0 <
Proof Let us consider the Cauchy problem for (5) in R, with the conditions
§ 6 The Cauchy problem for the fast diffusion equation w(O)
Let us see, whether it is possible to have tolill exlil/Cliol/ il/.!inile lime in the Cauchy problem for the fast diffusion equation II,
=
~II/IJ ,I > O.
11(0, x) :::: uo(x) >
X E
RN :
0, x ERN;
11/ E
(0. I),
SUpllO
< 00.
(I)
(2)
The situation here is more complex than for a boundary value problem in a bounded domain; however, it can also be analyzed using self-similar solutions.
= fL. w'(O) = 0,
(8)
where fL > 0 is an arbitrary constant. Let us prove that every solution of this problem delines. under the above assumptions, a required function Ws· Local solvability of the problem (5), (X) for small T/ > () is established by considering the equivalent integral equation. ·\)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/N-I 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
I-Ill
""Ixl
:~/II-Ilil.
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 left-hand 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 self-similar 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. Self-similar solutions (3) provide us with the following law of motion of the half-width 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 self-similar solutions of equation (I) in R+ x R N , which do not have that property:
= expl-ldT + 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 self-similar 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(I-IIl)
,
(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 self-similar 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 self-similar solution (19). which can
be written down explicitly: IIS(t.X)=(T+t)N I1 2.fNIIII.III{
(I-Ill)
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 self-similar 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 self-similar 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 non-trivial and "non-obvious" 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 self-similar 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 right-hand 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' N-I/'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 riMht-lwlld 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(T-21IT + 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 one-dimensional 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)
= 1-41In(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 "multi-dimensional" 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 one-dimensional 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 + -l-II~, 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'u-k (~) 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 self-similar 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 super-slow 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 non-negative 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 (lo-t) 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) ~ (IXfl-lxl)II-ITI/" 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 multi-dimensional 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 self-similar solutions for it in the half-space 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 self-similar 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 self-similar solutions are asymptotically stable in the sense indicated
*
above (see l, 2). We shall consider more closely solutions evolving in a blOW-Up 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 self-similar solution has the form
From this section we begin to analyse specific self-similar solutions of quasi linear parabolic equations with an additional term Q(u) (either a source or a sink) in the right-hand side, Some examples of such equations were given in Ch. I. First we consider self-similar solutions of travelling wave type in active media with a source, This problem was studied first. and in an exhaustive manner, in the well-known 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-'-o----x
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 blow-up regime). equation (6) is easily integrated. The corresponding self-similar 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 blOW-Up regime, Heat does not leave the localization domain, and for x > Xo the homogeneous temperature background remains the same (Figure 8). The spatio-temporal structure of the self-similar solution (5) indicates that if n < -I j IT, the inlluence of the blow-up 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 (LS-regime). 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 hlow-up 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 Kolmogorov-Petrovskii.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 well-posed, 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):
°
1I0(X)
Then it is clear that
== J, x
:s 0, lIo(.n
==
0, x > (),
In addition. at the asymptotic stage of the evolution the prolile of the thermal wave coincides with the function O~({j), the solution of the problem (8) for A :::.: Ao, This means that the similarity representation of the solution of the original non-selfDC to a shift of the similar problem, 0(1, {j) :::.: l/(I, {j + x"r(!)). converges as t function O(~ ({j), that is, -)0
110(1,') - f}~(')IIClJ(1
{j
=x -
2 Upper bound for the penetration depth of the wave Proposition 16. 111 Ihe pro/JIelll (I). (5). (6) Ihe penetralio/1 depth oj Ihe w{/ve: x"r(l) :::
+ AH:, + Q(O\)
:::.: O. {j
E
R: fJ,d -(0)
:::.:
2..}a t -
I r;:;
2", a
11'1'
hm'e Ihe ,/il/lml'illg estimale jill'
In 1+ O( I).
I
-'> (Xl,
(\ 2)
(7)
At,
Pro(lt:
By condition
(3),
where A > 0 is a constant (the speed of motion of the wave), Substitution of (7) into (I) gives us the ordinary differential equation f/~
(II)
Below we treat in detail several simple questions related to this problem, which at the same time illustrate methods of analysis to be used later.
shown that the asymptotic behaviour of the solution of the determined by a self-similar solution of (I) of the lravelling
= Os({j),
0, 1-'>00.
-'>
(6)
the thermal wave will start to move to the right as shown in the law governing its motion'? What is its spatial profile for
us(l, x)
(10)
dx"I!dl:::.:2..}a+o(1),1-'>00,
Fig. 10.
11(0, x)
8'!
I.
0,(00) :::.: (),
(8)
The boundary conditions here were chosen based on the form (6) of the initial function, The similarity equation (8) reduces to a first order equation, It is not hard to check 1255] thal this problem has a solution for any
the function v(l, v, = v"
+ avo
x),
which satislies the equation (\ 3)
I > O• .r E R,
and the same initial condition (5), (6), is a supersolution of equation (I) (by the comparison theorcm, Theorem 2 of Ch, I), Thcrefore u(l, x) :5: V(l' x) in R , x R. The function v can be easily computed (the change of variable v :::.: e"'w reduces equation (13) to the heat equation for w): uU
By this inequality the required half-width x,,/(I) will not exceed s(l), "half-width" of the wave that corresponds to the function lJ, i,e., the solution of the equation n
/ -I = -e 1 2
2
27T
/
1% , .\(11//' ,
{.,.,2 } d.,." exp-4
(/4)
It is of interest to note that the exact value of x''i(l) docs not differ significantly from the expression in the right-hand side of ( 12) (see 1195 j):
r:. = 2'10'1 -
Lemma 2. Under Ihese asswllpliolls, Ihe .I'olwiO/l of equalio/l (18) is crilical: (-)/(I,£,,):::O,I>O,£"ER. Proof The function::.
3 2y
I-:.Inl ct
+ 0(1).
I -+
x,
= (-),
satisfies the linear parabolic equation
::., = ::'u + Ao::' t + Q'«())::.,
o
Hence we obtain the estimate (12),
x"./(I)
I >
Let us show that the self-similar solution (7) is asymptotically stable; stability is not necessarily with respect to small perturbations of the initial function uoC..-) ,
0 E (0, I) such Ihal
Q(OIl) ::: OQ(II),
u
(Ihis (,(}/ldilio/l is ,l'lIli,\}ied hy Ihe source (4)), prohlem .lilr ( I) lI'ilh inilial.limclion
E
( 15)
(0. I)
The/l Ihe solllliO/l or Ihe Callchy
u(O, x) = I/o(x) = 8(}~(x), x
E
R.
(16)
c())lver~e.l' aSVlIlplolically 10 Ihe simila,.iIY.limclio/l (lU£,,) in Ihe jillloll'i/l~ .I'e/lse: Ihere exi.l'l.I' a cO/l.l'I(//1I go, sl/ch Ihal
( 17) .Iii,. all
gE R
Proof The proof is based on the lemma stated below (similar assertions in a more
general selling are used in Ch. V). As a preliminary step. we pass from equation (I) to the equation satisfied by the function ()(I. g) = 11(1. t; + Ao/}: 0/
= Ott + Ao(Jt + QW),
I > 0,
g E R. = (-)(0, £")
E
R.
(/9)
E R.
Taking into consideration (18) and the form of the initial function Bo(O == uo(£") = 8O~(£,,), we obtain that it is necessary to verify the inequality
3 Asymptotic stability of the travelling wave
('O/l.l'I(//1I
O. ~
which is derived from ( 18) by di fferentiation in I, In view of sunic ient smoothness of () and Q, this manipulation is justified: however, one can weaken the requirement () E C~/ (sec eh, V). Therefore by the Maximum Principle ::,(I.~) ::: 0 in R+ x R if this inequality holds at the initial moment of time, that is, if ::.(0. g) == 0/(0,0 ::: 0. £"
Proposition 17. LeI Ihere exisl a
91
~ 9 The Kolmogorov-Petrovskii-Piskunov problem
II Some quasi linear parabolic equations
90
( 18)
Under this transformation the initial function (-)o(g) does not change. Comparison of (18) with the ordinary differential equation (8) shows that the problem of asymptotic stability of the travelling wave self-similar solution is reduced by the transformation to the analysis of stability of stationary solutions of the new equation (18).
(-)/(0, £")
= ({Jo)u + A(j{8o)t + QlOo)
==
"" 0 + 00(-),1' ) + Q(u(-)s)
0"0'
0«(-),1'
:::
o.' ~ E
R • (L""0 )
The function (-)~(g) satisfies equation (8) for A = Ao. Therefore (20) is equivalent to the inequality -oQ(B~W) + Q(0(J~(£,,)) ::: 0, g E R, which holds by assumption (/5),
o
To conclude the proof of Proposition 17, it suffices to observe that the function £") is non-decreasing in I for all g E R. and is. moreover, bounded from above:
H(I,
(21 )
since this inequality holds for I = 0 (see (16). where 0 E (0, I), and O~ is the solution of equation (18)), Therefore for any g E R there exists a limit (j(I, g) -+ f)* (£,,), ( -,-> 00. Passing to the limit as 1-+ 00 in the integral equation equivalent to (18), we see that ()*(g) is a stationary solution of the equation (18). that is. a solution of the problem (8) for A = Ao, As was mentioned earlier, it is unique up to a shift. 0 Remark. It is not hard to estimate just how different are the solution u(t, x) and the corresponding limit function (-)~, which depends on the magnitude of ~() in (17) (that is, on the amount of shift). First of all, by (21) go ::: o. Second, let us compare the asymptotics of the function H(~(£,,):
§ 10 Self-similar solutions of the semilinear parabolic equatioll
II Some quasi linear parabolic cquatiolls
92
= 08~W
=0
oC o exp{ -- .Ja~)
+ ... , g
(here Co > 0 is a constant). Taking into account the fact that by Lemma 2 R. we obtain from the inequality (J~(g + go) ::: Oo(t;), or, which is the same, from the condition
+ go))
::: oCo exp( - Jag} as t; -,
00.
an upper bound on the magnitude of t;o: Co exp{ - jago I ::: oeo, that is, go ::: -a-JilIn 8. If 8 > 0 in (16) is small and therefore Oo(t;) is very different from 8~~(t;), then the difference between this function and the limit function, to which O(t, g) converges as I ~> 00, can also be large. Let us emphasize that the initial function 1I0(.r) in (16), with which 11(1, x) stabilizes to the self-similar solution, is substantially different from (6): it docs not have a finitc front, and (the main difference) lIo(.t) -+ 0 < I as .r -, -00. However, the law of motion of the half-width of the wave is in this case closer to that of the self-similar solution. It is not hard to deduce from (II) that XI' f (I) = 2.ja1 + O( I), I -l- 00 (compare with (12'), where there is another term, which grows logarithmically as I -l- (0). To conclude, let us note that using the proof of Proposition 17 under the assumption of criticality of 110, we can demonstrate stabilization of (17) to the minimal function O~ without the restriction on the source term Q'(II) ::: a, II E (0. I). In this context, let us quote some examples of stable travelling waves 1I.\(I, x) = O~~(t;), ,t = x - Aol, which can be written down explicitly. If Q(II) = II( I - u")( I + (/I + 1)11"), 1I E (0, I): 1' := const > O. then for A := Ao = 2.ja
:=
= 1I0(X) > 0, x E R N : Q( II) = 1I1n II is a source
11(0. x)
(JIUO :::
00.
(2)
Here the function (Q > 0) for II > I and a sink (Q < 0) for low temperatures II E (0. I). Equation (I) is intcresting in that it admits a large two-parameter family of selfsimilar solutions. which allow us to give a detailed description of solutions of the Cauchy problem and, in particular. to determine conditions of asymptotic stability of the principal self-similar solution (it will be the first to be defined below). The source Q(II) = II In II > 0 for II > I satisfies the condition
h --. = I"" e>..
. 2
till
Q(LI)
. 111 2
dry ry
--:=
00.
.
Therefore all solutions of the Cauchy problem are globally defined, thaI is, they exist for all I > O. Since the function Q(II) is differentiable everywhere apart from the point II = O. in a neighbourhood of which it is a sink, the solution of the problem exists. is unique, and satisfies the Maximum Principle. Moreover. using the self-similar solutions constructed below, it is not hard to show (as in the proof of Proposition 4 of Ch. I) that every solution of the Cauchy problem with 1I0(.t) '!- 0 is strictly positive in R 1 x R N and is a classical one.
1 A one-parameter family of self-similar solutions
We shall seek the principal·1 separable self-similar solution in the form
= 1T- 1 sin(1TII)(2 -- COS(1TII»),
arccos {(e2~ -- I )/( I + C2~)}, t;
c·~ as t; --, 00.
:=
1!I,(I)(J,(x), (J,(x)
>
= exp(-lxI 2 / 4 j.
(3)
0 the equation
liE (0, I),
from which ~/,(I) I
x)
Then from (I) we ohlain for ~/,(I)
such a solution is
In both cases
sup 110 <
2 a solution of the problem (8) is
For a souree of "trigonometric" type,
fJ~(t;) = 1T
93
In this section we consider thc Cauchy problem for a semilinear equation of the particular form N (1) II, = all + II In II. I > 0, x E R ,
u~(I,
Q(II)
+ II In II
00
-l-
0(1, g) ::: Oo(t;) in R+ x
Co exp( ~ Ja(g
Llu
§ 10 Self-similar solutions of the semilinear parabolic equation Ut = .:1u + u In U
and of the initial function OIM)
II, ""
E
R.
= exp{Hoe' + NI21 1I~(r, x)
and
= exp{Boc' + N12) exp{-l x I2 14}.
.JThe sense in whieh "principal" is
III
(4)
be understolJu will become clear from the following.
••
•• •
•• •
•• ••• •• •• ••• •• •• •• •• •• •• •• •• •
•• ••
•• •• •
•• •• •• •• ••• •• •• • •• •• ••
•• •• •• •• •• ••
~..
~.
* 10 Self-similar solutions of the semi linear paraholic equation
II Some quasi linear paraholie equations
94
Here Bo is an arbitrary constant. This solution corresponds to the initial perturbation lIO(X)
== u~(O, x)
= cxp{B o + NI2 - Ixl~ 14), x
E
RN .
(5)
IISS(X)
= cxp(Nl2 -
Ix1 2/4l, x
E Ri\'.
(6)
Thus, thcre are three types of esscntially different sclf-similar solutions (4): I) a growing solution (Bo > 0): 2) a decaying solution (Bo < 0): 3) a stationary solution (Bo = 0). All these solutions can occur if wc usc quite a restricted sct of initial functions (5). What is the domain of attraction of each of the three types of the principal sclf-similar solution: for what lIoCr) will each type of cvolution occur':?
2 A two-parameter family of self-similar solutions We can givc partial answers to the questions posed above by constructing a larger family than (4) of self-similar solutions of equation (I). Wc shall look for these solutions in the self-similar form (now the variables do not separate as in (3)): 11.\(1, x)
where (lo is a constant: here for (10) to make sensc for aliI::: 0 we must have the inequality 00 < I. Thcn (8) gives us the following expression for the amplitude of the self-similar solution: 0
- (lOC")]} ,
95
(10) -
-(10
4-:-I (
IXI2 __ }. -
(10)
x ERN,
(12)
3 Condition of stabilization to the stationary solution Let us consider the case B o = 0 in (12). Then it easily follows from (7) that the corresponding self-similar solution converges as I -> 00 to the principal jelfsimilar solution (4). that is. in this case. to the stationary solution (6). From (10), (II) it is not hard to deri ve an estimate of the rate of stabilization to IISS(X). For each fixed x E R N (for Bo = 0)
Hence it follows that on any compact set K I. IP(l).
+ II In u
For a fixed Bo the one-parameter family of these initial functions (ao < I is a parameter) characterizes the domain of attraction of each of the three types of the principal self-similar solution (4).
(7)
The second equation can be easily integrated:
'/1(1)
==
II l1 s(l.')
24/(1) _ ~ __ I (pU) - (/)~ (I)
,
IIO(X)
Substituting the abovc expression into ( I) leads to the following system of ordinary differential equations with respect to the functions 1//(1). ¢(I): ,
= tou
where Bo is a constant (the same as in (4)). The constructed family of solutions (7) has the properties I )-3), however, it is a larger family than (4). as it depends on two parameters (/o and Bo. The corresponding initial functions have the form
N
It follows from (4) that if Bo > 0, "s(l. x) grows without bound in R as 1-> 00: if B o < 0 then u~.(I. x) -> () in R N as I -> 00. The valuc B o = () corresponds to thc stationary solution of equation (1). which is independcnt of time:
II,
(II )
_.. lIs.\(.)lIcIKr)
= O(e
') -> O. 1->00.
The process of stabilization to the stationary solution is schematically depicted in Figure II. Thus the stationary solution (6) is stable with repect to perturbations of the initial function. not to ones of arbitrary form. but to ones which make up the selfsimilar initial functions (12) for Bo = O. Here the perturbations can be arbitrarily large in amplitude; see Figure II. where IIS(t 1.0) is several times larger than llss(O).
This result is interesting. since with respect to arbitrary perturbations, no matter how small. the stationary solution lIss is /lllslable. This is demonstrated by the following simple claim.
~ I() Self-similar solutions of the semi linear parabolic equation
II Some quasi linear parabolic equations
96
II,
= 1111 -+-
II
In II
97
Therefore the family of functions in (12) with Bo = 0 is the attracting set of the unstable stationary solution in the space of initial functions. We note that it is unbounded in C(R N ).
4 Decaying solutions These exist if Bo < 0 in (12); then IIs(l. x) -> 0 as I -> 00 for all x ERN. Then it can be seen from (10), (II) that for large I 11.\(1. x) has practically the same structure as the principal self-similar solution (4). To be more precise, introducing the similarity representation of the solution (7) (the similarity transform corresponding to the principal solution (4)). 0(1. x)
us(l. x) 1/1(1) -
= - - = IJ
(
•
Ixl
-------
(I - aoe')I/2
) .
we see that Fig. 11. Stabilization as I stationary solution IIssLI)
Proposition 18.
--+ (X)
of a self-similar solution
LeI 1I0(X)
where 8 > IISS(')IIC1R N )
= olls.dx).
IIS(I. x).
Bo = () to the unstable
N
( 13)
x E R •
I is a CII/lslwll (de\'ililioll jimll Ihe slaliollar.\' solliliO/l Iluo(') l)e N /2 ca/l be arhilraril.\' sll/ilil if8 is c/os!' to I). Th!'11
= (8 -
lim 11(1. x)
= 00,
( 14)
x ERN.
f""··')..,
thaI is. Ihere is
110
slahi/blliol1
10 IISS'
Proof Let us take in (5) an arbitrary Bo that in that case 1I0(X) =: OIl\'.\(X) ::::
= B(;
E
II~(O.
N x). x E R •
(D. In 8). It is not hard to check
and therefore by the comparison theorem 1I(r, x) ::::
1I~,(t.
x) =:
explB(ie'
+ N/2} eXPI-!xl' /41
->00. 1-> 00 in R
110(1.') -1J.(·)IICiR'1
->
O. 1-' 00.
( 15)
(16)
Here o.(x) =: 1I~(I. X)!r/I.(r) has the meaning of the similarity representation of the principal self-similar solution (4). This estimate implies asymptotic stability of the principal solution with respect to perturbations of the form (12) of the initial function (5).
5 Growing solutions If Bo > 0 in (12). it follows from (II) that 11.1'(1, x) -> 00 in R N as I -> 00. Using the same formula (15) to introduce the similarity representation of the solutions liS (I . x). it is not hard to check that all these solutions (for any Bo > 0, ao < I) converge in the sense of (16) to the principal self-similar solution (4). It is important to emphasize the following point. Let us determine the rate of change of the effective half-width of the growing heat structure I,,{ (t) = !x" r (I)! defined by liS (I . xl'{ (I)) = 1//(1) /2. Using the explicit form of the function liS we obtain for the half-width the expression
N .
This concludes the proof of instability of the stationary solution (6) from above. N In a similar manner we can prove that it is also unstable in C(R ) from below: to any initial function (13) with 8 E (0. I) corresponds a decaying solution: 11(1, x) -> 0 as I -> 00 in R N , that is. again there is no stabilization to IIss· It is proved exactly in the same way by comparing u(l' x) with a self-similar solution (4), in which Bo = Bo E (In 8. 0). Then 1I(t. x) :::: 11;.(1. x) -> D. I -> 00. in R'\' since B(I < D. 0
l~f(r)=4In2.(I-aoe
')->4In2. 1-> 00.
that is, for large I it becomes practically constant. Nonetheless. the solution us(l, x) grows without bound on the whole space (compare with example 13 of ~ 3. eh. I, where the half-width being constant went together with localization of the thermal structure in space). To conclude. let us observe that it is very rarely that one is able to construct a large family of exact self-similar solutions which coincide asymptotically with
the principal ("generating") solution. Frequently one can determine generating solutions in the framework of the theory of approximate self-similar solutions (see Ch. VI).
•
II A nonlinear heat equation with a source and a sink
Remark. It is easy to construct a family of self-similar solutions of the form
(7)
for the equation lI, :::: !ill - II
In lI.
1
N
( 17)
> O. x E R .
[ER'
Therefore for large
*
lIs(t.x)::::exp{('LlrBII-~ln(UII("-I)Jl 2all
~?;I" } I:: : D(te 4all
LI )
-;.
O.
f -;. 00.
f
exp
11,(1. x) ~
Then. taking into account the fact that the function Q(lI) :::: -liin II > 0 for II E (0. I) is a source and QW) :::: Q( I) :::: O. we obtain a problem similar to the one considered in 9. However here Q'(O') :::: 00; therefore the speed of the motion of the thermal wave will not be asymptotically constant. Self-similar solutions of equation (17) have the form
99
As far as asymptotic stability of the family (18) is concerned. we have for all Bo the estimate sup IlIs(l. ?;cl/") - exp {-
IXI"} - . { -4aoc'
so that the principal self-similar solution here is a different one.
§ 11 A nonlinear heat equation with a source and a sink
Ixl"
.}
4(uoc' - I)
.
1>0. xER
N
•
Let us consider the quasi linear parabolic equation ( 18) II, :::: (II"U,),
where Bo. ao > I are constants. Let
Bo -- -
N
2all
In«(/o - I) ::: O.
Then. obviously. 0 < lIsW. x) ::: I in R N • and therefore. by the comparison theorem lIs(l. x) E (0. I) in R t x R N (this can be seen immediately from (18)). In this case (18) represents a thermal wave with nearly constant (as 1 -;. (0) amplitude that propagates in all directions (Figure 12). Its effective width has the form
+ II"'! I
-
lI. 1 >
O. x
E
R; iT> O.
(I)
It differs from the one encountered before (Example 13, Ch. I) by the presence of the sink -li. This can signilicalllly change the character of evolution of the combustion process. We shall look for self-similar solutions of equation (I) in the separable form 11.\(1.
x) :::: '/I(I)(J(X). I> O. x E R.
Substitution into (I) leads to the problem (2)
that is. the speed of motion grows exponentially as
1 -;. 00.
us(f,z) f ----------------------------
••
•• ••e,
~
II Some quasilinear parabolic equations
98
For convenience let us set A ::::- I / IT. Then we obtain for the function O(x) exactly the same equation as for (}s(x) in 3. Ch. I. Therefore we can take (J == 0.1" The amplitude of the solution '/1(1) is easily computed from (2). and as a result we obtain the family of self-similar solutions
*
·1 ) lIs(t.x::::('
(I '" +.11C')
(3)
;-;('
where CII is a constant. To each of those solutions corresponds an initial function
o Fi~.
12. A travelling wave in the Cauchy problem for equation (17)
lIlI(.r)
==
I,S(O. x) :::: (I/IT
+ CII )
1/"(JS(.r). x E
R.
Hence we have that it is necessary to impose the restriction CII > -I /IT.
(4)
100
§ 12 Localization and total extinction phenomena in mediawith a sink
II Some quasilinear parabolic equations
lOt
everywhere in the localization domain Ixl < (meas supp e.d/2. The perturbations do not leave this domain, even though the temperature grows without bound (see Figure 13). From (3) it follows that the solution liS grows according to the power law
11sft,X)
IIs(l, x) ~ (To - £)-I/"OS(X).
(7)
Therefore the stationary solution (5) is IIllstable: small negative perturbations lead to stabilization to a different stable stationary solution IISS == 0; positive perturbations lead to growing solutions which blow up in finite time. Let us observe that the spatial dependence of heat transfer processes in this nonlinear medium (combustion or quenching) are determined by the same function esLr); it is only the equations governing the change of the amplitude of the heat structure that depend on the type of the process. In this medium there is also a characteristic spatial scalc, which is common to all the processes, the ./illlt!a 1/1 ell tal lellglh L s = meas supp Os = 27T(CT + I) 111 I IT.
§ 12 Localization and total extinction phenomena in media Fig. 13. Evolution of sclf-similar solutions
for Co <
(:I)
n, Co == n, Co
>
with a sink
n
For different Co in (3) there exist three types of self-similar solutions having different spatio-temporal evolution, If Co = O. then (3) is a stationary solution (Figure 13): IISS(X)
==
(T1I"OS(X), x E
In this section we consider in more detail certain properties of solutions of the nonlinear parabolic equation with a sink III
where
(T
> 0,
j.J
= (1I"t/,) ,
- II", I >
(I)
0, x E R,
> 0,
(5)
R.
1 Localization of heat perturbations If Co > 0 then the solution
liS
decays (quenches):
We are already familiar with one of the important properties of solutions of this 3, eh. I): if the initial function 1I0(X) in thc equation, the localization property Cauchy problem is of compact support, then as 1/: ...... 00 hcat perturbations do not propagate beyond a certain finite length. As in example II of 3, eh, I. we can prove a more general assertion concerning localization conditions of solutions of the Cauchy problcm for (I).
(*
" IIs ( l,X ) =Co
I I" e '.Il} () (I L,\,x+oe),1
...... 00.
(6)
These solutions are below the stationary one on Figure 13, and their existence has to do with the presence in (I) of a heat sink, which for small II > 0 is more powcrful than thc source. On the other hand. if Co E (--I I(T, 0). thai is, if lhe initial function lies above the stationary solution. then finite time blOW-Up occurs:
/(.\(1. x) ~, 00, ( ~
T(l
I
--" -- In( -(Teo) > 0 (T
*
Proposition 19. LeI lIoLr) be U filllClioll lI'ilh colllJ!UCI ,1'II{J{Jorl UIlt! j.J < (T + I. Theil Ihere exisls a cOllslwl/ L > 0 slIch Ihul /((1, x) == 0 .liJr u/l Ixi > L .!iJr UIlY I > O. This result can be extended to an arbitrary number of space variables. It is not hard to carry through the same kind of analysis for equations of the type (I) with arbitrary coefficients k(u) ::': O. Q(te) :s O.
* 12 Localization and total cxtinction phcnomcna in mcdiawith a sink
II Some quasilinear paraholic cquations
102
The localization condition v < IT + I for heat perturbations is obtained by a simple comparison of the solution of the Cauchy problem with a suitable stationary solution; moreover, this condition is both necessary and sufficient. This is indicated. in particular, by the fact that for v :?: IT + 1 there are no non-trivial stationary solutions that vanish together with the heat flux in some finite point. We shall return to consider the character of the motion of heat fronts a bit later. while now we consider another curious property of solutions of equation (1).
In this case this phenomenon is related to the presence of heat sinks in the medium. Proposition 20. LeI
<
I'
I. sup 110
=
< 00.
M
Theil Ihere is To S T,
=
MI"/(I -1'). s/lch Ihalll(l, x)
==
Proof: Let us compare equation (I):
with the spatially homogeneous solution v(I) of
11(1. x)
*
4--6. Ch, IV in self-similar solutions can be studied by the methods used in the analysis of self-similar solutions with blOW-Up (the difficulties that arise in the process are on the whole of the same nature. and have to do with "singularity" in time of the solutions under consideration). See also the Comments. Both those properties. localization and total extinction. are illustrated by the following example. which demonstrates specificities of motion of heat fronts in media with volume (body) sinks. Example II. Let the equation
2 A condition for total extinction in finite time
0 ill RJilr IIIII:?: To·
= '-1"'(1).
I >
0;
v({})
= M.
By the comparison theorem, /1(1. x) S 1'(1) in R; x R. However. it is not hard to see that v(l) = 0 for I = T,. which completes the proof. D From these arguments it follows that if we replace the term -II" in the equation by an arbitrary sink --Q(/I) (Q(l1) > 0 for II > 0), total extinction occurs if ,1 .0 /
x)
= (To -
dTJ Q(TJ) <
00.
1)1 / 11
"IIs(I:).?
= x/(T o _nur+1
"'l/lcll-"11 E
R.
(3)
1(4 )
where I/I(r) :?: 0 and 1/)(1) :?: 0 are. respectively. the amplitude and the width of the heat structure, while the compactly supported function ()(~) :?: 0 has the form 8(~) = 1(1 - ()+]'/". ~ E R.
Its regularity properties are satisfactory from our point of view: at the front points ~ = ± I the heat nux is continuous. Substituting the expression (4) into the original equation, we obtain for the functions !//. 1/) a system of ordinary differential equations:
_~l2 + NIT_) + 21/)1// _ 4)c l // (T
(2)
Setting here where To > 0 is a constant (the total extinction time) and Is(1:) :?: 0 satistles the equation
(r" t)' ..\ ..\
(0. I). Let us consider in R, x R N the Cauchy problem for
Let us assume that at the initial moment of time I = 0 all the heat energy is concentrated at the point x = O. that is L1(O. x) == 0 in R"\IO} and that 11(0, 0) = 00. This is a typical "self-similar"' statement. containing minimal information about initial data. We shall look for a solution of the problem in the form
By the comparison theorem, the same result holds in the multidimensional case. Formally. we may asume that the asymptotic stage of the total extinction process (I ~ To) is described by self-similar solutions liS (I ,
IT E
lI\(I. x) = !//(l)()(~). ~ = Ixl/(I)· v'(1)
( IT
+
I) ...
2(1 -- II)
I'
I
I"·" + - I - ..\ I'" 1-//"\
..\~
= O.
~r E R.
(2')
However. this fact depends strongly on the existence or nonexistence of non-trivial solutions of this ordinary differential equation which satisfy the condition Is ~ o. II:I ~ 00 (similar problems are treated in I. 3, Ch. IV). Then the expression (2) shows us the evolution of the extinction process. Asymptotic stability of these
*
103
If
= Y(I), ."
1/1'"
(}Z)
c
ITIP"
I/J"+ I
=0 .
(5)
= Z(I), we obtain the system
2(2 + N(T) I }"Z" 4 =- - - , --IT - + YZ-c = 2' if
if
which can be easily integrated. Let us write down the expressions for the amplitude and the width of the thermal structure: (4')
~ 12 Localization and total extinction phenomena in mcdiawilh a sink
II Some quasilinear parabolic equations
104
lOS
3 The motion of the thermal wave in the absence of localization For II :::: (r+ I in (I) a compactly supported initial perturbation will not be localized. This is because at low temperatures the strength of heat absorption is not sufficient to halt the thermal wave. The nature of the motion of a front point in these cases is determined from the analysis of exact or approximate self-similar solutions of equation (I). If IJ i f + I, then (I) admits a self-similar solution of a relatively unusual form: (6) IIs(l. x) = (I' + t) 1/"f(7/). 7/ = x - A In(T + I). where
1'::::
and A > 0 are constants. The function (f"()'
Fig. 14. Thc ucpcnucnce of the amplituuc Ijl(r) and half·width (4). (4') on time; I To is the total cxtinction lime
Here C/o
=[
2(2
+ NrT)'J rT
h
o
,N"il"+N"J
_.M
= W'~"Nr~)
[?_(2
. _[2(2"+(T N
• (0 -
:/'rT)r l
of the localized solution
(rl] "il"'
iN"liI2,N",
=[ (2
A_..._ ] ,2,
N"Ji!"IIINrTi!
+ NrT)h o
the width of the structure (/)(1) grows; subsequently the surface of the heat front starts moving back towards the origin .I = O. and finally. at time
the functions 1//(1) and 4)(1) vanish simultaneously, that is, we have total extinction. The self-similar solution (4) is localized: at every moment of time the diameter of the support does not exceed 24)(1 fl. Observe that (4), (4') imply that as I ..... To the solution has the following asymptotic extinction behaviour: IIs(l,X)
= [ rT(To-I)(I-'~")j ]
= O.
7/ E R.
(7)
*
VT
-
I
rT
satisfies the equation
To formulate correctly the boundary conditions for this equation. let us consider the following analogy with the results of Y. The quasi linear parabolic equation
= (v"!',), + ,Lv_v"'l.
T
> O.x E R.
N"I
A > 0 is a constant. Graphs of the functions 1/1(1), 4)(1) are sketched on Figure 14. It is interesting to note that the size of the support of the generalized solution Mdoes not change monotonically with time. On the interval (0. 'f), where I,
+ AI' + ~-f -- 1'"
f(7/) ~ 0
contains in its right-hand side the function Q(v) = vlrT - 1',,11 > 0 for l' E (0, (r,·li") and Q{O) = Q(rT 1iff ) = O. Therefore we could formally consider a Kolmogorov-Petrovskii-Piskunov problem for that equation and try to find a travelling wave self-similar solution. V(T, .I) = f(7/). 7/ = .\" - AT. Then the function f :::: 0 is a solution of equation (7). ami therefore it is necessary to impose the boundary conditions (-OO)=(T
li".I(oo) =0.
We restrict ourselves to deriving estimates of the size of the support of the generalized solution of the Cauchy problem for I' = rT +- I. when 110(.1) = 11(0, x) is a function with compact support. First of all, it is easy to prove the following claim: Ihere exisl COIISlilllls A > 0 (/Ild I' :::: I, sllch I!J(II meas SUppll(t,
x) :::
A + (T
1
In(T
+ I).
':::: O.
(X)
To prove this. it suffices to check that the function
II"
. (1+-0(1»
with i; = l.rl/do(To - 1)1 0 and d~ = 2(2 + NrT)T o. Thus it is not self-similar (cL 1 (2) with II = ]-(T) and is governed by the equation without diffusion. 11{ = _11 "
is a superso!ution of equation (I), and to choose the constants T :::: I and 7/0 E R so that I,IO(X) ": /If (0, x) in R.
13 The struclure of attractor for the semi linear equatiun
107
that is, unlike the case II = IT + I, for lJ > (T + I tbe thermal wave moves according to a power type law (faster than in (8)). [n this case (9) determines also, for example, the relation governing the change in time of the amplitude:
The proof procceds via construction of a suhsolution of a different form:
sup 11(1, x) :::: gs(O) 1
1/(1·· II, I~' 00.
\ER
II
=(I+tl I/"H(I-7]"/l/")I/",7]=7]o+x-Aln(l+tl.
where fI > 0, l/ > 0, A E (0, 1) satisfy certain inequalities; if they do, they can he chosen to be arbitrarily small. Therefore there exists an 7]0 E R such that 110(.1') =:: II. (0, x) in R. In view of the last estimate on, compactly supported solutions of the Cauchy problem are not localized for I' = IT + I. A sharp estimate of the support of any compactly supported solution can be derived by llsing a different particular solution. Namely, using the equation v, = A(v) == VV" + (I/IT) (v,)" - lTV", where l' = u", we observe that the quadratic operator A admits a linear invariant subspace W" = ,'fi { I, cosh(Ax)}, A = (T/(IT + 1)112. Therefore substiting v(l, x) = Co(l) + Cdtlcosh(Ax) E W", yields the dynamical system (ef. [49j) " C;, =-(T«(ii" + - -+IC j). 1
C"
IT
1
C' = - IT(IT+2)C' IT + I 0 I·
1
§ 13 The structure of attractor of the semilinear parabolic
> 0,
equation with absorption in R N
which can be easily studied. By using weak solutions of the form (v)+ in comparison with u(l, x) from above and from below, we derive the following estimate: meas suppu(l,
xl =
IT(IT
2
+
I)
II' Inl -
(I + () (_1_)) In I
as
1 ---+ 00.
For IJ :> IT + I asymptotic behaviour of the process is described by a self-similar solution of the usual form,
"
"
I' -
(IT
+
I) ,
I
+ --i(;;=:-"j')-' gs( + ~-_I gs
ll(O) ,= 0,
[:.1'(00)
In this final section we study in more detail the asymptotic behaviour of solutions of the Cauchy problem for heat equation with absorption in the multi-dimensional case: N (I) II, = tlu - u/3, 1 ;:-. 0, x E R ; {3 = const > I, u(O,
x)
= uo(x)
=:: 0 (¢O), .IE R
N
;
supuo <
00.
(2)
N
The initial function I/o is uniformly Lipschitz continuous in R This is a semi linear equation (IT = 0); however, the same analysis can be carried out for the more general quasi linear equation considered in 12. N Equation (I) is one of the few nonlinear parabolic equations in R , whose asymptotic behaviour as I ---+ 00 has been studied in sufficient detail. At present there exists a relatively complete, and for some parameter ranges, exhaustive, description of the attractor of the Cauchy problem for (I) as the manifold of asymptotically stable states (eigenfunctions of the nonlinear medium, e.f.), to each of which corresponds its attracting set l'f in the space of initial functions. A more detailed discussion of e.f. of a nonlinear medium can be found in Ch, IV (see
*
where the function gs =:: 0 satislies the problem (gd~s)
The situation for II > IT + 1 is simpler. The problem (10) has no compactly supported solution. For l' > IT + 1 the heat ahsorption on the wave front is so small that it exerts practically no influence on the speed of its movement for large I. As a tinal result we have that as I ---+ ::xJ the character of the motion of the front does not depend on absorption and is determined solely by the diffusion operator, that is u(l, x) is in some sense close to the solution of the equation VI = (v"v,)". But for this equation we know a self-similar solution, which describes the asymptotic stage of spread of the heat perturbation (see Example 8, Ch. I). Therefore 1.1,·/(1)1 11/1"t"I,1 -> 00, and furthermore sUP,cR u(l, x) ~ ,.-I/I"t"1 as 1---+00. Heat perturbations penetrate arbitrarily far; there is no localization.
l'
- gs
= 0, (
:> 0,
( 10)
= O.
A non-trivial compactly supported generalized solution of this problem exists for IT + I < I' < IT .!- 1 (similar problems are considered in Ch. IV). [n this case using the comparison theorem, we obtain from (9) an estimate of the support of the generalized solution:
1162,268,2691J.
Below we present a simplified description of the structure of the attractor of equation ( I ), which determines the asymptotic behaviour of solutions of the Cauchy problem as I -> 00.
~ !3
II Some quasi linear parabolic equations
108
We pursue two aims in concluding this introductory chapter with a detailed analysis of a particular problem. First, this is a result of a complex study of a rather complicated nonlinear problem. It turns out that the process of heat conduction with absorption in this case can evolve as I - , (X) in many different ways. In particular, a measure of this variety is the fact that the allractor of the equation is infinite-dimensional.) Secondly, we want to emphasize the essential difference in the structure of the altractor of the nonlinear equation with absorption (I) and of an equation with a source, which admits unbounded solutions. Analysis of the laller takes up a major part of this book. Without entering into details, let us indicate the main difference. If, roughly speaking, the equation with absorption has, for nearly all values of parameters, a "continuous" allractor, then in the case of an equation with a source term the altractor is "quantized" in a special way, and consists, apparently. of several collections of discrete states, combustion eigenfunctions of the nonlinear dissipative medium. Principles of constructing a discrete altractor are discussed in Ch. IV. Let us return to the problem (1), (2). The first "candidates" to be elements of the atlractor of the equation are. of course. its self-similar solutions.
109
The structure or allractor ror the semilincar equation
or an "exponential" one: H\(t)
= D(/IIH
1 N
eXPI-( /41
+ .. ,. t
-7
00:
D >
O.
(8)
Actually (8) is the limiting case of (7) for C = O. /.1. The set of sill1i/aritY ./ill1ctiolls (H s } ill the 1'1/.1'/'.1' 13 ::: 1+ 21N lIlId 13 < I + 21 N. The sets of the functions {O\ lin these rangcs of the parameter 13 are signiflcantly different, which eventually leads to differences in the asymptotic behaviour of solutions of the Cauchy problem (I), (2) for 13 ::: I + 21 Nand 13 < I +2IN.
Proposition 21. Let 13 ::: I + 21 N. Theil there exists all ill/illite IIl/lI1her (!f Solllliolls of the prohlell1 (4), (6) \\'ilh {}()\I'/,rll/\I' aSYlI1plotics (7) alld Ihere are 110 solutiolls Il'ith Ihe expollelllial as\'lI1plolics (8). /11 the case 13 E (I. 1 + 21N) there is all ill/illite collectioll ol'.!illlcrio/ls Hs(g) oI the fimll (7) alld 1/1 lewl 0111' soll/tioll Os of expo/leillil/I./im/l (8). Proof It is based on constructing super- and subsolutions, H j and H., of the problem (4). (6). We shall first seek them in the form
I Self-similar solutions and conditions for their asymptotic stability It is not hard to check that
Below we consider. for simplicity. radially symmetric self-similar solutions of equation (I) of the form (3)
where l' ::: 0 is a constant, while the function Os :> 0 satisfies the ordinary differential equation A H( H.1')
==
NI!' I, I Ii t I N(t" 0\.) + -0\1; + ---0, - H\, = 0 . 2' 13-1 ..
c:> O.
~
and therefore A,!({J-t ) ::: 0 in R; (that is, () 1 is a supersolution), if
(4)
(9)
It has the obvious homogeneous solution (5)
We shall be intercsted in its non-trivial solutions. satisfying the boundary conditions
Similarly. A/i( /I; ) ::: 0 in R! (()
is a subsolution) in the case (10)
(6)
A formal asymptotic analysis of equation (4) as t -, (X) (that is. Hs the possible asymplOtics of the prohlem (4). (6): a power law onc. fJs(t)
)For N
>
= ct
1/1/1
II
+ .... g
-7
I: for N = I it is al It:asl two·dimcnsional.
00:
C :> O.
->
0) yields
Varying the constants lit.. k 1 :> () which satisfy (9). (10). we can lind an infinile number of distinct pairs of functions (J 1 ::: /I _ in R l - Then. using a well-known principle in the theory or semi linear elliptic equations (see e.g. 1356, 357. 37X j), 10 each pair (0 (J ,} corresponds at least one positive solution (),d~) of the prohlem (4), (6), such" that. moreover. 0::: (}\(~)::: /It in R j ,
13 The structure of allraclor for the semilinear equation B(!' ;fl)
Let us try now to find OJ of exponential type:
~~----------~-------------
(II) Then AI!(Oj)
==
Ai
expl-ujfl x
x [at:!411'J -
1)~2 + [3 ~ 1- 2Na± - A~ exp(-ll':t(([3 - IllJ.
which gives us the following restrictions on the values of the constants at. A j ; > 0: llt
A~I>
Fig. 15. The case (3?: I + liN
< 1/4.
(_I --2NCJ:,)ex p { [3-1 (_I__ [3 - I I - 4a [3 - I +
a
2N CJ:+)}'
=1/4.Af3 '::SI/([3-I)-N/2.
(12) (13)
From the last inequality it follows that the subsolution of the form (II) exists only if [3 < I + 21N. In this case we can always pick, without violating (12), (J 3), the constants (YL. A j so lhat (J, ::: (J. in R, . which proves the existence of 0 a function ()s(~) with exponential asymptotics for [3 < I + 2/N. Non-existence of such a solution for [3 ::: 1+ 2/ N is established by the following lemma, which will play an important part below, In preparation. let us formulate a family of Cauchy problems for equation (4): A/1((J)
+ 21N.
h(/ve eXIJlJIlelltial asymptotic.I'.
Proof:
Let us multiply equation (14) by ~N
I
°
and integrate it over the interval
IfJ'({;)
+ ~fJ(~)~N = {( 2
.In
1/N '0(1/)
[~ 2
- -'[3 - I
+ 1if3 1(1/)J
d1/.
(15)
For [3 ::: I + 21N (N12 - 1/([3 - I) :::: 0) the right-hand side of this equality is strictly positive. If on the other hand Ii(~,.) = 0 ({) > 0 on (0. ~.).~. < 00), then.
+ 2/ N),
Theil Ih('/'e exists (/ v(/Ille J.1.1 E (0. fJl/). such
that Fir all J.1. E (0. J.1.1) the SOllitioll or the prohll'm (14) \,{/Ilishes (/t some poillt
=
~ ~ I' < x, For J.1. E 1J.1.1. 01/) th('/'I' exists al lea.H olle positive solll1ioll 0 with I'xpollential (/sl'IIlIJfOtic.l' (lnd an infinite nUIIlIJ('/' or solutions satis(.'·ing (7).
The second assertion has already been proved. The first one is established by "linearizing" equation (14) around the trivial solution () := n, Setting I)l(~) = f)(~; J.1.)1 J.1.. we obtain for the new function II' the equation
Proo(:
F/,,(,(',,) ~
(0, ~):
~N
curves in Figure 15 cannot intersect. In the case [3 < , + 2/ N the functions O(~: J.1.) have a more varied behaviour. Lemma 4. Let [3 E (I. I
Theil (}(~: J.1.) > 0 ill R; Jill' all J.1. E (0.01/). alld
C(l/1I101
=
fJ(~: fl) in the case f3 ::: I + Figure 15 shows schematically the functions fJ 2/N for different values of J.1. E (0.0 11 ), From (15) it follows that for [3 ::: 1+ 21N the function ()(~; fl) is monotone increasing in J.1. for ~ E R ,. so that differ~nt
( 14)
= O. ~ > 0: Ii'((n = O. 0(0) = J.1..
where J.1. E (0,01/) is an arbitrary parameter (clearly, 0 = 0(~:J.1.) -+ 00 as ~ -+ 00 in the case J.1. > 01/). Naturally, if (J = (J(~: J.1.) > 0 in R+ for some J.1. and 0(00; J.1.) = 0, then the solution O(~; J.1.) defines the required similarity function Os(~). We have the following Lemma 3. Lei [3 ::: I
obviously, Ii' (~*) ::s 0 and the left-hand side is non-positive for ~ = ~., so that ( 15) cannot hold, The second assertion of the lemma also follows from (15). If () has exponential 1 asymptotics (it is easily shown that in that case the derivative 0' will have the same property), then, setting ~ = 00 in (15). we arrive at a contradiction in a similar 0 way,
==
".'IN(t: N . 1 ("
"
"
. I'
)'
+ ~2'(" 1" ~ + __ r # _.I .._I./IL
:::::"f! ,.-
I
(I!
. I'
(16)
(it is clear that III (0) = I. I~ (0) == 0) with a small parameter J.1. f3 - multiplying the nonlinear term. The corresponding linear problem for J.1. = 0 has the form 1
* 13 Thc structurc of auractor for the scmilincar equation
II Some quasilincar parabolic equations
\12
liS itself, by liS (I , x; 1'). By asymptotic stability we mean, as usual, convergence of the similarity representation of the solution of the original Cauchy problem (I),
e(~;p)
e
-
H
-
-
••• ••• •• •
113
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
(2),
(),.(I.~) ::::
(1'
+ 1)1/1#-11 11 (1, ~(1' + 1)1/2),
I;> D,
~
(18)
ERN,
to the corresponding function Os :::: f}(I~\; IJ). The quantity l' ::: 0 is conveniently determined from the form of the initial function 110 E 'lr w It is clear that representation (18) of the self-similar solution (3) gives us precisely the function
8sW· The question of asymptotic stability of self-similar solutions is very easily solved in the case f3 > 1+ 21N.
.....,::",..:>...------="---------'-~
o
L-
Fig. 16. The casc {3 E (I. I + 'lIN). The thick linc dcnotcs the function (ls(~) with "cxponcntial" asymptotics
0=
(I(~: 1-'-.)
Proposition 22. For f3 > 1+ 21N Ihe a[{mcling scI 'lV" corresponding to a given self-similar solllliol! (3) has Ihe form
'WI' ::::
(110 :::
013 l' > 0:
lIo(X)- l'
I 2 ,I/I/J·\ If!.I(lxI/T / ) :::: o(ixl'2/I{J
I)).
Ixl
Using the change of variable ~ :::: 2( -7]) I j2, 7] < 0, it reduces to the boundary value problem for the degenerate hypergeometric equation:
->
xl·
(19)
From Proposition 21 it follows that this attracting sel is defined in an optimal 7] j '"0
+ (N'2 -
7]
)''.10 -
I. 73=-i!o:::: D,
7]
< 0;
fashion.
fo(O):::: I.
all solutions of which for f3 < I + 21N vanish at some point (sec. for example [35]). By continuous dependence of solutions of equation (16) on IJIJ·\ this is also true for all sufficiently small IJ E (0. IJI). 0 Figure 16 shows schematically the behaviour of solutions of (14) for different in the case f3 E (I. 1+ 2IN). To conclude, let us write down the solution of problem (4\, (6), for the case :::: 2, which has the explicit form
Let IIIl Elfl" Let us set 11'1: (x) :::: max(lIo(.r). 11.1'(0, x; Tll. WI1 (x) :::: min{ lI o(.r), 11.1'(0,.\;1')1, and let us denote by 11'"(1, x) solutions of equation (I), N w±(O. x) :::: w~(x) in R N . Clearly. 1I,i ::: liS. 1/' :: 1IS.)1l :: II:::: Wi in R , x R The function ;1 :::: 11' t - liS ::: 0 is such that PrtH!(
(20)
IJ E (0, ()n) W
f3
AN
()s(~) :::: - -...-~ (aN
AN == 48(·-(N
+ 14)·aN
+
~-)-
+ ---",
aN +~-
I(H 1+ NI2)112). B N
= 2(N +
It has power law asymptotics
BN
(7)
as
14
+
IO( I
i:
._>
x.
and therefore ;I(I.X)::
> O. ~ ::: 0,
= 24(2 + (I + NI2)1;2),
+ NI2)'J2).
2 Sw/Jilitv 01' sel(sil1lilar sullltiu/lS
The 1n D were ordered by introducing the parameter IJ :::: 8 1 (0) E (0, fl ll )· We shall denote the attracting set corresponding to a self-similar solution (3) by 'WI" and the solution
Vj1
/.'
(47Tt)' .. ,
( 17)
n'
ex p
{_1·4,f};;;(x+\,)d,\,.
By condition (19). ;I:(X) :: ,/>(!x!) in R N , where (Mlxl):::: o(lx! x. Then we obtai n from (21)
;~l~,
; i (I, x) ::::
0
(I
Deriving a similar estimate for;
N/2.f" = 11.\
(21 )
I
-
",v IJl
1 exp
•
{-
*~
}
4>(\,)
2/I/J
II)
as Ixl--->
til') .
we have
and it is not hard to see that the right-hand side goes to zero as t --->
00.
0
•• •• •• •• ••• •• •• ••
••• ••• •• •• •• •• •
•
••
•• •••
•• •• •• •• •• •• ••• •• ••• •• •• •• •• •• •••
~ 13
II Some quasi linear parabolic equations
114
A simpler estimate of the rate of convergence can be obtained under different restrictions on /./0. For example, from inequalities of the form (21) it follows that in the case uo(x) - l' 1/1f! II O,dlxl/T I / 2 ) E L1(R N ) we have the estimate II0r(l.·) - O,\(I·llllc(I{" = 0(1
N/2il/I/JII) ->
O.
1->00.
(23)
It is interesting to note that from (Il») we can deduce uniqueness of the similarity function Os with a fixed principal term in power type asymptotics (7). As can he seen from the estimate (22), this method of proof does not work for f3 :s I + 2/ N. In this case (as, in fact, for any f3 > I) the question of asymptotic stahility can be partially rcsolved hy using information ahout superand suhsolutions. 0 1 and {I • or equation (4), Let us write down the equation for the similarity representation Or = Or (T. ~) in a new time variahle T = In( I + I/T): iJO r
-'j(T
IJr(o,~)
= AWr). T > O. ~ ERN. = TII(/i IIUO(TII2~). ~ E
is a super- or subsolution of equation (24), then fh'(T.~) is monotone in T and hounded: therefore by a standard monotone argument for semilinear parabolic equations (see [22, 357. 378]) there exists a solution f)s = ()s(I~1l of the problem N (4), (6). such that (Jr(T.~) -> Os(I~1l in R as T..-.+ 00. In the general case the prohlem of determining attracting sets is related to the prohlem of uniqueness classes of similarity functions Os = Odl~I)· Namely, we have Proposition 23. Let 0 +, O. E C' (R N ) be. respectively, mdiall)' syl/ll/letric superalld subsolllliolls of equatioll (14),10 Il'lIich correspollds thl' sa1lle similarityfullc, V tioll Os = O(I~I:jL), f) :s Os :s O~ ill R . Theil the set
)(," = 1110::: 0\31' is
COli willed
N
f)
> 0:
ill '111'1"
In the case f3 <. I property of solutions,
(24)
Lemma 6. LeI RN
115
The structure of allractor for the semilinear equation
f3
E
+ 2/N
(I. I
it
+ 2/ N).
IS
important to note the following asymptotic
If' I/o
t= 0,
thell for some T > 0
(25)
(27)
Here A is the stationary operator A(O) :=
I
/1f.{I
iJO I L ~, + - - 0 2 / il~i f3 - I
+-
N
J
Il.
(26)
Proof Without loss of generality we shall assume that HO(O) > O. Then it follows from Lemma 4 that there exist sufficiently small l' > 0, jL > 0, such that
I
All the functions Os = Os(l~1l satisfy the equation A(H,\l = 0 in R N . Therefore it is necessary to study asymptotic stability of stationary solutions of equation (24). An important part is played by Lemma 5. LeI OJ (fJ .. ) bl' sOl/le sU!JCI'SOllllioll (subsolulioll) of equatio/1 (4), Ilial is Au(lJ, ) ::: 0 (Ali({l ) ::: 0) ill R N nlell tile sollllioll of equatioll (24) witll initial.lil/lctioll Or
Ho(r)::: T1!1f! l'O(\x\/T1l':jL).!x!
<.
T1/'meas suppO(I~I:jL).
(28)
where IJ(I~I;jL) is a solution of (14). Therefore by the Maximum Principle (the function f) in (28) is a suhsolution of equation (24)) 11(1,.1):::
I>
cr+n
O.lxl
<.
1!(/JIIO(lxl/(T+nll';jL).
(1' + nl/2meas supp()(I~I; jL):
and consequently
,Wr/ilT :s 0 (iJOr/iJT::: 0). T > O. ~ ERN.
P/'(}(!{ The proof is hased on the Maximum Principle. Indeed. the function (f}rJT satisfies a linear parabolic equation. ;: :s 0 in R N for T = 0 and so on. 0
Let us note that the subsolution 0 it is sufficient, for example to have 0 the radially symmetric function
in Lemma 5 does not have to be smooth; E C 2 wherever it is positive. Therefore. if
which ensures that (27) holds.
o
This is an important result. Practically, (27) shows that for f3 E (I. 1+2/N) the asymptotic hehaviour of all small solutions is descrihed precisely by self-similar solutions of the form C\). WC shall show below that the situation for f3 ::: 1+ 2/N is different. Let us indicate another interesting property which follows from the method of proof of Lemma 6.
•
116
~ I ~ The structure or attractor for the semilinear equation
II Some quasi Iinear parabolic equations
Proposition 24. For f3 E (I. 1+2IN) amol/g solutiollS Os(I~1l of the problem (4), (6) there is £I solwioll O~ havil/g the expollelltial asymptotics (8). whieh is mil/imal among £III (il/eludillg rudially nOIl-svmmetrie) solwiol/s o{ the elliptic eqllatioll A(O) = 0 ill R N The aI/meting set 'W 1'.' fL. = H,~(O), COllwins all sl~tficielltlv small il/itial./illlctiolls 11(1, alld, ill !Jartieular, the set
We shall need the following self-similar solutions of equation (30): (31 )
Here y > 0 is a parameter; the function 7) I N ( ry N
Pro(l! From Lemma 5 it follows that the solution of the Cauchy problem (24). (25) with the initial function Or(O. g) = O(ig!: fL). where O(igl: fL) is given in N the proof of Lemma 6. is critical, that is Wr
(,(0)
If')' "
f,
> 0 solves the problem
+ ~2'f",ry + y."f' = O.
2 Non-self-similar eigenfunctions (approximate self-similar solutions) Self-similar solutions with spatio-temporal behaviour (3) do not exhaust the set of elements of the attractor of equation (I). The remaining clements of the attractor are a.s.s .. which do not satisfy equation (I), unlike the exact slJlutions (3).
It is well known 135. 3171 that for y
==
f,(ry) = Mryc)'
If y
= N 12.
E
(0. N12) this problem has a solution with
+., .. ry
->x;
o(lxl
cflr!
xl·
= Ll.II.
t >
(33)
For y > NI2 (32) has no positive solutions. Let 110 EW o . It turns out that self-similar solutions lJf the form (33). (34) determine the asymptotic behaviour of almost all solutions 1I(t. x) in the cases 110 tfc L1(R N ) and lIo E LI(R/\'), respectively. Let us consider first the case Uo tfc L I (R N ). to which cOlTespond the values y < N 12. Let us introduce the similarity representation of the solution of the original problem (I). (2):
Proposition 25. Ll'f f3 > I
+ 21 N
({lId
f3(N - 2) S N.
(36)
that is, f3 E (I + 'lIN. (0) filr N = I or N = 2 alld f3 E (I + 21N. NI(N - 2)) Fir !'! ~ 3. Let there exist y E (I I (f3 - I). N12) alld positil'e COIlSWlltS T. M. A, sllch that
(37)
(37')
(30)
n. x E R N .
If 110 E 'Woo then the sink --liP becomes negligible as t the diffusion tel'ln.
0,
(34)
(2lJ)
If lIO E ' \JV o, then Or(t. ~~) --> 0 uniformly in f. E R N as t-".. 00, It is interesting to note that this simultaneously proves the known assertion (sec Lemma 3) that for f3 > I + 21 N there are no functions Os with exponential asymptotics. Therefore in the sel "Wo the solution 1I(t. x) does not evolves according to sclfsimilar rules. The asymptotic behaviour of 1I(t. x) for 110 E 'Wo is determined by self-similar solutions of the heat equation without a sink: III
= const >
(35)
O. we obtain the attracting
11).lxl ->
M
then the only appropriate solution is
set 'W o = !lIo ~ 0 II/oLd =
(32)
power law asymptotics:
I Conditiolls of asymptotic degellerucr of the absorptio/l process fill' f3 > I + 21N Let us return to Proposition 22. Setting in (19) 0\
'( > O.
'YJ
= O. f,(x) = O.
i'-· ""'-
the exponential asymptotics (8). Monotone stabilization Or -> 0'S(g) as T -> X N ensures that Os is minimal among all solutions of the equation AW) = 0 in R • as well as the inclusion 'Jf/-l. C 'W/ I •. This follows from the fact that we can provide the estimate (28) for any lIo(X) ¢O. 0
* 13 The structure of attractor for the semilinear equation
II Some quasi linear paraholic equations
ll~
= maxlO. zl ::: 0,
z~
=-
IIz(l)lIr'IRN)
Since by (37')
E
l
II
. I{'
l/ ::::
We shall assume that
j
= IIz+ (1)1I/liR s l + IIz-(l)III. ' IR').
and. furthermore. by (37) z±(O.·) that d II,_ t (I)III.lm"t:::: -, ( I
zl ::: 0 in R x RN . Clearly
min(O.
L1(R N ). From (38) it follows immediately IJ (I . .nd.\, . . -11,. d -' (I)III.'IR'I:::: O.
where C > 0 is a eonstant. From (40). invoking the Maximum Principle. we obtain h(T, Tt):::: C exp(-ITtI"/41 in R, x R
(39)
dt
ei'/I'/~iI,;,'
. R'
:=
"/JtNOAIJII/',1113U'IR','
(T+t)
.1
I>
:::: const . (T
+ t) NI'-yl,,
I >
+ ~2' r'lellll'/~
r'-llell'/8~fr(T'·)11"
./,
I >
O. Taking
we obtain now an estimate of the rate of convergence: y < (N y:=
(N
y> (N E
(1/((3'- I). N/2).
Ir
->
I,
as t
-> 00
+ 2)/(2f3), + 2)/(2(3). + 2)/(2(3).
in L1(R N ).
o
(40)
BUr) == 6.,/./1 +
N
~
/e",1
iJrt
N
7:L ":)-71 + "'i lr . 'Ttl 1
~
l'IJeiIJ!'/~
. r
.
,IT
dT <
00.
1.'(11')
It allows us to prove that every partial limit h·(T. Tt) -. I.(71) as T -. 00 N in L"(R N ) is a solution of the stationary equation BU.• ) = 0 in R . As IT :::: Cexp{-ITtI"/4j in R j x R N , we have that I. = I, = Mexp{,,-\TtI"/4\ (see (34)). The fact that the limit h(T. Tt) -> I.(TIl for T = T, -. 00 does not depend on the choice of the sequence (Til follows from a kind of monotonicity of the solution: d -/ III,.(T. ·)IIL'(I{s) :::: 0, T > 0 ( T
The restriction (36) in fact is not significant. and we can get rid of it by analyzing the stabilization Ir -. I, as t -. 00 in the norm of L',+I(R N ). where /) > 0 is a constant (ahove we considered the case fJ = 0). . Next we shall touch on the case 1I0 E 'W o n L1(R N ) if y = N/2. It is not hard to derive the equation for the similarity representation (35) for y = N /2 in the new time T = In( I + t (fl;
I
_ (T(',)'"IIJ 1,,\'(2
from which, after taking the scalar product in /}(R N ) with (f r ),. we easily obtain the important estimate
O.
The second estimate in (39) means that 1Iz'(I)llr"R" :::: const for now into account the fact that
•
(el'/I'/~'\71/Ir) +
'\7'/'
O.
It is easy to check that for y > 1/((3 - I) and under condition (36). we have the inclusion II E LfJ(R N ) (see (33)), and therefore
'ddr liz + (I)III. ' IJ{s,
N
so that h is unifonnly bounded. The differential operator in (40) is easily reduced to divergence form by multiplying both sides of the equation hy expll71I"/4}:
Au, in R, x R N • from the tirst estimate (39) we deduce that
::::AIJ!.· 1I~(I.x;Tldx:=
Therefore if y
119
(here we have used the fact that the functions I, == M exp{ -ITtI" /41 are monotone in M in R N ). It remains to show that II f, 0 (that is, M > 0). This is easily done by constructing a special subsolution of equation (I): lid
(I,
x)
:=
III(I)(T
+ t)"NO exp(-Ixl" /[4(T + 1)11·
It is not hard to check that (li ), :::: J.II 1//(1) ::::
For (3 > I
+ 2/ N
-lllfJU)(T
.. II
+ I)
IJ
in R+ x R
N
NII'-II/". I>
.
if
O.
we can take as 11(1) the function
1/'(1):= 1110
+ A(T + n'.
where € := N(f3 - 1)/2 - I and €A > appropriate restrictions 011 1I0(.r), we have 1/0exp{-!TtI"/4j for any T> 0, 71 ERN.
I> (1/'0
0; 1110> O. A> O.
+
/I ::: tL.
AT"')fJ. Therefore. under the N in R, x R , that is. fr(T, Tt) >
•
* 13 The structure of allractor for the scmilinear equation
II Some quasilinear parabolic cquations
120
and, moreover, assume thai in
2 Conditions for asymptotic degeneration of'the diftils ion process
a
00;
2
= const >
(41)
O.
Then above we considered the case a ::: 2/([3 ~ I): a = 2/{[3 - I) is the "resonance" case, while for a > 2/([3 - I). [3 > I + 21N, the absorption process degenerates. Thus it remains to consider the case a < 21 ([3 - I) in (41). It turns out that in this case as I 00 diffusion degenerates, and as a result the asymptotic behaviour of u(t. x) is expected to be described by self-similar solutions of the first order equation u, = -u i !. t > O.
X E
RN ,
Then there exisls P ::: 0 sl/ch that for allY illilial{unction I/o, which sati.l:!ie.I'./iJr sOllie T > Othecollditiollluo(·)-II,(O.·)II/IIII/2 E HI (R"'), stahili::.atioll frC) ---> I,(·) in U,jl(R"') occurs as t ---- 00.
z, =!lz -
(42)
g(l, x)
= [3 t
.10
(43) > O.
Substitution of (43) into (42) gives us the following equation for I " + -1 . --,'L aI, - f ~ (Ii = 0 a([3 - I) at;, .' [3 -' I " " " i.1
-l;
~ER
.'
I, ::: 0:
IM) = 1([3 -
I)
+ CH'(E;/I(:I)IC:I"IIi-11r'/I{.HI,
II, satisfies in
(7]u(t. x)
+ (I
+ h(l, x),
- 7])11,(1,1')/'
(48)
I
drl > D.
= !ll/,(I. x).
By assumption ::.(0,') E U+I(R''Il. VI::.(D. ')III,jll/2 E L 2 (R N ). Let us take the scalar products in L 2 (R N ) of both sides of (48) with Izl/'z. Then using natural assumptions concerning the regularity of the generalized solution of the linear equation (48), we obtain
N
,
(49)
(44)
It is easily integrated, and the general solution of the problem (44) has the following
form:
::.g(t, x)
h(t, x)
= (1' + t) 1/(Ii'III,(~), ~ = 1'/(1' + nl/I"lf! Iii E R'I; T = const
11,(1, x)
=u-
Pro(i{ We proceed with a formal analysis, The function z x R N the equation
R,
which can be conveniently written in the form
N
(45)
C(w) > 0, w E S: C(w) E C (S).
Let us summarize brietly the above results. Let unix) ~ lxi'''. II'I ----
121
Let the condition N-(a+2)(p+ I) <0,N+la([3-I)-2\(p+ I) >0
E; ERN,
where C(w) ::: 0 is a sufficiently smooth function defined on the unit sphere S = lw E R N Ilwl = I}. Let us note that in the general case the functions (45) are not radially symmetric. Let us move on now to determine attracting sets con'esponding to each of the a.s.s. (43). Below we denote by
be satisfied. Then it is easy to check that !ld, inequality
E
U II I (R N ). Using the Holder
as well as the identity
(46)
the similarity representation of
u(t, x)
()o =
defined using (43),
N - (a It{ [3
+ 2)( + I) + I)
- 1)( /1
< 0,
which follows from (43), we obtain from (49) the following estimale
Proposition 26. LeI (2 -
N)
[3 _ I
I
<
a < min
{")}
73=--)' N
(50)
(45)
(47)
d .. -, Ilz(l)II/J"'iJ{'1 S (l ( I
+ r)
.\
1/1,. I>
0:
•• •• •• •• ••• •• •• •• •
••
•• •• •• •• •• •• •• ••• •• •
•
••• ••• •• •
•• •
••• ••
•• •
•• •• •• •• •• ••• •• •• •• • .~
~
II Somc quasilincar parabolic cquations
122
111.,
= l16. c rJu"I!{"1
1.1 Thc slruclurc of allraClOr for thc semilincar equation
123
< 00.
Taking now into account the fact that a = const > 0, H = H (a) > 0 can be arbitrarily large. This is easily verified by substituting thcse functions into ( I). It can be shown that for all sufficiently small lIo(x) (for example, for lIo ~ exp{-I.\fl. Ixl ~ (0), after a tinite time tl > 0, the condition /I :.:: II :.:: 11+ will N hold for t = t I, x E R N , and therefore II :.:: II :::: II + for any I ::: II, x E R . Then from (54) we obtain the estimate
where
we obtain the final estimate of the rate of convergence: O(l~+ I
(51 ) Therefore this is also true for any possible limiting function g.(f) and the estimates
8 < -I,
()(I t),
{
have the form as t -, 00. One can see that 8 to (50) that E > 0, that is,
+
I -
E
< 0 if 8 > - I, Let us require in addition
N - alP
+
(52)
I) > O.
Then it is not hard to see that (51) guarantees stabilization of Ir to I, as t ~ CX;', and that thc system of inequalities (50), (52) is compatible under the condition (47).
The precise form of X. (q) becomes clear by using the equation satisfied by the similarity representation XI = Xr(T, ?;), with T = In( I + tiT):
o (58)
3 A.s.s. .li)r thc critical vlIlIIC or paramctcr f3 = I dcclI." lit il(finity
+ 21N,
110
has cxponcntilll
In this casc there ariscs probably the mosl unusual a.s.s.: 11,(1, x)
= 1(1' + t) InlT + 1)1
NJ2I.(. (
,
(1'
+x t)
1/')' -
l' > I.
(53)
By uniform bounded ness of grIT, q) (see (57», from the last equation (which can be put in divergence form by multiplying by expll?;1214\), it is not hard to deduce boundedness of expllql" 18l(grlr in L"(( I, (0) x R N ). This estimate allows us to pass to the limit as T ~ 00 in (58). As a result we obtain for g.(?;) the stationary equation
wherc the function g,(q) > 0 will be defined below. This a.s.s. corresponds to thc similarity represcntation (54)
The fact that for small 1I0(X) > 0, 11(1, x) evolves as I ~ 00 according to the spatiotcmporal structure (53), follows from the existcnce of a super- and subsolution of equation (I) of the following form:
and by (57') M E lA, HI· Below we only prove uniqueness of the limiting function g,(q).
Proposition 27. Let griT, q) ~ g,(?;) as
f3
= 1+ 21N,
T -+ 00
110 = lIo(lxl) ~ exp{-!xl") as Ixl ~ 00. Let IInif{mnl." on el'e,'Y compact sct in R N . Theil
Remarks amI comments
II Some quasi linear parabolic equations
124
36511. Proposition 4 is proved in 11191; see also the more general statements of
Proof By (59) we only have to show that M ::::: (N/2t 12 ( I
+ 2/N)N!14
(61 )
Integrating (58) over R N • which can be done by (57). we obtain
d
dT IIgr(T, .)II/ ' (R'1 :::: T
==
I
+ I;T
(" 1
[N
I 1-: -;:;"lIg,(T. -----1 T + n .!.
(grJ(T) ')1l1 1 iJ{')
== --
(62)
1 IIgr(T. ·)II/I-!'IR'I . T > l'llN .
'v C;'
. I
.,
(J (g,)
T.
H.T)
---..-'''--------- d T T + In T
must converge. Since under the conditions of the theorem C;'(gr)(T) we obtain the condition IIg.II/1m'·, > 0 (sec (57')) and
--+ G'(g.)
N + !IN == '2l1g,II/"IR" -- II g, 11 /.1'!·',R'1 '
as T
--+ 00.
= () . o
from which follow (61) and (60).
Remark. An elementary analysis of the behaviour of trajectories of the "ordinary differential equation" (62) as T -.... 00 also allows us to prove the stabilization gIlT. t) --+ g.(t) in R N as T --+ 00. where g. is the function (60). Under the stated conditions a.s.s. (53) satisfies the linear equation
1187] and § 4. Ch. VI. § 2. Concerning Proposition 5. sec 1386. 3841. The presentation of subsection 2 follows 11871. Another method of proof of a statement similar to Proposition 6 is contained in 12341 (let us note that the method of 1234] cannot be used to derive an estimate of the convergence rate of H --+ Is as I --> 00. which is of importance
in applications).
1.
From (57) we have that Ilg(T. ·)III'IR'1 is bounded from above uniformly in Therefore it follows from (62) that the integral
j
125
(63)
.
which differs considerably from the original one. To conclude. let us again remark on the curious transformations the semi linear parabolic equation (I) can undergo at the asymptotic stage. Depending on the magnitude of f3 and the initial function lIoC,). it is equivalent (in the sense of a.s.s.) to one of three types of equation: linear equation without sink (30). first order equation without diffusion (42), or. finally. equation (63) with a linear sink.
Remarks and comments on the literature § 1. Basic material needed to derive the elementary results of subsections 1-3 is contained in the well-known monographs 1282. 101.338\ (see also [378. 357. 361.
§ 3. Existence of solutions of the problem (5) under quite weak restrictions on the coefficient k(u) has been proved by different methods in 123. 24. 68]. The proof of Proposition 7(' uses the method of 11871: justification of the transformations used
there can be found in 1330 I· The self-similar solutions (14) of subsection 2.1 were considered first in 132. 29. 301. where existence and uniqueness of solution of the problem (16). (17) were established (subsequently. they were established by a different method in 1205]). Asymptotic stability of the solutions (14) with respect to perturbations of the initial function. boundary regime and the equation (the coefficient u" --> k(II)) is proved in 1119. IH41; in this context. sec Ch. VI. where similar solutions arc used to construct families of a.s.s. of nonlinear heat equations with non-power type coefficients. Solvability of the problem (19') and uniqueness of Is arc established in In. 29. 30. 32. 2051. Questions related to asymptotic stability of the solution (19) and construction of the corresponding family of a.s.s. of a large class of boundary value problems arc considered in II X4 J (these questions arc also briefly discussed in 3. Ch. V\). The localized self-similar solution (21) is studied in 1351. 393. 3521. Its asymptotic stability is proved in 1119. 153\ (sec § 4, Ch, III); the corresponding family of a.s.s. is constructed in 1119. 184. 187] (see 3. 4. Ch. VI). Analysis of self-similar solutions (23) is the subject matter of much of Ch, III. where additional information and the relevant references can be found. The exact solution (24). which is invariant with respect to a Lie group of transformations (sec 1322 Il. is constructed from general considerations by the methods of 1134J. 117(11. though. of course. it is well known; sec [441·
*
*
*
4. Existence and uniqueness of the self-similar function Is in (5) has been established by different methods in 17. 211. The estimate of the convergence rate (8) for arbitrary initial functions 110 ¢ 0 has been obtained in 1211· In subsection I we present a different (and. in our opinion. simpler) method of proof of convergence. The bountlary value problem with the condition (13) has been thoroughly considered in 1101. where solutions of variable sign were also studied. It is of interest. that in that case. in distinction to Proposition 9. it is possible for a solution 11(1. x) to stabilize. in a special norm, as I _., CX) to a spatially inhomogeneous function.
('Other estimates of the convergence rate are obtained in 1360. :126].
§ 5. First existence and uniqueness theorems for fast diffusion equations of general
form are proved in [343] (N = I). where the total extinction effect was discovered. Similar studies of the multi-dimensional case for 11/ E (0, I) incluoe, for example, [19, 43, 53, 328 J. In [43[ a method of proof of total extinction in the boundary problem (I )-(3) is presented. The proof of Proposition 10 uses a different approach. Asymptotic stability of the self-similar solution (4) under some restrictions on n c R N was established in [47[. Let us note that in the course of the argument of [47[, conditions of stabilization to an unstable stationary solution of a quasi linear parabolic equation 5.7, Ch. IV). are found (such problems arise in
**
§ 6. Here we mainly follow 11681. The first result concerning the total extinction
phenomenon in the Cauchy problem (I). (2) was obtaineo in [43[ in the case < 11/ < (N - 2)/N. N ::: 3. 110 E L1(R N ) n U'(R N ), Ii > (1 - 11/)/(2N). Conditions imposed on 110 in Proposition II arc weaker. In case 0 < 11/ < (N - 2) equation (I) is shown [251[ to admit a unique self-similar solution (3) with finite mass (this determines the exponent 11 > 0) which is asymptotically stable [166 J. Example 2 is taken from 1431. Proposition 12 is proved in 11681 by constructing a strictly positive in R t x R N subsolution of the problem. Propositions II. 12 provide a fairly precise oescription of the boundary between the sets of It/o I for which there is, or is no. total extinction in tinite time. Non-occurrence of total extinction for all 11/ > (N - 2), /N is proved in 119[ (see also 1328]).
o
*
§ 7. The elementary transformations II - f E(II) (Example 3) will be used in 2, Ch. V, to present the special comparison theory for solutions of two different
nonlinear parabolic equations. The substantial simplilication of equation (3) in Example 4 (the right-hand side of (8) is independent of r) obtains in the case IT = -4/3. N = I. when equation (lJ) is invariant with respect to a five-parameter Lie group of point transformations [80. 811. Concerning the transformation (10) and other properties of solutions of equation (9), sec [278[. For an application of the transformation of Example 5 see 7, Ch. IV. as well as [112.114.150[. The fact that it is possible to linearize equation (15) has been known for some time (for related results, sec the references in [12 j). Group-theoretic aspects of the ability to linearize this equation were analyzed in 1511 (note that (15) is the only nonlinear heat equation II, = (1\(11)11,), invariant with respect to a non-trivial Lie-Blicklund group: see 1221, 51. 262]). Example 6 is taken from 151[. Example 7 demonstrates new properties of the "multidimensional" equation with the coefficient k (II) = 11- 2. Proposition 14 is proved in 157j; we remark that that paper uses the same transformations as in [262. 221[. Equivalence of equations with k; = t/"', ITI + IT2 + 2 = O. was established tirst in1310j; using group-theoretic methods the same result was proved in 12211· In Example 8 we present a new particular solution of equation (36). which cannot be obtained by known group-theoretic methods. Proposition 15 is a natural general-
*
127
Remarks and comments
II SOIllt: quasilillt:ar paraholit: cquatiolls
ization of the result of [57 J. An exact solution of the super-SlOW diffusion equation (37) is constructed in 1250]. using the approach of 1571. In 1191 J it is shown that precisely this solution describes the asymptotic behaviour of an arbitrary solution with the same initial mass. Estimates of solutions of the super-slow diffusion equation of general form are obtained in 199\. Asymptotics of solutions of the boundary value problem for equation (37) with the Dirichlet boundary conditions is contained in [1451. § 8. Existence and uniqueness theorems for solutions of degenerate equations of
the form (I). (2) with lower-order terms are proved in [375. 296, 224. 371 J7. Continuity of the modulus of the gradient of the solution (that is. of the heat flux) of an equation of the form (2) has been established in [9J. The self-similar solution (3) in Example 9 is a particular instance of solutions of quasilinear equations of a more complex form, which were first constructed in [281. By the change of variable II, = V equation (I) reduces to the previously considered equation VI = (iv!"v).\\: therefore all results concerning asymptotic stability and a.s.s. extend with minor modifications to the case of gradient nonlinearities (this refers also to the localized solution of Example 10). § 9. For some generalizations and extensions of the results of 1255] see [241, 242. 95, 195.357,358.3611. The proof of Proposition 17 illustrates the technique of derivation of pointwise estimates of solutions of parabolic equations. as well as one of the simplest methods of comparison of different equations; more complicated examples are given in eh. V. § 10. The families of self-similar solutions (3), (7) were constructed in 180. 811:
presentation of the main conclusions uses the results of [82]. The result formulated here. concerning instability of the stationary solution (6) and the existence of the 5. 7. Ch. non-trivial attracting set corresponding to it. illustrates the ill1alysis of IV (where in principle we present a method of constructing a large attracting set of an unstable stationary solution of a quasi linear parabolic equation). The idea of using a two-parameter family of invariant solutions ( 18) to study the properties of travelling waves in the Cauchy problem for (17) is due to V. A. Dorodnitsyn.
**
§ 11. The one-parameter fami Iy of solutions (3), fcasibil ity of construction of
which was oiscussed on group-theoretic grounds in 180, 811. is consioered in [82 J.
=I in (I» is constructed in [302 J; for details on that see the survey in [162 J. A more general statement than Proposition 19, is proved in 1231[. The tirst mention of total extinction in a medium with a absorption is contained in [3431; an analysis of this phenomenon for equations of general form was undertaken in [231]. To study asymptotic stability of self-similar solutions (2) and solvability of the problem (2')
§ 12. The first example of a localized solution in a medium with a sink (Il
7S ee also tht: rdnt:nccs containcd therein_
•
Rcmarks and commcnts 128
we can apply the methods of ~~ I, 5, Ch. IV. See also 1192 1. A particular solution of equation (3) (Example II) in the case N = I is contained in 1248\; its multi-dimensional analogue is to be found in [3011. This explicit solution admits a natural N-dimensional generalization. Selling in (3) /I" = v yields tbe equation VI = vtiv + (I /(T) IVvl 2 - IF =' A(v), where the quadratic operator A has an (N + I)-dimensional invariant linear subspace given by the linear span WN+I == .cJ'{ L xf . .... x~, j, that is. A(W N + Jl c;:; WN+ I· For more general examples of invariant linear subspaces for nonlinear operators see 11361· Substituting into the equation V = Colt) + C 1(t)xf + ... + c'v (t)x~ E W N .; I, we arrive at a nonlinear dynamical system for the coeeficients. By analyzing its properties we derive. in particular. compactly supported solutions exhibiting /lo/l-syml/1etric extinction in finite time 1167]. Asymptotic extinction behaviour of solutions of (3) is studied in 1188 J. Some of the results of subsection 3. which deal with estimating the size of the support of a generalized solution. are proved in 1481, where references to earlier work can be found. More details concerning properties of solutions of nonlinear heat equations with a sink term can be found in 1230. 231. 237. 248, 1I. 50, 89. 208,2531 (sec the survey of [162] and 12331l. § 13. Presentation of all the results of subsections I and 2.1 follows 1162 1. Solution
(17) was found in [341· Let us note that in the case T = 0 every non-trivial self-similar solution (3) is generated by a singular initial function uo( x) of the following form: if I < f3 < I + 21N and (Js(g) is a solution of the form nn. then llo(.r) ~ D'8'(.\), where / = 2IN(f3 - I) > I (by a different method existence of such \'1'1".1' singu/ur self-similar solutions is proved in \551: uniqueness is proved in 1240\): if Ifs(g) has the power law asymptotics (7), then I/o(x) = Clxl 2/1p-11 in RN\{O), where C > 0 is the constant of (7). For I < f3 ::: I + 21N all these functions are not in 1. 1 (R N ). Therefore the self-similar solutions we construct seem to indicate the optimal degree of singularity necessary for the existence of a non-trivial solution of the Cauchy problem. The above conclusions agree well with the results of 1541, where it is shown that for f3 ::: 1+ 21N and uo(x) = 8(x), a non-trivial solution does not exist, i.e .. u == 0 in R+ x R N (let us note that for f3 ::: I +21N there is no self-similar solution (3) with Os =1= 0 of the form (8)). An assertion, which is stronger than Proposition 22, is proved in 12351, where for f3 > 1+ 21N the authors prove existence and asymptotic stability of an infinitedimensional set of asymmetric self-similar solutions (3), g = xl(T + 1)1/2 ERN. A generalization of Proposition 25 to the case of more general initial functions 110 (f3 > 1+ 21N) is contained in \208, 2351. In 12351 the reader can find conditions of stabilization of u(r. x) to stationary solutions of the form (31), where T/ = N x(T + r) 1/2 E R N and I, rf. LI(R N ) (Uo f/. LI(R N )); the case I/o E L1(R ), is considered in 12081·
12<)
11 Somc quasilincar parabolic cquations
Results of subsection 2.2 are contained in 11631· The stabilization If ---> f" x. of Proposition 26 means, in particular, that ,I/IP-IIU(r. x) ---> (f3I)I/I{HI, , ---> (X) for any x E R N This result was proved first in 12081 (obviously. it does not give any information on the spatial structure of the thermal perturbation). Let us note that since the function G(w) in (45) is sufficiently arbitrary smooth, the set of asymptotically stable a.s.s. (43) is infinite-dimensional. A more complicated situation of extinction in finite time for the porous medium equation with absorption. when the limit profile satisfies the first order HamiltonJacobi equation is studied in [1881. Proposition 27 and all the auxiliary statements required in its proof are established in 11621. A lower bound for the amplitude was discussed previously in 12081. The same phenomenon of the appearance of unusual logarithmic perturbations of the asymptotics arises in the Cauchy problem for the quasilinear equation II, = tiu" i I - ull , (IF> 0), I/o E I. I (R"') has compact support, for the critical value of the parameter f3 = f3. = IT + I + 21N. See the results of [I n. 1791 for the one-dimensional equation (N = I) and IllJOI, where this equation is studied by a different method for arbitrary N ::: I. In this connection, let us also mention the paper [23lJl, where the occurrence of logarithmic perturbations of the asymptotics for f3 > f3. is related to the behaviour of lIo(x) ~ Ixl-'" as Ixl ---> 00. A general classification of asymptotics of solutions depending on the parameters IF > 0 and f3 > IT + I is more or less contained in the papers [236.237.238. 23lJ. 240. 178, 17lJ. 188. 190,741. See also the references in the last papers and in the survey 12331. For the equation with absorption with gradient diffusivity /I, = V'(IDI/I"DII)-/lI3, IT> 0, the critical case f3 = IT+ I+UF+2lIN is considered in IllJOI (a uniqueness theorem for the asymptotics was proved earlier in 1178. 17lJll. Therefore existence of a critical value of the parameter. for which the nonlinear interaction of various diffusion operators of diffusion equations with absorption generates an unusuainon-self-silllilar aSylllptotics. is of a fairly general
This entire chapter is devoted to the study of unbounded solutions of parabolic equations. Here we consider the character of heat transfer in a medium. the temperature on the boundary of which follows a blOW-Up regime. We deal with the cases of power law dependence of the thermal conductivity coef1icient on temperature, and of a constant coefllcienl. The study is based on an analysis of unbounded self-similar solutions and docs not require any special mathematical methods. The main effort is directed towards the study of physical properties of boundary blow-up regimes. We consider in detail different types of propagation of thermal waves, establish conditions for the appearance, and the physical meaning of the heat localization phenomenon, which retlects a kind of inertia of strongly nonstationary diffusion processes. Analysis of the heat transfer processes is conducted in dimensional form, which immediately allows us to usc the obtained results to derive realistic physical estimates. These results are used in Ch. V, VI in the study of heat inertia in media with arbitrary thermophysical properties.
= [/,(1)
> O. 0 < ( <: T:
lit
E CI([O. 7')): [/,(1) --> X. (--. T.
(3)
The compatibility condition III (0) = lIo(O) is taken to hold. The main goal is to study the behaviour of the solution of the problem as ( --> 'I' . We shall be particularly interested in the conditions for heat localization, a paradoxical property of the heat conduction process. which shows itself at the asymptotic stage of a blow-up regime.
2 Examples of localization in boundary value problems
Example 1. A standing thermal wave (sec ~ 3. Ch. I). Let us consider the problem ( I )-(3), where k( II) 111(1)
= kOll".
IT
= const
= As(T·- I)
1(". (
>
O. ko = const > O.
< T. As
= const
> O.
(4)
0<: x ::: .Is.
§ 1 The concept of heat localization
This problem has a separable solution:
o<
1 A boundary blow-up regime
x ::: .Is.
(5)
x >- .Is.
We consider a one-dimensional process of heat propagation in a medium that occupies a half-space Ix >- 0) with thermal conductivity coefficient which depends on the temperature: k = k(lI) >- 0 for II > O. k(O) ::: O. On the boundary x = 0 the temperature follows a hlow-lIp regime. that is it becomes inlinite at a certain linite moment of time I = T > 0 (Tis the hlow-lIp lime) .
The process is described by the first boundary value problem for the quasi linear parabolic cquation lit
= (C/) ( II ) ) == (k ( /I ) II, ) ,. 1\
k
E
C2
«o. (0)) n C (I (). (0)).
(
I)
where (6)
Let us indicate the main properties of the solution (5). (6): a) for 0 ::: x < Xs the temperature liS (I . x) goes to inlinity as I .... 'I' - ; b) udl ..\') == () for all (E (0. n for any x~: Xs (Figure 17): the point x = x.\. in which both the temperature and the heat !lux arc zero, is a tixed boundary. which separates heated mailer from cold. The process of heat transfer is localized in the finite domain () < x < xs. even though in that domain the temperature grows without bOLind as I -. T .
• * I Thc concept or heat localization
III I·kat localil.ation (incrtia)
132
contained in the domain (XI:::: X < ool, is bounded. As in Example I. practically all the energy is localized in the finite domain (x < = xs(k o. Ro)}. The difference is that the temperature to the right of the point Xs is nonzero (Figure 18). but is uniformly bounded during the entire course of the process.
/2
x,
8 3
Fig.
3 Definition of localization and its physical meaning
17. A standing thcrmal wavc. Thc paramctcrs arc:
As "0 (1.354, .\s
T
0.5. T "'" I: I: T r",-c2.9·10·'.4:T-{ 9·10'\
2.
'T
1/ ~"
T ... {
1.25·10
.. U.S. ko = 0.5. 5.3 10'. 3:
= kll
>
0:
111(1)
Definition 1. The problem (I )-(3) exhibits slricI hcat IOCll!i::.alio/l if there exists a constant 1 > 0 such that 11(1. x) = () everywhere in (D. n x (I. (0). We shall call the smallest such number 1 the IOCl/!i::.alio/l deplh /'. and the set (0 < x < I'} will he called the loca!i::.alio/l domai/l . This definition has content if the following two conditions are satisfied: I
= Asexp(Ro(T .. t)
I},
N.o = const > 0:
1111(.1)
== O.
has the solution
.0
I
(8)
I I I
< T by
I I I I
I I
the function
As
lim I .r
l
c) for any
XI
>
X,'
the energy
E(I.
XI)
~
~
I I
=
')-112
(x)
Xs
= - 1-
-fiT
11(1,.1)
:u(T~.r) ~ ~,x<.r.s I
the temperature goes to infinity as I -+ T : b) for x > .1'.1' the temperature is non·zero and is bounded for all 0 :::: II(T. x) =.
d ~ < ex;.
l;
bances in processes described hy equation (I) (see ~ 3. eh. Il. Therefore Definition I makes sense, for example. for k(lI) = koll". IT > 0 (but is not applicable to a medium with constant thermal conductivity k == ko > 0). b) 110(.1') = D for x> 10 .1 0 < 00, that is, the function lIoe,) must be of compact support (there must exist a region of "cold background" initial temperature in the medium). ~u(t,.r)
satisfying the following properties: a) for
[k(g)] _.--;:-
~ which is a necessary and sufficient condition for tinite speed propagation of distur· a)
Example 2. Effective heat localization in a medium with constant thermal con· ductivity. The problem (I )-(3), where k(lI)
133
I
I.,
(9)
. 'X,
= .!~ 11(1. g) d§
('
l'
112 dv <(X);
.;I/I.lk,,1'1
o <
cons!. IE (0.
n.
.r
Fig. Ill. Thc conccpt or clTectivc heat localization
Example I demonstrates strict heat localization, and in this case the depth of localization is I' = Xs (see (6)),
1 Formulation of the problem
Definition 2. Problem (I )-(3) exhibits et.(eclil'e locali;.taio/l if the set (UI.
= {x
>
0111111,./
1I(f.X)
=x}
is bounded. We shall call the quantity L' = meas WI the eIll'cli\'(' locali;ario/l deplh. while the set {O < x < L'l will be called the e[(ectivt' locali;ario/l dO/l/ai/l. In Example 2 we have heat localization in the sense of Definition 2 with effective depth L' = Xs (see (8». For most physical problems the function 1I(t, x) (temperature) is never zero: therefore Definition 2 is the more natural one. Heat localization (inertia of heat) makes it possible to attain any temperature and concentrate any amount of energy in a bounded portion of space, and to contain them for a finite time practically without loss from the localization domain. This unusual property of the heat transfer process can be used in many applications. The concept of localization for meuia without heat absorption makes sense only for boundary blOW-Up regimes. If. on the other hand. insteau of (3). we have in the problem (I ). (2). a boundary regime without blQw-up, that is 1I(t, 0) = II I (t) ---+ 00 as I ---+ 00. then. as is easily shown, 1I(t, x) ---+ 00 for all 0 < x < 00. that is, there is no localization. The proof of this fact proceeds by comparing 1I(f, x) with self-similar solutions of equation (I) of the form IIS(t, x) = (J(x 2 II) :::: O. which exist for arbitrary functions k(lI) (see 3. eh. I). There are two possible regimes (moues) of heat propagation with localization. We shall say that an S-regime obtains if L' > O. Then the temperature anu the energy grow unboundedly in the localization domain as I ---+ T. If L' = O. then we have the LS-regi/l/e: 1I(f, x) --) 00 as I -) T only on the boundary x = O. The amount of heal that enters the domain (0, xJ,
*
E(t, x)
=
r ./0
111(1,
?;)- IIII(?;)] d?;.
0 < x < 00, IE (0.
can in this case be either bounded or unbounded as If. on the other hand. we have
lim
,·r
11(1, x)
I
---+
n.
*
It(l.
0)
= AoU' -
The required self-similar solution It,
I)". All
Its
= (kolt"It,),.
=::
const > O.
11
< O.
satisfies the following problem:
-00 < 1< T. x>
It( -00. x)
= 0,
= Ao(T -
I)". --00 < I <
u(t, 0)
x >
(I)
O.
(2)
O.
(3)
T.
Its solution has the form
(4) where I;
-
.r
= --,--.,-------- :::: kl/-A"I-(T _. 1)11+>IIrJ/2 o
(5)
0
0
is the similarity variable. The heat flux W
=::
-k(lt)u, has thc representation (6)
Here I(?;) and w(?;) = -I"(?;)f'(?;) ((' = dlld~) are dimensionless functions of temperalure and flux, respectively . The function I(?;) :::: 0 is determined from thc equation (("t)'
= 00
Itl (I)
=::
T
-1(1 + Il(T)/21f'~ + III = n, 0
<
?; <
00.
with the boundary conditions
for any x > O. then Ihere is JlO loca!i;ario/l and we say that the HS-regime obtains. In the present chapter we consider the case k(lI) = kOIl". (T :::: 0, kll = consl > O. We study the heat localization effect both in the strict sense 2. 3) and in the efTcctive sense 4). There is a close relation between the two definitions; it is established in 4.
(*
In this section we construct self-similar solutions of the problem (1.1 )-( 1.3) in the case k (II) = kllll". IT > O. Together with comparison theorems. they provide an efficient apparallls for studying the localization property. For k(u) = klllt". equation (1.1) has power law self-similar solutions corresponding to boundary regimes with blOW-Up:
(**
I(O)
=
I. f(oo) == O.
(8)
We rcmind the reader that in order for the solulion to make physical sense. we are seeking a non-negative. continuous solution of the problem (7). (8), while the function (v(?;) has to be continuous and bounded.
-I §
1Il Heat localization (inertia)
136
On the straight line ~I ==
2 Construction of self-similar solutions
(I)
and, using the change of variables
reduces to the first order equation
d11
1_
= __ [~(~. qlT 1/1
IT
IT
+1)
4/'+1
+(~+3) 4/"/I+IT(//'
1'//
IT
III _
a
~_JUT ~/l. 2
(II)
We shall say that a point!; I E (0. x) is a front point if I(!;) =: O. ~ ::0: ~ f· f(l;) :> 0 for I; <: {f. In the (41, 1/') plane a front corresponds to 4) =: O. the value of l/J is not known a priori. The behaviour of integral curves of the equation ( II ) is different in the cases I + JUT <: 0 and I + JUT> O. In a neighbourhood of the line 4) = 0 the integral curves of equation (II) have the form ( 12) ~I
2 , = -_.·.·~·~II) + ()(c/)'). I
(13)
+ fUT
Only the curve (12) with A = 0 satisfies the second of conditions (X) for I +JI(T <: 0 (the heat flux be continuous at a front point only for this choice of the constant A). For I + fUT > 0 the condition on the front is satisfied by the curve (13), Therefore if the solution exists. it is unique. From (10). (12), (13) we obtain the asympllltics of the solution in a neighbourhood of the front: for I + fliT <: 0
IW = ( -
+
I JUT ""'7-'''- IT I'
( 14)
for I
+ fliT>
()
f(l;)
=:
C(" 11 11/11'1 + (' I 1;1211
21/11 ;/11'1
-2¢/IT
+ .... I;
-->Xl.
(
15)
where I;r = (I(fl, IT) < 00 is the similarity coordinate of the front. (' = ('(11. IT) > O. C I =: _C" I 1[211(211 + JUT -. I) 1/( I + II(T)2 <: O. In general case the values of {r and C are to be determined numerically.
137
we have the inequality
dl/I
2
JUT
IIl/I
IT
24J"
-' = -- + --
Equation (7) admits the similarity transformation
d 4)
2 Blowing-up self·similar solutions
2 <: ---. rr
that is. as 41 is increased the integral curves intersect the straight line I{I = -2c/)/IT with a slope larger than that of the straight line itself. Since for c/I :> O. I{I <: -2¢/(T there are no isoclines of infinity. the integral curves do not leave that region. The desired trajectory (see (12» for A = 0 lies below the straight line I{I == -2Ip/IT. and therefore we may restrict ourselves to the analysis of the equation (II) in the region II) > 0, 1/1 ::: -2q)/rr. Then it follows from (10) that f~ <: O. { <: ;; J' that is, the required solution is a monotone decreasing function (monotonicity of the solution can be easily established directly from equation (7». In the IC/). ~/)-plane. III the boundary point;; = () correspond 1/1 = aG, 1/1 == -x. In the domain ¢ > 0, 1/1 < -2Ip/IT there is a unique direction. 1/1 == -21/I/rr, along which there is a bundle of integral curves ( 16)
where B <: D parametrizes the bundle, which enter the point c/I == x, 1/1 = -oc. The required solution corresponds to some value W == CO(I1, IT) < D. In the plane fl. to any curve in the plane of 1/). 1/1, there corresponds a family of similar curves. obtained by the transformation (l»): the solution I(l;l is choscn using the lirst of conditions (X). Integrating (16). taking into account (10) and (X). we obtain thc first terms of the asymptotic cxpansion of the solution in a neighbourhood of ;; == 0:
(r
where C II (I1. IT) == /'(0) < D is computed numerically. The expression in square brackets in (11) is a quadratic polynomial in '/1 with a non-negative discriminant. Therefore (II) can he rewritten in the form
wherc 1/1 = IV I (I/J) are isoclines of I,ern of the equation. The continuous curvc 1/1 = ~l (1M is wholly contained in the domain 1/) ::: D. '/1 ::: ---24)/IT. The isocline I{I = 1// 1 (1/) lies above the line I{I == --21/1/IT. Once the critical points corresponding to the front and the boundary, are analyzed (the other critical points are of no intcrest). it is not hard to construct thc whole field of integral curves. using which we can prove existcnce of the solution, a trajectory that connects the front point with the boundary point.
Solutions of equation (I) and of similar equations describing diffusion of heat from the boundary x = 0 in a half-space are usually called Ihermal waves. Let us review some concepts related to thermal waves (see eh. I. II). Thefrolll of a wave is the point with the coordinate x/(t). such that
f
1·
11(1. x)
3D Fig. 20. NUlllerieal solution or the problem
40 (7),
(X) for
lT
= 1,
1/
=
~O.15
= O. x>
11(1, x)
XI (I);
> 0, x < xf(t)·
The quantity x I (I) determines the depth of penetration of heat into the medium. The point with the coordinate x"/(ll. such that 11(1. x,or(l» = 11(1,0)/2. is called the hallwidlh point of the thermal wave. For the self-similar solutions, from (5) we have ( 17)
Figure 19 shows the "phase portrait" of equation (II) in the case I + IICT > O. The thick line denotes the required solution. while the dOlled one shows the nullisocline 1/1 = 1{1 (c/J). In Figures 20. 21 we present results of numerical solution of the problem (7). (X) for I + IICT > 0 and I + IICT < O. respectively. Thus. the solution of the problem (7). (X) exists. is unique and monotone. For I + IICT < 0 the front of the solution is at a tinite point. If I + IICT > O. then f({j) > 0 for all 0 < {; < 00.
Remark. In the case II = -I /(IT + 2), equation (7) has a hrst integral E = fiTI' - f{j/(CT + 2). E = const. It is easy to establish existence. uniqueness and monolOnicity of the solution for some E = E(CT) E (-00.0).
. '(1) ,1,'1 -
t:
1.II2 A iT !2(7'
<""Jl\o
0
_ 1)(I+l/( r l/:'
( 18)
'
where the constant g,'1 > 0 is such that f({j,,/) = 1/2. The amount of heat contained in wave at lime 1 is E(t)
=
j''''''
.10
11(1. x) dx.
For the self-similar solution we obtain L(I)
,., «»A lIr +:'I!2/.I!2
= :~ __o_,_~J)_(T ..• t)11+lIlIr'1211!2I',x' I
where
ctJ(O)
+ lI(cr + 2)
= -('(0)
.
I
II
f. - - - , CT
+2
> 0 (the heat l1ux at the boundary is positive).
(19)
.. 140
Xo
~
11\ Heat localization (inertia)
Below we shall make use of the following fact. For a fixed coordinate 0 < < 00 the similarity coordinate ~ = ~u. xo) changes in time according to (see
(5) ) 1'(( ~
\.) _
.·0
-
,.
"0
T
I [k ()I/2 Atr0 /2(T _ if
()I I
. usU.
01)::::
,.". . I)
Ao(l -
11
> -lilT and g(t.xo)
,."I) [I +
./(~):::: Ao(l -
IUT
•
"~(T<;/(g
.
/
._+
o as
. ] I/tr
-<;j).f
+ ...
IThis result docs not contradict the linite speed of propagation of heat in a mediulll with k(ll) ko//". IT 0: infinite time elapses rrom the start of lhe process till tillle
n.
/
( .....,. T- if
2) The width x r U ) and the effective depth of heat penetration x<,/(I) grow without bound as time approaches the blow· up time. In the limit the thermal wave covers all the space. 3) EU) < 00. ( E (-00, nand E(t) _.-> 00 as ( ~. T . 4) I/s(t, .Io)/l/s(l, 0) ......,. I as I .....,. T", that is, in time, at every point of the space the temperature behaves essentially as on the boundary x = O. Thus there is no localization for n < --I liT, and the HS-regime obtains. In some sense the l'IS-regime is similar to regimes without blOW-Up: with time. the influence of the boundary condition is felt in more and more distant regions of the medium (compare this. for example. with the self-similar solutions of subsection 2, ~ 3, Ch. II and of ~ 3, Ch. I, which correspond to boundary regimes without blOW-Up: 1/(1.0) = Ao(", Ao > 0, 11 > n, I > 0). However, infinite values of temperature arc reached not for ( = 00, but at the finite blow-up time. The HS-regime (HS comes from "higher" than S) is a "superfast" way of heating the medium: the boundary heating for I ......,. T is "faster" than in the S-regime: see Figure 22. Let us enumerate the physical properties of self-similar solutions for 11 > ~ I liT. I) It follows from (15) that the front of the wave is at an infinitely far point I : x /(1) :::: 00, ( E (-00. T). This result is a priori obvious fmm physical considerations. Indeed, from (17) it can be seen that under the assumption /; I <00, x/(I) would go to zero as ( --, T . This would mean shrinking of the domain encompassed by the thermal wave, which would be impossible.
( E (--00.
II (t,O)
(20)
f/I<TI/2] .
that is, gU . 11 < -lilT. Let us now analyze the physical properties of the self-similar solutions. In the case 11 :::: - I I IT, the solution of the problem (I )-(3) (the S-regime) is given in Example I. The front of the thermal wave and the half-width are constant, E(r) < 00, ( E (-00. T) and E(r) .....,. 00 as ( .....,. T . For 11 < -I I IT the solution has the following characteristics: I) The front is at a finite point gr < 00 and in a neighbourhood of the front we have the asymptotics .Io) .....,. 00, ( - ,
••
141
2 Blowing-up sclr-similar solutions
/ /
I I
l-"o!...t..,J0':_ _""I:!Lozu...:_ _
~.
L..._
T
Fig. 22. Thc ficld of boundary blow-up regimes
2) The half-width decreases as ( .....,. T at the rate given by (18). Energy which enters the medium is concentrated in a part of the space, which becomes sn1all~r with time. We shall call solutions of this kind thermal H'lI\'es (!( decreasing c./jec(/I·e dlIllCIISilill.
3) Taking into account (20) and the asymptotics
W(I. xl, ./
'II
= _C",·I_I +
IUT
(15),
for all x > 0 we obtain
A"llkll '"J/llllln.I12IH/(f IJ
111(1,""1
+., ..
IJ
that is, the self-similar solution converges from below to a limiting curve. The presence of this limiting curvc. thc "tracc" of thc boundary n:gime for ( = T-. whieh restricts thc growth of heat-rclatcd quantities at each point of the material is cquivalent to the definition of the LS-regime, and is an important property 01' that regime (from that we also have shrinking of the half-width in the case of LS-regime). 4) From (19) we obtain thc following. For -lilT < 11< -1/(rT+2) = 11 •. the amount of energy is finite: E(I) <: 00. I E (-'00. and Ftl) .....,. 00. t .....,. T -, that is, the mcdium is imparted infinite energy, which is bcing concentrated in a neighbourhood of the boundary. For 11 > 11* we have E(t) = 00. ( E (-00. T); nei~hbourhoods of the front contain an infinitc amount of energy, However, E(7 .. ) -- f:'(t) < 00 for all ( E (--00, TJ, which means that only a tinite amount 01 energy cnters the medium as the blOW-Up time approaches. Finally,
Thus, for II > -1/lT self-similar solutions belong to the class of LS-regimes (LS comes from "lower," the boundary regime as I -> Y- is "slower" than the S-regime; see Figure 22) and we have effective heat localization. Remark. Suppose it is not the temperature, but the heat !lux on the boundary, that blows up in finite time: , iJlI W(t,O)=-kou'(I,O)~I\ rlx
The problem
(I), (2)
,(
- k
.J -
(22)
Theorem 1. LeI Ihe Iwwlliary WI/diliolls ill Ihe problem (1.1 )-( 1.3) satisfy the ilIeqllalilies AsT·I/tr( 1-·· x/xs)c/", x :S Xs, (I) lIoL\} < 1 - { O. x > .Is = (2koN~(IT + 2)/(T)I/.,
0
.
As(T
-I)
1/1'.0 < 1<
(2)
T.
(22)
also has a self-similar solution:
1/ltr+cIwc/ltr+CI(7' _1.)ICII,+II/ltr+21/;( J)
143
1 A class of boundary regimes leading to heat localization
111(1) <
o=W O (T-t)"',1I1 <-1/2,I
with condition
.) -_ k 0
11,1 I, _t
3 Heat "incrtia" in media with nonlinear thermal conductivity
where As > (J is a cOllslall1. Theil fo//owillg estimales hold:
The function 1"(.1) :::: 0 satisfles an equation which reduces to (7) by the change of variable 11= (2111 + I )/(IT+2), and the conditions 1"(00) = 0, -F"(O)F'(O) = 1. Taking into account (9), we have for F(i)
Therefore self-similar solutions of the second boundary value problem (I), (2), can be expressed in terms of already analyzed solutions and have the same properties. For III < -(IT + I )/IT, III = -(IT + I )/IT and 1/1 > -(cT + I )/IT, HS-, S-, and LS-regimes obtain, respectively. Analysis of self-similar solutions that blow up in finite time is the first important step in the study of the localization phenomenon. The ideas of three types of thermal waves, of "fast" and "slow" solutions, will be frequently used in the sequel. Comparison theorems, which express continuous dependence of the heat conduction process on boundary data, together with self-similar solutions, allow us to map out the classes of blOW-Up regimes with differing physical properties. (22)
By comparison theorems, validity of the estimates (3) follows immediately from the properties of the self-similar solution 11.1'. The self-similar S-regime defines a class of "slow" boundary regimes, Which ensure heat localization. It is interesting to note that the estimate (3) of localization depth is independent of the period of action of the boundary regime. Figure 23 shows the dynamics of the thermal wave in the case 1I0(x) == 0, 111(1) = Ao(T - I) II". The half-width of the wave (crosses) increases initially, and then stabilizes. The front of the wave does not penetrate beyond the localization depth I' = XI = 0,5, The dashed line in that Figure shows solution (1.5). Another assertion concerning localization is established using the self-similar solutions of the LS-regime constructed in 2. Let us consider first the case 1I0(X) = 0 for x > O.
*
Theorem 2. If ill Ihe pmh/em (1.1 )-( 1.3) 111(1)
:s
Ao(T - 1)".0 <
I <
T;
11
= const E
(-I/(T, 0),
(4)
IhclI we hlll'c hcal /oca/i:atioll ill Ihe LS-"cgillle, alld Ihc ./iJi/owillg eSlimates hold: (5)
§ 3 Heat "inertia" in media with nonlinear thermal conductivity
*
In this section, using the self-similar solutions of 2 and comparison theorems, we study the influence of boundary blOW-Up regimes on a medium with a power law tr dependence of the heat conductivity coefficient on the temperature: k(lI) = koll , (T > O. Physical grounds for heat "inertia" arc discussed in the framework of studying the evolution of an initial temperature distribution in the Cauchy problem. Localization of heat in multi-dimensional problems of nonlinear heat conduction is also considered. '
lim, .r
11(1, x)
:S
C(II, (T)(Aoko")I/(1
1""'.1 2"/11,'"".
(6)
Proot: Since 11 > -I/(T, (2) follows from (4) for some constant As. that is, the boundary function is majorized by the self-similar S-regime. To determine the smallest constant A,I' in (2), we insist that at t = 0 the temperature on the boundary does not exceed the boundary value (Figure 22) of the solution (1.5): III (0)
:S AoT"
= AsT
I
/"
(7)
•
* 3 Heat "inertia" in media with nonlinear thermal conductivity
145
III Heat localization (inertia)
144
Figure 24 shows Ihe results of numerical computation of solution of the problem 1I0(X) == 0 and the boundary regime III (I) = Ao(T - I)" corresponds to the self-similar LS-regime. The half-width of the thermal wave first increases and then begins to shrink. The solution is bounded by the limiting curve (6) (dashed line): the front of the wave docs not penetrate beyond I' ::: Xs = 0.87.
U(t,I)
(I.I )-( 1.3). Here
16
12
2 Conditions for thc abscncc of localization Thcorcm 3. If ill the pmhlell1 (1.1 )--( 1.3) the bOlllldur." regime s(/tisfies the illB
eqllulit."
6
III (I) :::
thell there is
110
Ao( T - I)". 0::: t' ::: t < T:
loc(/li~llIioll (HS·regill1e) (/Ild us t
lim I
o
.~w_w _ _ ._.....
L
0.2
1I{1.
x)
·r
->
11 < -
1/(1'.
(8)
T - we hm'e the estill1(/tes
= 00 e\'eryll'here ill R!. (9)
_.. J~_",»-
0,1/
IS
0,6
x
Fig. 23. Heat localiz.ation in the c;"e 01 a boundary S-regime. The parameters arc: IT == 2. II == .-0.5. ko ==. 0.5. A c"" 0.354. T O~ 0.1125: I: T-t == \.105·10 I, 2: T .. / == 4.04· IO'c. o 3: T _ t == 3.05. IO-c, 4: T t \.0:; . 10 c. :;: T t == 6.5· 10 '. 6: T·- t == 2.5· 10 1
where Ali > 0 is SOIl1/' ('OIlS{(/lIt.
t 1J(t,I!
7: T - / O~ 5 . 10.. 4
Taking into account the fact that /.IO(X) == 0, we obtain localization from Theo· rem I, while from (7) follows the estimate (5) of the localization depth. For x ::: 0, 0 :::: I < T the solution 1I{1 •.\) is majorized by the self-similar solution 11.1' {I , x) for the LS-regime. which corresponds to the same values of the parameters (I', II, Ao. ko (this follows from (4) and the condition lI.dO. x) > 11(0. x) == 0). Then from the inequality II :::: liS in (0, T) x R t we obtain the estimate 1I{1. x) :::: IIs(T. x), which is the same as (6): sec subsection 3. ~ 2. 0 The theorem is true for any initial function 1I0(X) with compact support. since we can always lind a constant As > n. such that tlo(x) :::: IIS(O •. tl in R I · Then the estimates (5). (6) would depend not only on the parameters of the boundary regime, but on the initial data as well. Thus, if condition (4) holds, we have localization in the LS-regime. there exists a limiting curve. and the half·width of the thermal wave decreases. Since the boundary regime acts for a tinite time. unlike the case of self-similar LS-regime (see subsection 3. 2) the thermal wave has a finite front. The estimate of the magnitude or r in terms or the localization depth of the majorizing S-regime depends on the length of time or the heating process.
*
,...__..1 .
0,6
---
)-
Z"
Fig. 24. Ilcat localization in the LS-regimc. Thc paramc(ers arc: (T "'"' 2. II "'" ·-0.2:;. I k = 0.5. An == \.06. T = 0.1125: I: T / 1.02· IW • 2: T ..· t :1.1. IOc, 3: n T t = I. OS . I 0 c. 4: T I = 3 . I 0 .j. :;: T .. I ...... 2.4 . I () 'i. (): T -- t == I . 10 (,
3 Heat "incrtia" in media with nonlinear thcrmal conductivity
the inequalities
u(t,X)
WU.O):::: Wo(T-I)/II.
W(I.O) ::: WorT -
-(fT+ 1)/fT < 111< 1)/1
1
•
III
<
-(if
+
-1/2:
I )/fT,
arc satisfied. respective analogues of Theorems 1-3 hold. These results follow from properties of self-similar solutions of the second boundary value problem. which blow up in t1nite time.
3 Physical basis for heat localization. A class of temperature profiles with inertia
:r.
Fig. 25. Hcat propagation in HS~rcgimc. The paramcters arc: (T == 2. II == - I. ko = 0.5. Ao = 0.12. T = 0.1125. uot.1) = O. \ > 0: I: T - ( == 1.02:1· IW 1.2: T - I == 4.11 . 10 ·c 3: T - ( = 3.0l)· 10 ".4: T -- ( = 2.07· IW". 5: T~· 1 = 1.05· Ill'"
Here fr = {;I (II. fT) > 0 is the dimensionless coordinate of the front of the corresponding self-similar HS-regime. Let us show that under the assumptions made II ::: u, in U*. n x R+. where liS = IIsU, x; A~) is some non-localized self-similar solution of the HSregime (2,4). (2.5) for A o = A~. Without loss of generality let /((1'. x) > 0 on an interval (0, x'). x· > O. Let us choose the value of Ao == A~ > 0 in the self-similar solution (2,4) for II < - II if so small that
Results of subsections I. 2 show that localization is conditioned not only by the speed of the process. but also by "internal" properties of the heat conducting medium. Let us consider the evolution of a thermal perturbation in a medium. whiqJ1 is not acted upon by any boundary regime. or the Cauchy problem for equation (1.1) with the initial condition 11(0. x)
Prof)!
II.IU', x; A~) :::: 11(1*, x). x:::
o.
(10)
Existence of such a constant A~) > 0 follows from the obvious conditions (see (2.4). (2.5»: IIs(l', x;AI~) -'> 0, SUPPIIS(t*, x;A(~) -'> (O} as A~ -'> 0". Then by (8). (10) and the comparison theorem we have that 11(1. x) > us(l, x; A~) in (I', n x R j , which proves (9). 0 Numerical solution of the problem (I.I )-(1.3) in the case IIdl) = Ao( T < -lifT (I-IS-regime), is shown in Figure 25. Theorems 1-3 allow us to classify boundary conditions that lead to blOW-Up. and establish important properties of regimes of heat propagation. The "boundary" between different regimes is the boundary condition conesponding to the selfsimilar S-regime. Let us note that if on the boundary we arc given not the temperature. but timedependent heat nux that blows up in flnite time: WU, 0) -'> 00. I -'> T. then if
1)/1, II
The function constant.
/(0(.1')
= 1I0Cr)
:::
0,
(11 )
-00 < x < 00.
has compact support and supp Uo
= (--xo, xo).
Xo
> 0 is a
Definition. There is heal localizatioll (illerlia) in the problem (1.1), (11), if there exists (,. such that suPP/((I. x) = SUppliO for all 0 < I < I,. In other words, heat contained initially in the domain Ixl < Xo does not propagate out of the domain during the finite localization time I, (,(fT. ko: lt~). In the following theorem we characterize a class of "inertial" temperature profiles.
=
Theorem 4. 1(Ihe illilial he(l[ profile ,wli,ljies Ihe cOlldilioll (12)
Ihell Ihere is heal localizalioll ill Ihe Callchy problel/l (1.1), ( II )
WId
Ille localiza-
lio/l lime .l'lIri.l:!ies Ihe eSlimate
I, ::: I'
= .I~fTI [2k o(fT + 2)11;;'] .
( 13)
Proof: By the Maximum Principle.
11(1.0) :::: II",. I >
n.
(14)
•
*3 Heal "inertia" in media with nonlinear thermal conductivity
III Heat localization (inertia)
148
The function in the right-hand side of (12) has for x > 0 the same initial data as (1.5), the self-similar solution IIs(t, x) of the S-regime, if we set there T = 1 x1J(T/[2k o«(T + 2)11;;,], As = (xfl(r/l2k o«(T + 2)ll /" (here Xl' = xo, IIs(O. 0) = 11 m ). Let us compare the solutions 1I(t, x) and IIs(t, x) in (0. n x R+. From (12), (14), we have that 11(0, x) = lIo(.r) ::: IIs(O. x) for x::, 0 and LI(t. 0) ::: Ll II , ::: LI.dl. 0) for all 0 < I < T. Therefore by the comparison theorem 1I(t.
Hence, using the properties of the solution 1I{/, x)
= 0,
11.\
T
= t'.
( IS)
I'.
( 16)
Similarly, it is proved that tI{/, x)
=
o.
x
<:
-Xo· ()
Combining (15) and (16), we have that which concludes the proof.
tI(t.
<: , <:
x) == () for Ixi
> XII. 0
<: I <: I',
0
Thus, there always exists a class of initial temperature protiles which have the localization property. The estimate (13) of localization time depends on the parameters of the medium (k ll . (T) and the initial temperature protile (parameters xo,
Ti(t. x)
of equation (2,1) at time
I
tim)'
The heat inertia phenomenon has a simple physical interpretation. The rate of temperature growth at any point of the medium is determined by its spatial proli Ie ~n a neighbourhood of thaI point. If the temperature protile is sufficiently "convex" (in the case IT = 2 convexity has the usual meaning), the temperature docs not change, or changes only slightly in a neighbourhood of the front points x = ±xo. Therefore immobility of the thermal front depends on the behaviour of tllI(X) in a small neighbourhood of the front, though, of course, the length of localization time is determined by the global spatial structure of the initial perturbation (in other words, by the "degree of convexity" of its profile everywhere in supp tlo: this is rellected in the estimate (12)). It is clear that in a medium without absorption a convex temperature profile can exist only for a finite time 2 . Thermal energy enters colder regions from the holler ones, a "concave" temperature profi Ie is formed. and the wave starts to Illove. This is easily seen by comparing tllI(.r);E () with the function ( 17)
- 2Lo~alizat;;~;~~f·"tI;~I~I~~~I--;~~:;;II:;;~ltions coming frum thc prcsence in the mcdium of hcal sinks, which was hriefly considercd in Ch. I. II. is of a differcnl "physical" nalurc, In particular. in the presencc of ahsorption. localization for any Icnglh of timc is possihle.
= Ti,II(t)( I
..,
")
I liT
( 18)
- x- /C((I))I
= 0: here
'(1) -C' ( )'k l / I 'r+2I Q'''/1''121( .\ I I (T 0 0 I
-
(sec (1.5)). we obtain
x:::: XI), () <: , <
that is, the self-similar solution
C· (
LIm (I) - , } (T
x)::: Lls(t. x). x _ 0.0 < I < T.
149
)k
0
1/I,n2'{)~}/ilTI}'( ~o I
+ I 0 )1/lIT,2, ' + 10 )'"
1/1,,,2,
.
and C I. C} are some positive constants. This solution of "instantaneous point source" type describes the evolution of a thermal perturbation of energy Qo > 0 concentrated at the point x = 0 at time I = -10 < 0 (sec ~ 3, Ch. I). Choosing the magnitudes of 10 and Qo so that lIo(.r) :::: Tin(x) in R by the comparison theorem we have that 1I(t. x) :::: li(f. x) in R, x R and therefore meas SUppll(t. x) :::: meas suppTi(t. x) ~ 1 1/ 1,,+2, -> X as I -+ 00. Localization in the Cauchy problem is possible only for a finite period of time. The initial function (17) is an example of a "concave" temperature profile, which docs not have the inertia property. Here the thermal wave is in motion for all I > O. By the comparison theorem the same is true for all initial perturbations LlnCr). which majorize (17). when the fronts of the perturbations lIo( x) and lin ( x) arc the same. Figure 26a shows the results of a numerical computation, whieh illustrates Theorem 4 in the case lIo(x) = IIm(I-lxl/rn)~/". Till time heat is localized in the domain (-Xo. xo). In the course of time. the temperature profile reaJTanges itself into a concave shape, and the wave starts to move. Evolution of an initially concave profile is shown in Figure 26b, where the initial function has the form (17) (the size of both perturbations and the amount of energy they contain is the
,= ('
same in both cases). Figure 26c shows the dynamics of a "combined" profile: for x > 0 we have taken Llo(.r) = 11 11 ,(1 - x/xo)}/", while for x <: 0 we have taken the function (17). As a result the right side is localized, while on the left the wave starts to move immediately, that is, for a finite time we have directional heat conductance. The above properties of temperature profiles allow us to explain the physical nature of the localization phenomenon under th~ action of slow blow-up regimes on the medium (subsection I). Boundary S- and LS-regimes expose the inertia of the heat conductance process by creating and supporting localized temperature profiles. In the S-regime the rate of energy supply into the material is so adjusted to the properties of the medium that the heal is distributed over the whole profi Ie (see (1.5) and Figure 17). With a slower energy "provision," heat is mainly concentrated near the boundary, the profile is more "convex" (compare Figure 24 with Figures 2.1. 17), and the LS-regime is brought about. Formation of inertial protiles takes place at the localization depth, wbich is determined by the parameters of lhe problem.
•• •• •• ••
••• •• •• •• •• •• ••
•• ••• •• •• •• ••• •
••
•• • ••• •• •• •• ••• •• •• ••
u{t,.!:) U(I,I)
lSI
at the boundary x = () vanishes). then during the period leading to blOW-Up of the original boundary regime. the thermal wave hardly propagates (see Figure 26. a). During heating which is faster than that of the S-regime. a concave temperature prohle is formed (compare Figure 25 with Figures 24. 23); the domain occupied by the thermal wave expands. there is no localization. and we have the HS-regime. Let us note that the action of boundary regimes that do not blow up always creates concavc prohles and there is no localization (see the Remark in subsection 3.
* Therefore the heat localization phenomenon is related not only to the speed I).
of the process. Interaction betwecn the rate of heating of the medium and its properties determines thc naturc of the temperature profiles being formed. inertial or otherwise. This conclusion is also truc in the casc of arbitrary media (see Ch. V). including media with volumctric energy sourccs (see Ch. IV). D ~-'-'L-_ _-'----
a
o
......lJL,..
2
4 Heat localization in multi-dimensional problems. The "thermal crystal"
u{t, I)
The main properties of blOW-Up regimes in heat conducting media established" in subsections 1-3 for one-dimensional media. also characterize the case of many spatial variables. The mcthod of analysis of multi-dimensional equations is also based on the construction of certain particular solutions and the use of comparison theorems. The new clement, in comparison with one-dimensional geometry. is the shape of the heat localization domain. which can he quite contrary to intuition about
•• •
••• •• •• •• •• •• •••
*~ Heat "inertia" in media with nonlinear thcrmal conductivity
III Heat localization (incrtia)
ISO
diffusional dissipative processes. Let us illustrate this remark using easy examples.
First of all let us lind a
particular solution of the equation JII =~ iI. ( k(II)i.lll) , L-
c Fig. 26. Heal Ioc.dization in the Cauchy problcm. Thc parametcrs arc: a) IT = Xo = 2. ko =, 11 m = /' = I; I: / = 0.25, 2: / = I, :1: / = 2.5. h) IT = X ((0) = 2, ko = I, um(O) = D.62; I: / = 0.5,2: / = 2, ~: / = 5. c) IT = Xo = 1"((0) = 2. 11 m = um(O) = k o = I; I: / = D.5. 2: / = I, :1: / = 2
As the thermal wave approaches the boundary of the localization domain. the concave propagating profile rearranges itself into a convex form. which is easily seen in Figures 23. 24. From that moment the localization effect becomes manifest: the size of the heated domain docs not change signihcantly, the half-width is either constant or decreases. heat does not penetrate beyond the localization depth. If. after the formation of the inertial proille. energy is no longer supplied (the heat flux
iJl
.
I~
I
iJx,
(19)
iJx;
which is the multi-dimensional analogue ()f the self-similar S-regime. In equation (19) x = (XI, ... , XN) E R N arc the spatial coordinates, II = 11(1, X) ?: 0 is the temperature; k(lI) = kOIl". IT= const > O. is the thermal conductivity coefllcienl. As in (1.5). we shall seek a separable solution of (19): 11\(1,
x)
= V(I)(}(x).
(20)
Substituting (20) into (19). we obtain the following equation for the functions V (I), H(x):
I V" t
IV
_ _ _._1_
I lil
N 'J =._I '" -.!._
()
L
I:::·'
1
(lx,
( koH"-~ 'Jf) ) (lx,
= C > n.
(21 )
•
~ :>
III Heat localization (inertia)
152
Hence
= As(T -
V(I)
t)
lilT,
= (Ca)
As
1",0 <: I <: T <: 00,
(22)
that is, the required solutions will blow up as ( --'> T· at the same rate as in the one-dimensional case, The elliptic equation (21) satisfied by the function H has a solution of the following kind:
=L
a,x,: ai
:::'c
const > 0:
I
Then
(I('T)) .:::
L /0
Let us clarify the relation between the solution constructed above with the one-dimensional one (Figure 27): (25) XI
I
and, for example,
=
sian ding (hermal lI'a\'e.
a~ :;:,: I.
0 satisfies the one-dimensional equation
iJ(YJ)
(I _~!~) ~itr
,
X,I
==
153
the apex of the pyramid and decreases to zero as we approach its hase (YJ -> xs), which is the stationary boundary of the localization domain. By analogy with (1.5). the solution (24) can be called a mlllli·dimensional
tV
N
H :;:,: O(rl), YJ
Heat "inertia" in media with nonlinear thermal conductivity
(2k(}A:~ ~~~; 2)
I'
(23)
Therefore the desired solution has the form (24)
The spaticHemporal structure of this solution is the same as that of the one'tiimensional one, It can be considered as the solution of the boundary value O. i =:; I, ., .. N) with the corresponding initial problems in (0, T) x Ix E R N lx/ and boundary conditions. Let us indicate the main properties of the solution. first of all in two·dimensional (N == 2) geometry. The boundary temperature is prescribed on the XI . .12 axes. is equal to zero for XI 2:. .Is/ai, .12 .::: xS/a2 and blows up in tinite time for o ::: XI <: xS/al' 0 .::': .12 <: xs/a2. Nonetheless, unbounded growth of the temperature as I -> T takes place only in a tinite localization domain. the Iriangle with vertices at (0.0), (xs/al.O). (0, xI/(2)' In the rest of the medium (for YJ > X,d the temperature is zero for all 0 ::: ( <: T. The localization domain is separated by the stationary front, which is the piece of the straight line connecting the points Us/ai, 0), (0 . .I,ll (2). In the three-dimensional case the localization domain is the prramid with the apex at (0. 0.0). triagonal base and vertices at (xl/al. O. 0), (0. xS/a2. 0). (0. O. xs/in). On the lateral boundaries of the pyramid the temperature blows up in finite time in accordance with (24). while outside the boundaries it is equal to zero. Inside the pyramid 11.1'(1, X) ~> 00 as ( .-> 'I'm The temperature is maximal at
> XI'.
which does not depend on other spatial coordinates in the planes XI = consl. In Figure 27 the coordinate axes arc oriented so that the .\.1 axis is perpendicular to its plane. Let us rotate the X I . .12 axes by an angle {3 E (0. 7T /2) with respect to the Xl axis and let us consider solution (25) in a triangle with vertices at the points (0.0), (xs/nl. 0), (0. xS/(2): al == cos {3. (Y2 = sin {3. Inside it the temperature depends on (I. XI, X2). therefore 1/,1(1. X) can be considered as a solution of the two-dimensional equation (19). The boundary conditions and the solution itself are easily computed from (25). with rotation of the X I, X2 axes taken into account, and coincide with (24). The three·dimensional solution is obtained in a similar way after an additional rotation of the Xl axis. This methc1jl can be used to construct multi-dimensional standing thermal waves in more complicated domains (dashed line in Figure 27). Boundary conditions are determined from (25) if the houndary of the domain is prescrihed. If the boundary regime in the multi-dimensional problem is slow (majorized by the S-regime boundary dependence). then we have heat localization, and upper bounds both for the solution and the localization domain can be obtained using the function (24). Figure 28 shows the results of numerical solution of equation (19) for N = 2 in the domain IXI > O. X2 > 0) with l.ero initial conditions and boundary conditions II (I. X I,
1/(1.
O.
() ) == A. 0 ( I _ ()"( I
-
(t
= Ao ( 1-· t)"( I
-
2/lr ll'2 x 2j.r.l) I .
X2)
)2i" . I.\' I / X,I',
which are majorized by boundary values of the solution (24) for AI 0= O.S, al = a2 = 1/ J2. T == I. The localization domain is the triangle with vertices at (0,0), (J2. 0). J2). The results illustratc hcat localization in the LS-regime, when n > - I/ (T. The thermal wave for all 0 <: I <: I is inside thc domain or localization of the majoril.ing S-regimc. Fl'Om some moment of time onwards. a concave temperature profile is formed, and the effective dimensions of the hot domain shrink. As in one-dimensional geometry. under the action 01' fast blOW-Up rcgimes there is no localil.ation. Figure 29 shows thc evolution of the multi-dimcnsional HS
* 3 Hcal "inertia" in mcdia with nonlincar thermal conductivity
III Hcat localization (i ncrtia)
r
155
u (t, .x, . Xl)
,/xz ·T.l=·l:slaz
5,1
t,J
\ '-
.T,
fl
Fig. 27. Gcomctric intcrprctation of thc solution (24)
blow-up regime under the same ,initial and boundary conditions as in the previous computalion, but for 1/ < -II (T. Herc a convex tcmpcraturc profile is formed and heat propagatcs into infinilely far rcgions during thc finitc timc of existence of the solution. Therefore the main results and intuition concerning the influence of boundary blow-up regimes in a heal-conducting medium extend to the multi-dimensional case. Let us nole that the rate of tempcrature growth in the S-regime. which separates slow and fast blow-up regimes, does not depend on the dimension of the space and is determined only by the properties of the medium. A new feature of multi-dimensional geometry is the shape of the heat localization domain, which can vary considerably. Let us give some appropriate examples. Lct us consider a solution u(r, x, . .12) of the problem for equation ( 19), N = 2. in the domain (0.1') x (XI > O. .12 > 01 with the initial function (24l for t = O. Let us prescribe zero heat lluxes on the axes XI • .\"2: knll"u" = 0 for .12 = O. kOIt" u,~ = 0 for XI = n, t E W. 1'), so that no energy enters the medium. Since heal fluxes on the boundary. corresponding to the self-similar solution lIs(r, xl. arc positive. by the comparison theorem we have u(t, x) ::: IIs(t. xl in
c Fig. 28. LS-rcgime in two-dimcnsional gcomctry. The paramcters are: IT = 2. /I = -0.25. = (f2 "" 1/J2. .Is '" I; a) T-/ = 0,22. h) T-/ 9,1.10- 5 •
k n '" I. An = 0,5. T = I. lrl c)l' /,=4,10- 7
=
~ 3 Heat "incrtia" in mcdia with nonlinear thermal conductivity
1lI Heat localization (inertial
156
url,x"
157
(0.7')
Xz)
X (XI:::
5,7
O.
x2 :::
01. and therefore
u(t. x)
= O.
I E (0.7'). O'ixi
+ O'2x:,
(26)
::: Xs·
1,9
Moreover. by construction u(t. x) > 0 for I E (0. T) and lYj x,
+ lY2X2
< Xs.
O. Xc ::: o. Solutions of equation (19) in the spatial domains (XI> O. X2 < 01. (XI < O. X:, > OJ and (XI < O. X:, < 0l with the same houndary conditions have the same properties. Therefore u(I.X) = 0 for / E (0.7') for all lYllxiI + a21 x 21 ::: Xs· However. by symmetry all these solutions coincide. in their respective quadrants. with the solution of the Cauchy problem for equation (19) in R I X R:' with initial
XI :::
function (27)
.7-',
b
U(I,X,,:!.;,)
where II", = AsT 1/". XII = X.I· = (2kIlN~(IT + 2)I(T)I!:'. Thus. the initial temperature distribution (27) is localized in the r!loll/bus (diamond) alixil + u2lx:,1 ::: XII for time not less than I,::: /' (TX6/(2ko(IT + 2)u:~,). The estimate of localization time is identical to formula (13) for one-dimensional
=
geometry. Performing the same construction for the three-dimensional case. we see that the initial perturbation
113 (28)
is localized for a linite time
Imlroll L~, a;J.\,1 :::
(rJ
is estimated using a similar formula) in the (Jcta-
XII·
For a finite time a I!lerll/(/I crY.I'/ol. an octahedron lhat preserves its shape. exists in the medium. Inside it. the h2mperature is different from zero. while on its boundaries and outside. it is zero for all/ E (0. I,). A numerical simulation of a two-dimensional analogue of the thermal crystal is shown in Figure 3D. Initial daw is the function (27). For a finite length of time thermal energy is 11JI:alized inside the square. With time. the temperature profile inside the localization domain rearranges itself into a convex shape. and heat starts
Fig. 29. HS-regime of heal propagation in two-dimcnsional gcomctry. The paramctcrs are: IT'" 2. /I cc; -I. k ll I. All 0.5. T I. aj '" Q'2 1/J2: a) T - I == 0.27. b) 2 T-I;::,X.7.10- 2 .c)T 4 ..'.I0 2 .d)T 1=2.7.10
to spread (compare with Figure 26. a). By the comparison theorem. the functions (27). (28) define a class of inertial temperature profiles in multi-dimensional geometries. The temperature distribution is localized if for each of its front points we can find a function of the form (27) or (28). which majorizes this distribution and has the same fronts. By this method it is not hard to construct localization domains of various shapes (ball. ellipsoid. etc.) and to determine the correspondll1g initial temperature profiles.
First of all let us establish the connection between two solutions 1/") (v = I, 2) of problem (1.1 )-( I,3) corresponding to different constant initial functions, 11;;'\1;) =: CI'-I = const ::: 0 and the same boundary regimes as t -> To', I/'-I(t, 0) = 1I\"I(r) >
0,
IE
(0,
n.
Lemma I. Lei C III ~ C 121 • and assllme Ihat Ihefllnctions t E (0. n. and that there exisls T E (0. T) slIch that III
III
(I)
= III121 (I)
=: III (f). t E [T.
11\"1
do /lot decrease in
T ): (I)
III <" 121 III ..:. III .
t E (() , T ) .
Then
i,
.r,
rz
(2)
b
where IlEI")(1)
=
r"'' (11 1"1(1, x) -
./0
C(I'11 dx
E
[0. (0),
IE
(0, T):
I'
= I. 2.
The functions IlEI/'1 (I) have the mcaning of energy supplied to the medium up to the moment t E (0. n. First of all let us note that by the Maximum Principle 11(1·1 ::: CI"I in (0, T) x R so that IlE"'I(I) ::: O. Under the assumptions of the lemma, equation (1.1) can be" integrated over (T, I) x R+, t E (T, n. This follows from an integral identity satistied by the generalized solution for a particular choice of a sequence of test functions with compact support f = f(x/CY) -> I as a ---'> 00 everywhere in R I ' I.('(t) I ~ I, and known regularity of the generalized solution (see 3, eh. I). The result of formal integration of equation (I. I ) ovcr (T, t) x R+ is: Proot:
*
c Fig. 30. Thcrmal crystal in two-dimcnsional gcomctry. Thc paramctcrs arc: 11 m =: 0.5,.1'0 cc, I, rtl =: cr2 1/J2, I' I; a) I 0, maxu(O, .1'1 . .1'2) =: maXII(t, .II' .12):= 0.24, c) 1:= 4, maxII(l.rl . .1'2) 0= 0.18
IT =:
0.5,
2, ko := I. h) 1:= 0.8.
§ 4 Effective heat localization Independence of effective localization on the initial state
*
In the study of the thermal inertia phenomenon in 2, 3, we approximated the "zero background," by taking the initial function to have compact support. Let us consider the original problem (1.1 )-( 1.3), in which the bounded continuous function 1I0(.r) is arbitrary. In this casc heat localization is to be understood in the sensc of effective localization,
(modulo sign, the integrands are heat nuxcs at the boundary). By (I), we have for all t E IT, T) !lE(f)
IlE(T)
=
£'
[iJIIIII
k[III(S)!-;---(s, 0) ,]x
• T
By the comparison theorem for I E [T. T), we obtain
11 111
~
11
121
121
-
ax
]
dx
in (0, T) x R I . Sincc
alii II il1I (2 ) - ( f , O ) < --(t,0).
ar
a1l
-.---(.1',0)
11
111
(t,
ds.
0)
= 11 121 (t, 0)
§ 4 Effective heat localization
III Heat localization (inertia)
160
Therefore the integrand in (3) is nqn-positive, which is equivalent to (2).
0
Decomposing this integral into the sum of integrals over > [} Ih is an arbitrary constant, we obtain
161
(0 . .1'0)
and
(.1'0,00),
where
.1'0
Therefore, for the same heating regime, the amount of heat entering a colder medium is not less than that supplied to a warmer onc (if !lE(T) ::: 0). Of course this lemma can bc extended to cover a wider class of initial perturbations. If lIi;'1 are non-constant, but. for example.
('O lTi I11 (r. x) -
.10
0
<' 11 111 • X ..:: 0
then if conditions
(I)
> ()..
) 11 1"1 E. 1_ I (R +, 0
hold, instead of
(2)
0 ,-->.
x
-->
!""
(4)
oc.
we derive the estimate
0::: (""[lI(2I_- II III\(r.Xldx::: .10
(UI (11 0un )]
r.. II/11_IIII'I(T..Ild.l.
(5)
.10
or. in other words, 1111(11(1,.) -- 11 111 (1. ·)III.'IR. I is non-increasing in I E IT. n. The lemma we provcd abovc expresses a kind of stahility in L' (R +) of the heat diffusion process to perturbations of the initial function. Using it. we can establish the following assertion concerning independence of effective localization depth from the initial function. Theorem 1. LeI 1/1"1 (/1 = 1,2) /Je solllliolls (!(pro/Jlelll (1.1 )--( 1.3) lI"ilh /Joll/ulary . Iy, .1'1/1'I1 Iwl I I' . (I ) III! II I Iel · lOll . .I' I/oII'I ,1/ 1"1 ' respecllve COIl(I II COIllIIIOIlS (. [' 'urllenllorc, 1 [{I
(I) --..
00,1-->
x)} dx-
l'"
C dx
• (J
•
11 II I
UI!I(I.
filll(r.
x)
dx ==
II -
+
j''''' (fi111(1, x) -
Cl dx -
• .\Il
11
+ 11 -1.1 :::
const, I E IT.
n.
\1\
Let us prove uniform houndedness in I E IT. n of the integral 1,. Since by the Maximum Principle Ti (1 ) ::: it lll in (0, n x R f . we have that /1 ::: o. Furthermore. /1 = CXo and thereforc / 1 ::: const + 1.1 for all I E IT. n. Let us consider the integral I ~. Since the solution lI 11 I is localized and .1'0 > III L ', there exists a eonstant M > O. such that 11 111 ::: M in (0. n x Ixo, x). By (7) this means that Ti lll ::: M in (0, n x lXI), X). Then by the comparison theorem Ti lll ::: liS in (0, n x (.1'0, (0), where IIs(l. x) == 0«.1' - xo)/II12) is the self-similar solution of equation (1.1). which satisfies the conditions IIS(O. x) = 0, x > Xo; IIs(l, .1'0) = M. I E (0. n. Concerning the existence and uniqucness of the solution liS ::: 0 see suhsection 4, ~ 3. Ch. I. as wcll as the Commcnts to Ch. I. Here (J ELI (R t ). Thus
T", (6)
,/"I(x) arc 1101l-incrcasillg ill x > 0; /1 = 1,2. LeI [{II I he elll'clive/y loca!i::.cd alld wlellhe deplh oj loca!iwlioll he L(II'. Thcll IPI is also loca!i::.ed, alld e" I ' = till'. Pro(lt: Let us consider Ti( II, nl11 , solutions of problem (1.1 )-( 1.3) satisfying the conditions nlll(O, x) == 0,
-111«) _ - mdx,sup . f· 1/1)II). II •.1) =C ..,sup [(I)111} =Till I(I . 0) == Ti l11 (I . 0)
I
. Ul)III ( () ),1/1)11101 1l1
== 1/1 (I) for lEI T, n.
Let us extend the functions nil') (1,0) in 10. T) so that Lemma to the solutions TiI"1 and nl!I(I, 0) ::: 1/1111(1), //1,"I U ) ::: nl11 u, 0). From the comparison theorem it follows that n lll ::: 1/111,//111::: Ti l11 in
(0,
n
x R!.
Applying the lemma to the functions Till'). we rewrite inequality e l1l = C in the form
."" -111 . (II (I ..n - C. .- II
/
.11
_n(
II
. (I ..x) I d.1.
could be applied
for aliI E IT,
n.
Therefore /1 ==
!""
•
{n I1I (1. x) - C} dx ::: const
for any I E IT, T). From this wc immediately deduce uniform boundedness of Ti l11 in !T. n x (XI). x). Indeed, by monotonicity in I E (0, n of the boundary function n ll" I (r), iirst, fil11 ::: C (that is. 11 ::: 0). and, second, nl11 (1, x) is non-increasing in x for any I E (0. n (see ~~ 1.2. eh. V). Therefore for any II :> III ~
By the second of inequalities (7) we have that for any Xo > LI II ' there exists a constant M > 0, such that 11 m < M in (0, Tl x (xo, (0). Therefore the solution um is localized and L(2}· :::: LIII'. Exchanging 11 11 I and 11 (2 ), and using the same argument, we obtain the opposite estimate LII1 ':::: L(2}·, so that LI II ' = L I21 ', which concludes the proof. 0
163
u(i,.r)
Let us now consider a case of absence of localization. Theorem 2. Under Ihe condilions IIf Theorelll I lei Ihe slIllIlioll i::.ed (HS-regillle). Theil 11 121 is not IlIca!i::.ed either.
1/ II
he
11111
local-
Prollf If we assume the contrary, viz.. that 11 121 is localized, then by Theorem I 11 111 is also localized, which contradicts the assumption. 0 J
Remark. The requirements (6) on the boundary data can be substantially weakened. Actually, for Theorems 1, 2 to hold, it is sufflcient to satisfy the first of conditions (6), and to have the continuous initial functions IIi;') uniformly bounded. Thus, the properties of regimes that blow up as t -> T" do not depend on the initial temperature profile. If a boundary dependence ensures heat localization (in either strict or effective sense) for any initial condition, then localization will occur for any other bounded initial perturbation. Depth of localization and class of regime (S- or LS-) are also preserved. In particular, many asymptotic properties of heat diffusion processes in the problem (1.1)-(1.3) remain the same. These were studied in 2,3 for k(lI) = kou" and a function uo(x) with compact support. For example, we have
:r.
Fig. 31. Effective heal localization in a medium with nonlinear heal conductivity. The parameters arc: IT:= 2. ko == I. /IoCr) == I. As == 'I' == I. L' = 2: I: T -- 1 = 1.7.10-- 1,2: '1'-1=5,48.10 2 ,3: T·"1=2.37·10 2 ,4: 1'-1=8.14·10'.5: 7'-'1=3.,)·10-'. 6: '1'-1= 1.58.10- 3 ,7: T-I=6.:\7·10~
Figure 31 shows the results of a numerical computation, which illustrates Theorem 3. Asymptotic stability of self-similar solutions of the HS-. S-. and LS-regimes is proved in eh. VI, where we also derive convergence rate estimates. For example. Figure 32 shows the '"arrival'" of a non-self-similar solution at the spatio-temporal structure of the self-similar HS-regime (thick line). Without introducing. a precise notion of closeness, let us observe that the influence of initial data in a domain covered by a thermal wave becomes neglig.ible if the medium is supplied with an amount of energy, which is at least an order of
Fig. 32. "Arrival" of a SOhHIOI1 al a self-similar HS-regime. The parameters are: IT == 2, II==-I.k o 0.5,A o 0.12,1'=1.12·I0 1 :I:T 1=').2·10 1,2:T 1 6.15·10-'. 3: T - 1 := 1.05 . I0 3:
f
({
(I,
"1<')12)
l:) == _"'-'. ..2'-~!L:;.---.:,---'-'-..:.
magnitude larger than the initial amollnt. For example. for the S-regime. when the characteristic size of the resulting thermal wave is constant, convergence to selfsimilar structures occurs when the temperature at the boundary is approximately 10 times larger than the characteristic initial temperature. This fact is ret1ected in Figure 31,
~
III Heat localization (inertia)
164
Therefore the self-similar solutions that blow up in finite time, which were constructed in ~ 2, arc stable asymptotic states of thermal processes. Heat localization in a medium, or the lack thereof, is determined only by the form of the boundary regime, unlike some other phenomena of fl()fllinear heat conductance (for example, finite speed of propagation of perturbations), for the existence of which special initial data are required. These results extend the sphere of applicability of the phenomena we arc considering in various physical situations. However, the following question arises: is heat localization a property of a medium with precisely the nonlinear thermal conductivity k(lI) ;;;;;; kOIl". rT > 0, or is the localization phenomenon present in arbitrary media'l In particular. is it possible to obtain the different heat diffusion regimes in a medium described by the classical heat equation'.'
165
4 Efl'cctive heat localization
From (13) it can be seen that the most interesting class is of "exponential" blow-up regimes. Indeed, if III
then
11(1. x) .-> X
(I)
= A\(T -I)"
as r
->
exp{Ro(T -
I)
I},
(14)
Ro. As > 0,
T' , where x E (0. xs). where XI;;;;;;
For any x > Xs the temperature at r
:=:
2jk~. T
( 15)
is bounded: 1/
II
I'
In till
< X. (16)
2 Influence of boundary blOW-Up regimes on a medium with constant thermo-physical properties Let us consider the problem of heating a medium with constant thermal conductivity in a boundary blOW-Up regime. illl
il:'11
dr
dx-
-;-- ;;;;;; k()'~-·,. 0 < 11(0,
I <
(9)
T. x > O.
( 17) ( 10)
x) ;;;;;; 0, x> O.
11(1,0) "'" 11\(1) ~ O. 1 E 10, T); III E
C(IO. T));
III -> 00.
For r :=: T the heat flux and the amount of contained energy in the domain x > Xs arc bounded. The parameter II determines the nature of the change in temperature and heat flux at the point x = Xs: for l' > 1/2 (/I > 3/2) the temperature (heat flux) is bounded at r = T ,while for II ::: 1/2 (/I ::: 3/2) it is not. Solution (12), (14) is an example of the S blow-up regime. It is the analogue of the standing thermal wave (1.5) for the case of constant thermal conductivity. From the comparison theorem we obtain, that for boundary regimes majorized by (14),
(II)
r -> T ,
which is a particular case of the problem (I.I )-( 1.3). For simplicity, we have taken the initial temperature of the material equal to zero. which is not essential. due to the superposition principle (see also subsection I). [n processes described by equation (9), perturbations propagate with infinite speed, so that localization must be understood in the effective sense. Solution of the problem (9)-( I I) is expressed in terms of the double layer
localization of depth L' ::: XI occurs, and for x > x.\ the solution is bounded for all 0 < r < T by the limiting curve ( 16). Condition (17) distinguishes the class of slow boundary blOW-Up regimes in this problem. If III (I) :::
Ao(T -
x) ;;;;;; _......:\_.2Jk oTT
/'1 . Il
exp {.
To determine conditions for localization tn the problem (12) to the limit as 1 -, T:
1~.I]1l1(1.
x)
0=
11('1', x)
IIrT'.
.-!!L~~ tlr.
._I:'_o __ } . 4k ll ( l - r) (I -
exp!Ro(T - I)"}. 0 < II :::
1
< T;
( IR)
0, A o > 0
(for II :=: 0 we assume II <: 0), the integral (13) converges for all x > 0 and the function 11(1, x) is infinite only at the point x :=: O. [n the rest of the space it is bounded:
potential 11(1,
I)"
.. I <
..... x)
,/ \'/(~,,,TI
( 12)
exp { -II
,,"} II ." + -Rox:''' -II (4k ll )"
I.'
/- till.
(19)
r)l/.
(9)-( II),
= 2J~OTT.f exp { -4k;)(;~':::'-~)}
let
LIS
(Tl~(~;l/:'
pass
111
Therefore if condition (I R) holds. we have the LS blow-up regime. Finally. in the case of fast regimes /II (I) :::
clr.
( 13)
Aller -
I)" exp!RoU' - I)"j. 0 <
the integral (13) diverges for all R+. and the HS-regime obtains.
Thus, in a medium with constant thermo-physical properties. exactly as in the case k{lI) = kOIt" (cr > 0), there arc three regimes of heat diffusion. Heat inertia. and the appearance of a finite thermal process localization domain also occurs in a homogeneous medium with infinite speed of propagation of perturbations, Let us consider the question of localization in the multi-dimensional case. Here a great variety of localization domain shapes can be constructed. in particular. domains with a non-smooth boundary. Solution of the heat equation (k o = I) il 2 11
illl
-
iJxT
ill
iJ2
+ ---'1
iJ.\~ .
0 <
< 1'. XI > O. -ex.:: < X2 <
I
x. Fig. 33. Heat localization domain (da,hed) with a non-smooth houndary (P is a corner
with the conditions 11(0.
XI. X2)
point)
== 0;
11(1. O. X2)
= (1)(1, X2)
::': O. IE (0.
n,
X2 E
R,
where <1>(1. X2) -> 00 as I -> 'I' for all x2 E Eo S; R, Eo i= 0. has a representation in terms of the two-dimensional heat potential:
11(1 .
, .1'1.'\2)
XI if, XT + X~ } x = -47T exp {--'---, 4(1 - 7)
where b > 0 is a constant The inner integral in (21) converges for all equals 2(1' l(x2. 7 )=
to l' . wc obtain the limiting temperature distribution: XI II(T-.XI • .I'1)=-
(21)
- d)2 <
il 2 + 2dh,.I'1 > 0,
the boundary of which is composed of a segment of a straight line and a half-circle. Inside the domain the temperature goes to infinity as I -> '1". while outside it is bounded uniformly in time, Using (21) it is not hard to devise localization domains with boundaries given by any second order curve (parabola, ellipse. hyperbola). The principle of superposition allows us to combine domains corresponding to different boundary regimes and to obtain as a result localization domains with non-smooth boundaries. In Figure 33 the boundary consists of segments of a circle and an ellipse.
For example. let the boundary regime have the form
(1)(1 ,
.1'1) -
= exp {-4( l'.\~_-I)} r- (I • X1) -'
3 Asymptotic stage of development of blOW-Up regimes in a medium with constant thermal conductivity
JI
where exp {
{
2b.l'1 } --_:.-
4(T -I)
O. .1'2 > d.
X2
.
< O.
0< -
X,
.-
_< iI
= const
>
O.
Using the integral representation (12) of the solution of problem (9)-( II). it is hard to characterize in detail the asymptotic stage of the process. To do that. we shall construct self-similar and approximate self-similar solutions of the problem. Let us consider two important examples,
~
III Heat localization (inertia)
168
= A n(7' -
I)", 0 <
1
< 1',
II
= An(T -
r)l/fs(~), ~ = x[ko(T
/ --1)\1 "
-Xi
/,0
(22)
< 0,
lead to the occurrence of LS-regime. We shall analyse the problem the self-similar solutions lIS(I, x)
4 Effective heal localization
flt,V
I. Boundary regimes of power type, 1I1 (I)
169
(9)-(
II ) using
< 1 < T. x> O.
For the function r\(~) we then obtain the problem t~
I - it\~ + IIfs = 0,0 <
~ <
Xi: fs(())
= I. fs(Xi) = 0,
which has a unique positive monotone solution
f.d~) = 15:i~ -- II)
[X exp {_ ~".\} .1'''
I (I
+ .1')" ··In ds,
0 <
4
17 (-11).10
~
0/
<
Xi. ___ L . __.. j
From the Maximum Principle it follows that the difference between a selfsimilar and a non-self-similar solution satisfies the estimate
o S lIs(l, x)
- lI(1, x) S Lls((), 0)
=
(23)
AoT".
[)
0./'
Fig. 34. Convergence of a solution to the power law self-similar IS-regime in a medium with con,tant thermal conductivity. The parameters are: ko "" I.D. /I ,~ - O.S. An co l' = I: I: T ..._/ = 0.77.2: 1'-1 = IUD, 3: 1'-/ = D.SS, 4: 1' .. I,,,, OAI
Introducing the "similarity representation" of the solution, A similar expression is true for the quantity we obtain from (23) for all 0 <
1 <
IIf(l, .)_. fs(·)IIC1I{,I
W(I, x('I/(t))
I' the estimate
S 1'''(1' --Ir" .......
0,1 -->
r.
that is, asymptotic stability of the self-similar solution (for results of a numerical computation see Figure 34,) Stability ensures that all the main properties of the solutions lI(1, x) and liS (I , x) are the same at the asymptotic stage of evolution. For example, the half-width of the thermal wave X"I (I), determined from the equation I
.
I
lI(1,x,./(I))=2I1(1,O)=:iAo(T-I)",
1---1'.
satislies by (23) the inequalities
I = :::;W(I, 0),
defined by the equation
0 < 1 < T.
where W(t, x) is the heat nux. so that x"I/(t) is the coordinate of the point on each side of which the amounts of energy entering the medium are equal. Figure 35 shows the results of a numerical solution of prohlem (9), (10), (22) for II = -I. Dashed and dash-dOlled lines show, respectivcly, the trajectories of x = X,.((I) and x = x,.I/(I). Self-similar hehaviour is established once the temperature on the boundary becomes 5-10 times larger than the initial one (which is close to the criterion obtained for media with k(lI) = knll". IT > 0). Nonetheless, there are certain differences betwccn the solutions lI(1, x) and lIs(t, X). The asymptotic hehaviour of the limiting distribution of 11(1, x) as x ....... 00, •
II(
Let .1.\ I (1/2) = ~<'/ < 00, where .1.\ I is the function inverse to fs(~) (it exists by monotonicity of fs(t-)). Then we obtain from (24) an expression for the half-width,
X('I/ (I),
r , -Il -
A0_"__ 1'21/ ,--n 2" r;:;;k n 2 \I 1T
/'-
,
..
) ,\. , ... 1/ (.1
1/2
. d.1
, ,- /1·lk"l)
is exponential, unlike thc power law asymptotics of thc self-similar solution, ., IIS( I , x)
If we neglect the highest order derivative term in (28). we arrive at the degenerate problem
a.v ~ = ~ An
ill
\/,(1.0)
(ii.V ~) 2
0<1 < 1', x>
ilx
= AoRo(1'
-
0 <
I)",
I <
0,
1'.
which has the self-similar solution 11+"10
(31 )
The function 0, (tl 2: 0 satisfies the equation "
L -_ _..-L
--L-_
o
I
2,0
1,0
.
(=
2.3·10".5: T -- 1
= O.
t
> 0; 0,(0)
= I.
(32)
and wherever it is positive. is defined implicitly from the equality
.Z·
[/e :-,y::~~~
Fig. 35. Dynamics of the power law LS-regime in a medium with constant thermal conductivity. The parameters are: ko 1.0. Ao = T = I; I: T·- { = 0.113. 2: T - 1 = 5.H. 10 ". 3: T I.'" 3.5·10 ".4: T 1 T-I=I.2·IO".7:T 1==7.10.
1+11,_
wy -'-2-0,t + 1If1,
=
I.K· 10 ".6:
This. naturally. has to do with the faet that the self-similar profile 11.1'(1'-. x) takes the infinite amount of time I E (-00, 1') to form. 2. Let us consider now the asymptotic stage of exponential boundary blOW-Up regimes: III (I) = AolexplRo(1' - I)"} - II. 0 < I < 1'; /1 < O. (26)
, [~l :
n)' _ nfl,!,
'f I~ r"'' ''''I' ,
I :
'+
(33)
n
At all the other points we set O,(~) = O. The properties of the monotone function 0, depend on the parameter I) If -I < /1 < 0 (LS-regime). then H,(t) > 0 for all t > 0 and
/1.
For /' = 0, (26) differs from (14) by a constant, which is not essential. The problem (9). (10), (26) contains at least tWO parameters with the (limen10
sion of length. Iko(1' -1)1 1/" and IkoR o 1/"1 and. therefore. has no self-similar solutions. We shall show that the asymptotics of the solution of the problem as I -~ l' is described by self-similar solutions of a "degencratc" equation. It is constructed as follows. The change of variable V(I, x)
= A o Inll + 11(1, x)/A o ]
(27)
('(11)
2) For 11
1+11. 2 = ---·_.
2"/111'11(
11)1/11+,,,.
211 = -I (S-regime) the solution has the form
f)(~)= {(1_~/2)2. o < , .
O.
/j < 2.
/j 2: 2.
~
---_.---_._--'<
t:; '1\ -.
'-,.
I . l_ ~1,.\;
"~I
.
. \1.\
•
,
~
III Heat localization (inertia)
172
3) In the case O,(g) > 0 for D:5
11
O(t.
< -I (HS-regime) 0, is a function with compact support: gr = 2(_11)"/2(_1 - I1r'(ltlll12, O,(~) = 0 for all ~ 2: ~/,
g<
173
4 Effectivc heat localization
tJ
1,0
such that moreover OsW = -1(1
+ 11)/21~I(gI
--~)
+ o((~f
- ~)) as ~ -+ ~f
.
In all the cases O~(O) = -("11)1;2, O':(~) > D wherever 0, > D and O':(~) < O~(O) (1 ·····11)/4 for g E (D, ~I)' From the properties of the self-similar solutions and the Maximum Principle, we obtain the estimates (see 2, eh. VI)
=
*
-AoRoT" :5 V(I,
x) --
I' V,(t, .\) :5 AollfJ;'(~)IIClo~tlln - I ' ,I E (0. -I
Then for the solution of the original problem we obtain from
n, x>
O.
(27)
(34)
For the similarity representation of the solution V,
o
we obtain from the preceding inequalities the following estimate of the rate of convergence to the approximate self-similar solution: !lO(t,,) - O,(·)IICiR, 1= 0 «1'.- I)
"1 In ('I'
..
1)\)
-+ O.
1 -+ T.
(35)
In the case of the S-regime, convergence (35) is illustrated by the results of numerical solution of the problem (28)-(3D), shown in Figure 36. Significant growth of the temperature in the main part of the localization domain, as compared with the temperature for x > .Is = 2(k oRo) 112. occurs when the temperature on the boundary grows by a factor of 10-20. The estimate (34) makes it possible to analyze in detail the fully developed stage of the process. For example, in the S-regime we obtain, as 1 -+ I' , from (34) for all D < x < .1.1' = 2(k oRo) I /2 the estimates Aolexp(/?o(T -
I)
I (I
-- x/xd - RoT II - II :5 :5 AoT I 12(T --
I) 1/2
11(1 • .I)
explRo(T --
I)
x)
~ explRo(T - I)
I (I
- x/xdl, 0 < x <
In the case of the HS-regime, we have for all x > ()
which means that at each point in space the temperature grows to infinity, but more slowly than the temperature at the boundary. Using (34), it is also not hard to determine the dynamics of the evolution of the process at the asymptotic stage: .1,./(1)
= In2 [_"_'0_]1
12
(1' .. 1)11 "In
+0
[(I' -- 1)(1 11)12] .
RO('-II)
:5 1(1 -
.1.1'; 1 - ,
.1,_,,(1)=
x/x.d I.
whence it follows that inside the localization domain the temperature changes according to 11(1.
Fig. 36. The parameters are: 11 =: -I. ko = Ao No = l' "" I. .1'.\' ~c 2; I: T - I = 0.95. 2 2 2: '1'--1=0,47.3: '1'-1=0.2.4: 'I' 1=0.1.5: 'I' 1==5.10 ,6: 1'-/=2.5.10- • 2 7: 'I' - ( = 1.2· 10-
Unlike a medium with a power law nonlinearity, in all the cases here x"f -+ 0, 0 as 1 -+ T . including the HS-regime. when the temperature grows without bound in all of R I ' This shows, in particular, that shrinking of the half-width is .1"/1 -+
not a sign of localization in a blow-up regime. Let us note that the frequently used dimensional estimate of the half-width. xcI (I) ~ Iko(T - t) 1112. in this case does not describe the process correctly at the fully developed stage. Thus. under the influence of exponential boundary blOW-Up regimes there is a kind of degeneration of the parabolic equation (9) into the first order equation
all
(ilu/iix)"
- = k o --'--a/ Ao + II self-similar solutions of which provide us with the principal term of the asymplOtics as / --> T- . The function (31) is an approximate self-similar solution of the problem (28)(30). The general theory of a.s.s. of parabolic equations and its applications are presented in Ch. VI. Results of this chapter testify to the generality of the heat incrtia phenomenon and show that conditions for its occurrence are not hard to satisfy. Thesc results will bc used in Ch. V. VI in thc study of boundary blOW-Up regimes in media with quite general thermo-physical properties.
Remarks and comments on the literature
*1. The self-similar solution of the S-regime (5) was constructed in [3511 (existence of separable solutions for equation (I) was known before; see. for example 1331. where for the first time the standing thermal wave was studied; that paper also verified numerically its asymptotic stability. The paper [3511 led 1.0 detailed studies of the heat localization phenomenon in media with nonlinear thermal conductivity 1390. 264. 265. 2661. where all the main concepts and deflnitions are developed. A detailed analysis of the localized solution of Example 2 is presented in 1347, 348. 1491·
*2. The three types of self-similar blow-up regimes (S-. LS-. and HS-regimes)
were studied in 1352. 393. 267. 1651. In the presentation of subsection 2 we follow 11651. Bya different method the existcnce and uniqucncss of self-similar solutions
are proved in [205. 2061·
*3. Thcorems on presence or absence of localization (subsections I and 2) arc proved in 13041 (an interesting criterion of localization depending on the form of the boundary function which blows up in finite time has been obtained by [204J). A more detailed discussion of the physical basis of localization can be found in 1393. 267, 2681; these papers also discuss the possibilities of its experimental study. Localization in the Cauchy problem has been studied in 1352. 393. 2671·
Remarks and COlllments
175
An interesting example of a localized initial function was constructed earlier in [171. where the exact value of the localization time was calculated; it agrees with the calculations of subsection 3. Results of subsection 4 are contained in the main in [277. 331. 267J. A list of papers dealing with the analysis of local properties of the degeneracy surface in problems for quasi linear parabolic equations can be found in the Comments sections of Ch. 1 and II. ~ 4. Results of subsection I appear partially in [153]. The study of subsections 2, 3 was published in 1348. 347. 149J. An elementary presentation of some of the questions relating 1.0 the localization phenomenon can be found in [394J. Possible applications of the discussed phenomena were considered in [392. 350]. Blow-up regimes in compressible media with various physical processes are studied in [15. 70. 71. 72. n. 382. 387. 388. 389. 366. 3181. A more complete bibliography can be found in [267.268.26 9 1.
•
177
Chapter IV
Nonlinear equation with a source. Blow-up regimes. Localization. Asymptotic behaviour of solutions.
The present chapter deals with the study of spatio-temporal structure and conditions for the appearance of unbounded solutions of the Cauchy problem for quasilinear equations with power law nonlinearities: 11,0..-= V'. (II"V'II)
+ 11 13 ,
I>
(0.1)
O. .I ERN.
(0.2) where (r > 0, f3 > I arc constants. Equation (0.\) describes processes with a finite speed of propagation of perturbations (sec 3, eh. I). Therefore, if 110 is a function with compact support, u(t, x) will also have compact support in x for all 0 <: I < Til, where Til :s: x is the time of existence of the solution. The main question, considered in 1, 2, 4 is to define conditions of localization of unbounded solutions.
*
**
«Definition 1. An unbounded solution of the problem (D. I ), (D.2) is called (slricllv) localized if the set
n l . = {x
ERN
I U(TII'
x)
== Thill
'1" 11(1.
x) >
O}
is bounded. The set n l . is called the lo('ali;alioll dOl/will. Boundedness of U r means, in particular, that 1I(t, x) "" 0 in RN\U, for all 0 :s: I <: Til. This follows from general properties of solutions of parabolic equations with a source. A strictly localized solution grows unboundedly as I ~> Til in a domain wr = {x ERN I 11(1'11' x)
=x}
of finite size, which, in general, is different from U,. As in the case of boundary value problems (eh. III), localized solutions can be conveniently divided into two classes: S-regil11l' solutions, for which 0 <: meas WI. < 00, and LS-regill1l'
solutions, for which meas WI. = O. In the latter case the solution u(t, x) grows to infinity, for example, in one point, while at all the other points it is bounded from above uniformly in I E (0, To). In the most general case the classification of blowup regimes should be based on the measure of the blOW-Up set having the form BI. = Ix E R N 13 sequences I" -+ To and x" -+ x, such that u(l", x,,) -+ 00 as II -+ 001. Obviously, by definition of an unbounded solution B, =F (1 for bell-shaped data. Definition 2. There is no locaklilioll in the problem <0.1), (0.2) if the domain ill. in (0.3) is unbounded. We put non-localized unbounded solutions in the class of HS (h!ow-up) regimes. A combustion process is not localized if as I -+ Ti; heat propagates into arbitrarily distant regions. In a number of cases the condition of Definition 2 is equivalent to the requirement
that is, the non-localized solution grows to infinity as I -+ To in the whole space. In *~ I, 2, 4 it is shown that for f3 ::: (r + I the problem exhibits localization; the case f3 = (r + I corresponds to the S-regime of combustion, while the case f3 > rr+ I e<mesponds to the LS-regime; for I < f3 < (r+ I there is no localization (HSregime). The study is conducted by constructing unbounded similarity solutions I), as well as by the qualitativc method of averaging 2), which establishes their asymptotic stability in certain parameter ranges. In 3 we prove various assertions concerning conditions of existence of unbounded solutions of the problem (0.1), <0.2), and we show that for f3 > (T+l+2/N it can be globally solvable (for "small" data 110), which conlirms the qualitative
(*
(*
*
*
conclusion of 2. Rigorous results on the existence (f3 ::: (r + I) and non-existence (I < f3 < cr + I) of localization of unbounded solutions for N = I arc given in 4. The next section, 5, is wholly devoted to the study of asymptotic stability of similarity solutions. In 6 we show that for some 110(.\) in the case (r+ I < f3 :s: «(r+ I )N/(N - 2)+ the problem (0, I), (D.2) evolves in LS-regillle of blOW-Up, in which meas CUi. = O. There we also obtain bounds from above and below for II( To' x) in a neighbourhood of the singular point where a(T o ' .I) = ex). In 7 we use the above approach to study the selllilinear equation (0.1) for (r = o with a reasonably general form of source. There we obtain, in particular lJfeeti\'(! lo('(/li::'lIliol1 conditions for blow-up regimes, that is, conditions for boundedness of the set lll,. We consider in detail the phenomenon of degeneration of the equation III = !ill + (I + II) In/3 ( I + II), f3 > I, at the asymptotic stage of blOW-Up, Asymptotics of the combustion process is described by invariant solutions of a Hamilton-Jacobi first order equation. This degeneration phenomenon has already been considered in the context of boundary value problems (see 4, Ch. III).
* I Threc types of self-similar nlow-up regimes in comnustion
IV Nonlinear equation with a source
17H
§ 1 Three types of self-similar blow-up regimes in combustion It is convenient to start the study of the relatively complicated problem (0.1). (D.2)
by an analysis of particular self-similar solutions of the equation (0.1). Here we construct unbounded self-similar solutions. the spatio-temporal structure of which is substantially different in three cases: I < f3 < IT + I (HS blOW-Up regime). f3 = IT + I (S-regime; a solution of this type in the one-dimensional case was considered in Ch, I. Example 13 of 3). f3 > IT + I (LS-regime). Though these particular solutions arise only for a special choice of the initial function 1/(1(.\'). the analysis of their spatio-temporal structure allows us to make assertions concerning the character of evolution of combustion processes with finite time blOW-Up in the general case (see 5). Moreover. they can be used to establish conditions for existence of blOW-Up. that is. conditions for global insolvability of the Cauchy problem (see 3. 4). They arc also used to prove localization of unbounded solutions in the case f3 ::: IT + 1. The spatio-temporal structure of unbounded self-similar solutions contains important and nearly exhaustive information about general properties of evolution of unbounded solutions of the equation (D.I). Therefore it is not an exaggeration to call the particular solutions we construct cigellfillletiollS (e.l'.) of combustion of the nonlinear dissipative medium corresponding to the equation (D. I ).
*
*
**
179
According to ( I ). this solution corresponds to the process of spatially homogeneous (homothermic) combustion with blow-up, Below we restrict ourselves to an analysis of radially symmetric self-similar solutions: f,
= 1'/('1'0 -
1)'11. ,.
= Ixl.
(4)
Then (2) becomes the ordinary differential equation
The first operator can be written in the form (lJ'I~, (J'I', )'
N - I
+ -:-0': 11'", . f,"
and therefore. if we want the solution Os to be defined in R the symmetry condition (1'1(0)
=0
N
•
we have to impose
(lJs((l) > 0).
Moreover. we shall require the following condition to be satisfied: (7)
Os(OO) = O.
1 Formulation of self-similar problems
For any I IT > 0 and f3 > I, equation (0.1) has unbounded self-similar solutions of the following form: liS (I , x)
= (To -
where III
1/1/3110df,L f,
t)
= 1f3 -
(IT
+
= x/(T o -I)"'.
(I)
I ll/[2((3 - Ill·
Here the constant To > 0 is the time of existence of the solution liS, for I ::: To the solution (I) is, in general, not defined and the amplitude of the solution grows N without hound as I ---'> Til' The function Os(f,) ::: 0 satisfies in R an elliptic equation obtained by substituting the expression (I) into (0.1): 'V,_. f,' -.' _1_ 01 , (IJ'.I''VtIJ S ) - lII'VrOS' , (3 - I'
+ 0/3.1 = 0 •
t.. ERN.
(2)
This equation has the trivial solution fJs(f,l == O. as well as the spatially homogeneous solution Os(~)
-----'The case
IT
==
01/
--
*7,
= () is considered in
= (f3 -
I)
I/li!
II
(3)
In this section we mainly deal with a study of the problem (5)-(7). and with an analysis of the properties of the corresponding radially symmetric solutions of (I). The equation (7) is degenerate for Os = 0; therefore in general (5)-(7) admits a generalized solution. not having the requisite smoothness at the points of degeneracy. However, in all cases the self-similar heat flux. _f,N-.' (J:~ 0:\" must be 1 continuous (similarly, in the case of equation (2) the derivative 'V0S+ must be continuous in R N ). This means, in particular. that O~~O~ = 0 wherever Os = O. Any solution of the equation (5) can be considered in its domain of nonmonotonicity as some kind of oscillation around the homothermic solution 0 == Of!. This analogy has to do with the fact that the maximum of the function Os can be attained only at a point where -(ls/((3 .- I) + II~ > 0, that is, for Os > Oil. and the minimum at a point where 0 :::: (Js < Of!.
2 Localization of combustion in the self-similar S-regime,
f3 =
0'
+
1
In this case equation (2) assumes the simpler form (8)
•
§ I Three types of self-similar blow-up regimes ill combustion
IV Nonlinear equation with a source
180
I +u(t,:r:)
while the corresponding radially symmetric problem (5 )-(7) can be written as I ((N
(N
1
"
I O'Ui'I') , ••
_
~(il' + (I~+I = O. IT .
(tl(O) :;-..: 0 (OdO) :> 0). O~(OO)
/ Thl.' CIlse N
181
8
(> O.
(9)
= ()
( 10)
=I
In one-dimensional geometry the equation (9) becomes an autonomous one. and can be integrated. In particular. it is not hard to obtain the following solution of the equation (9):
10
(II) I
Z
where Ls =
As follows from (I), for ((I(T. x)
21T
-(IT IT
+-
I
(12)
1)10.
f3 = IT + I. i; == x: therefore (I) is a separable solution:
= (Til
-
t)"I/"(}I(X).
0
<. I <.
To. x E R
( J:1)
(the function Os is here evenly extended into the domain of negative values of x). The solution (13) looks unusual from the point of view of traditional ideas about propagation of heat in diffusional media. The point is that in (11) (1.1'(.1') is a periodic function: it vanishes at the points Xk = (112 ± k)L, (k = O. I, ... ): furthermore the heat flux -0:; O~ = 0 is continuous: O'Ut, -+ 0 as x -+ Xk· #Therefore a generalized solution of the problem (9), (10) will be obtained if take a function Os consisting from only one "wave" of the general solution ( II ), while at all other points we can set Os == O. Henee it follows that. in particular, the following function is also a self-similar solution: 11,,(1. x)
=
(1' _.. Il I;" 0
{
0,
,. .=.1!' -...t..1J. ( "I" I~'
) 1/" cos-,!U /.,
x
I.-
Ixi <: L I /2. Ixl ::: L,,/2: ()
( 14) <. I <.
To.
This is the elementary temperature prolile of the self-similar S hlow-up regime. which is loclI!i;,ed in the domain Ilxl <. L s l21 during all the course of its existence. Despite unbounded growth of the solution as I -, 1',; at all points of localization {Ixl <. L.1 /21, heal does not penetrate the surrounding cold space. The quantity L s is called lhe!illtdlll/lelllllilellgih of the S-regime of combustion in the nonlinear medium. It is shown by numerical computations that for practically arbitrary nOll-monolone initial perturbations. for f3 = iT + I. the unbounded
Fig. 37. Numcrical manifestation of thc S-rcgimc. Thc parameters are: IT "" 1. f3 = 3. N = I. L s ~o 11T{IT + 1)1r'/IT ~ 5.44: I: II ;;;;: 0.1: 12 = 7.91· 10 2.3: 13 = 19.6.4: 14 73.0. 5: I) = 74.9. 6: 1(, 74.951. 7: 17 "" 74.9548. 8: Ix = 74.9551
=
=
solution goes to infinity on a set of length /'.1" II". on the other hand. we have that lIo(.r) :> 0 on a small interval. mcas supp ((II <. L s , then we have strict localization on an interval of length L.\. Furthermore. Ls characterizes the maximal length of propagation of heat perturbations with compact support during the course of existence of the unbounded solution (see 4). We present here the results of two numerical computations. In Figure 37 we show the evolution of an initial perturbation flO(X) of small energy, distributed over a small region (smaller than L.I). It can be clearly seen that first the heat profile spreads to a certain I"I.'SOl/ltllCI.' (eriliCil/) lel/glh: only after that, starting with time 1-1. does the combustion process become intensive. and as I -, T() it evolves according to the self-similar solution ( 14). In Figure 38 the initial energy is large and occupies an extensive domain. inside which ((o(.r) is close to a spatially homogeneous profile flO == I. Actually Figure 3X shows instahility of the spatially homogeneous (homothermic) solution in the S-regime. In both cases an unbounded heal profile is formed, which follows, as I -, To (To is the interval of time for which the profile exists: it is di fferent for different profiles), the course of evolution of the self-similar solution (14). In the first case, as can be seen from Figure 37, the solution is strictly localized in an interval of length L.\. In the second case (see Figure 3X) there is no strict localization,
* I Three typcs of sclf-similar blOW-Up rcgimcs in combustion
IV Nonlinear equation with a source
182
183
Theorem 1. For 01/.1' N > I there exists a solutiol/ O.d~) olproiJlem (9), (10) with compact support. The ./imctiol/ Hs is mOI/OlOl/e decreasing where!'er it is positive. The proh/em has I/O /101/-11l0n()/Ol/e so/wiol/s.
120
First of all let us note that the fact that any possible solution Os has compact support follows from an analysis of the equation for small Os > 0 using fixed point theorems for continuous mappings (first (9) is reduced to the equivalent integral equation). This simple analysis provides us with the only possible asymptotics of the function Hs : it has compact support and if meas supp H,I == ~o > n. then
80 -
'10
w\(~)
== 0 for all ~ ::.: ~()), whcre E(~) -'" 0 as ~.+ ~I)' For the proof of existence, it is convenient to consider. side by side with (10). the family of Cauchy problems for equation (9):
.r
o
Fig. 38. Numerical manifestation of the S-regimc. The paramctcrs arc: (T == 2. f3 = 3. N = I. L.I' :::::: 5.44: I: II = O. 2: I" '" 0.43. 3: '-1 = 0.4464. 4: '4 == 0.4484. 5: I, = 0.449 I. 6: 1(, == 0.4495.7: 17 =, 0.44963. 8: IX ",,0.449699.9: II) = 0.449738. 10: 111I = 0.449747
however lInoolinded growth takes place with the fundamental lenoth scale L·. Similarity transformation of any non-statiunary solution of the probleem shows its .so.lution for ~II initial data is in a certain sense close to the corresponding sell-sllnllar solutIon (14), the spatio-temporal structure uf which is a fundamental property of the S-regime. The proof of this fact is given in ~ 5. For N = I there exists a countable set of different self-similar solutions, composed of an aroitrary numoer of the elementary solutions (14). which by the thermal isolation condition. burn independently of each other. Any elementary structure can be removed, without any consequences for the evolution of the neighoouring ones. [t turns out that a finite spectrum of similarity structures is also possible for f3 > (T +- I. However, in this case the principle of superposition, of combining elementary structures to give more complex ones, is not as simple (sec Remarks).
th~t
2 The multi-dimel/si()/w/ case N > I
A self-similar solution of the S-regime exists in spaces of arbitrary dimension, However, unlike the one-dimensional case here there are no non-m(;notone solutions.
0(0)
= p...
(nO)
= o.
(9).
(17)
where p.. > 0 is a constant. At the points where 0 ::.: o. equation (16) coincides with (9). We must find a value p.. = Oll > O. such that the solution of the Cauchy problem (16). (17). 0 = H(~:p..), is non-negative for all (;::.: 0 and satisfies the second condition of (10). that is, 0(00: p..) = O. Local existence and uniqueness of solutions of the problem (16), (17) for all sufficiently small (; > 0 is established by analyzing the equivalent integral equation using the Banach contraction mapping theorem. The main interest lies in the global analysis of properties of solutions O(~; p..), which is presented below. First of all let us note that every local solution O(g; p..) can be extended to the whole semi-axis ~ E R 1 : fI({;: p..) can go only to the values o or ±tT II" as {; . --+ 00. Proof of Theorem I is based on the following lemmas. Lemma l. Let (\ 8)
Theil 0«(;; p..) > () .!ill' (/1/ {; > O. Furthermore, .fill' allY p.. so/wioll is bOlll/ded in R,: If/({;: p..)! < p... ~ > O.
:>
01/
(T,I/"
Ihe
(\9)
•
~
IV Nonlinear equation with a source
184
185
I Three lypes of self-similar blow-up regimes in combustion
Let us show now that for some sufficiently large positive,
}J-
the solution 0 is not strictly
Lcmmu 2. 71,ere exis(s JJ- :::: j.L' > Oil. jilr which Ihe sO/lIIiOIl or Ihe Cal/chy proh/em (16), ( 17) hecomes ::.ero (/( (/ poilll g > 0, pro(~r Let us assume the contrary. Lei O(r j.L) > 0 in R t for all JJ- > Oil, The
problem (16), (17) is equivalent to the integral equation
(IT
+
[~I
l)glN /' 1]N-1 ./0
1T/(ITIII
d1], g>
0,
(23)
IT
where we have introduced the notation r/J(.<,) :::: IOI"O(g; JJ-).
=
Fig. 39. The function
V/~(.<,)=«(T+I)gl
1
(T+
N /' ./0
T/ I [LJJ-
"11/IJLIIT/("llll/IJL-I/IJL] d1], g>O,
(24)
(T
and by (19) Proof The proof relies on an identity, to derive which we multiply (16) by IHI"O' and integrate the resulting equality over the interval (0, g), taking into account conditions (17). As a result we have
1'/11'(/;)1 ::: 1,.<, ::::: 0;
2
+ (N
d1] - I) [ ' (iHI"O')"(1]) ----
1]
,0
+
I j"
, (T IVI,(g)I:::-;:-__ -
::::
l;N
I
(20)
I
,0
+I =-N'<' (T
where
1
-----JJ2«T + I)
2({+2
j,Llr-+-2
- (T«T -------, + 2)
JJ-::::: 0
(21 )
(the graph of this function is sketched in Figure 39). From (20) it follows that (1)(IO(g)l) ::: (I)(JJ-) for all {? > 0 (the equality is attained only in the case 0 fill, }J- :::: fill). Therefore if JJ- < JJ-., the solution of the problem satisfies the estimate
=
JJ- > fill,
[J
Remurk. By (20) the possible oscillations of 0 around the homothermic solution fi = Oil are damped, that is, if'<'l < 6 are maxima (minima) of the function 0 ::::: 0,
then (22)
I "+ I] (1]= /
TI N-I
[ -}1
[
I
I
+
(T
;;j.L
(26)
Ir]
•
.
( l; > l.
From (25) and (26) it follows that for any j.L > Oil the functions 1/11, and I/I~ are uniformly bounded on any compact interval [0, '<'11/1. Then from the Arzela-Ascoli compactness theorem it follows that there cxists a sequence j.Lk -4 00, k - 4 00, such that the corresponding sequence 1/IJLk(g) converges uniformly on [0, gil/I to some function w(g). The equation for w is obtained from (24) by passing to the limit j.L :::: j.Lk -4 00 (convergence of 1/1 1",' to 11" is established by passing from (24) to the corresponding integral equation). It has the form
1I,'(g) :::: where JJ-I > 0 is the second (different from /J-) root of the equation
(25)
Oil,
Moreover, from (24) we derive the estimate
+
I -(IIJI"O')"W
j.L>
-(IT
+
I)/;I N /' TIN
./0
IlIJ(1]) d1],
g > 0;
10(0)
:c;
I.
(27)
and 11' E C(iO, (0») nC I (R1)' Taking now into account the assumption that '/' 1' ;:. 0 in R i for any j.L > Oil, we obtain that 11'(1;) ::::: 0 in R" However, by (27) jtl(g) is a monotonc strictly decreasing function, so that 10 > 0 in R" This immediately leads to a contradiction, because (27) is equivalcnt to the boundary value problem
I Thrce types of self-similar blow-up regimes in combustion
Proof Let us consider equation (23), which is equivalent to the problem (16), (17), for Jl = JlI. The integrand contain the function 1C/;1,,!(rT+ Ilc/;, which is not differentiable at 0 on K, then we have continuous dependence on Jl. Let {; = (; I be the tirst point where O({;; JlI) = O. By assumption, (IHI" 0)' (~I: Jlil f. O. Then we have continuous dependcnce on Jl on any interval 10. (; I - f j, where f > 0 is a small number. In a neighbourhood of c/; == 0 the operator in the right-hand side of (23) is not a contraction. but thc term 1
Fig. 40. A sketch of lhe curves 0 O( ~; different fl- > fll/. f3 '0' IT + I (S-regime)
fl-),
solutions of lhe problem (16). (17) for
,>
whose solution lJl = C I ~I~ NIO J IN ~Ii~ (CT+ I ) 1/2 t) (C 0 is a constant. .J IN· 21/2 is a Bessel function) vanishes at the point {; = t I == ;<~ I; (cT + I) I /~. where .::~ I > 0 is the first root of the function .JUI! ~1/2. 0
Proof 0( Theorem I. It is based entirely on Lemmas 1-3. Let us introduce the set .M = {Jlo > 0 I O(t; Jl) > 0 in R I for all 0 < Jl < Jlo }. From Lemma I it follows that .M =I- lJl o .::: 0,,). By Lemma 2 ,M is bounded from ablilve. Therefore there exists 00 = sup.ill < 00. From Lemma 3 it follows then that the solution of problem (16), (17) is the required function Os, satisfying (10), with the asymptoti'c behaviour given by (15). Monotonicity of any non-negative solution of the problem (9), (10) follows immediately from the Remark to Lemma I. 0
3 Non-localized self-similar solutions of the HS-regime, f3 < Propertics of solutions of thc problem (16), (17) arc shown in Figure 40. To values JlI > Oil. Jl~ > JlI therc correspond solutions ()(~; Jl) that are strictly positive in R 1 (Lcmma I), whilc to a value Jl.l > Jl~ there corrcsponds a solution Ott; }-tl), which vanishcs at a point (Lcmma 2). Therefore thcre can exist a value }-t = 00 E (Jl~, Jll!' for which the function II" I I (t: ( 0 ) is "tangent" to the {; axis at some point (; = to. and this "tangency" allows us to extend O({;; ( 0 ) into the domain {; > (;o idcntically by zero. As a result we have a generalized solution of the original problem (9), (10) with a continuous hcat flux _(;N Ill/sl"O:~; it is markcd by a thick line in Figurc 40. But to bc able to use the above properties of the solutions O({;; Jll. we shall nced a condition of continuous dependence of O({;; Jl) on the parameter Jl. Observe that in general there is no continuous dependence. This is clearly seen in Figure 40.
=
Lemma 3. !-I'I Ihl' SOllllioll 0 O({;; Jlll. JlI > 0 he such, Ihm Oil Ihe cOllllwel .I'l'l K 10, ~'II/ Ilh('/'1' are 110 pOiIlIS!lJ!' whieh 11/1"0 (lfJl"IJ)' O. TIlell Ott: Jl) alld
=
Jl
= JlI
Oil K.
=
=
CT
+1
Here we use the same mcthod to prove the theorem concerning solvability of the self-similar problem (5)-(7) for f3 E (I, CT + I). Direct inspection of the equation shows that a solution Os( (;) can only be a function with compact support, having in a neighbourhood of the degeneracy point til = meas supp Os asymptotic behaviour different from thaI of (15):
Odtl . .
=
(CT+ [
I
-(3)CT
2(f3 -
Theorem 2. For allY I < f3 <
CT
I)
tolto . . .- .t)
+ I Iherl'
]1/"
(I
+ cu(t), -
(28)
I'xisls a cO/llpaclly .\·upp0/,Ied sollltioll
Os or Ihe prohll'/Il (5). (7), which is sl/'iClly decreasillg whc'rl'\'e/' (Js > O. The prohll'lIl has 110 1l00l-/IIOIlOlolle solUliolls. For N = I Ihe compaclly supported SOlllliol1
0.1' is ul1ique.
~
IV Nonlinear equation with a source
188
Let us briel1y describe the main steps of the proof. The counterpart of identity (20) has here the form
has the same form as in Figure :19. Therefore, taking into account that III = [(3 - (IT -+- 1)1/[2«(3 - I)] < 0, we have that
0 in R+ for all
011 = «(3 - I) 1/1(3-11 the solution Hence it also follows that for any fl Thus we have proved for the case is uniformly bounded: 10({)1 ::: fl in R+. f3 < IT -+- I the counterpart of Lemma I. To prove the counterpart of Lemma 2, problem (5). (7) (tirst equation (5) is extended into the domain of negative values of 0) is reduced. after the change of variable c/) = IfJllTfJ, to the integral equation
-+-
(IT
=1I1(fT
-+- I k i N
-+- I ).fl
,f
T)N
IT/IIT+ Il e/)-+-
I [(
/3-~-1 -
1
Ic/)II!!'(IT' III/IIF+ II}
assumes the form
x
l(
T) N 1 [(
.0
13.f1111"IIT/11T 111/1111
IN') III Il
M_'''_
f3 - I
-
" -+- N--I , - I ! ' -+~
(IT
1
I'~ .10
T)N
...... x, where w(~) satisfies
11I,I3/1ITil'(T))dT).
(;
=
I.
= n. 11'(0) =
I.
> 0: w(O)
-+-
11'(
I)w"!
'II
IT,'
:::::
O. (; > 0;
I!'
'
(0)
(31)
*
whose solution has a zero. This is proved in subsection 4.1 of 3, where the case of arbitrary (3 > I. Ir ::: n, is considered. Proof of Theorem 2 is concluded as in subsection 2. using an assertion analogous to Lemma 3. Uniqueness of the compactly supporled self-similar function Os, N = I, will be proved in 5, by analyzing a quasi linear partial differential equation. Dependence of fJ(~: fl) on Il for f3 < Ir -+- I is in principle the same as in Figure 40. Having convinced ourselves of the existence of a suitable function 0,\, let us now indicate the main properties of the self-similar solution (I) for I < f3 < IT+ I. This is the HS blOW-Up regime. and the unbounded solution is not localized. This last property follows directly from the change with time of the radius of the support of the unbounded solution liS in (4). Indeed, from (4) we obtain the following expression for Ix I (t) I, the radius of the spherical front of the propagating thermal wave:
*
1.1/(1)1:= .fo(T o - nil! I'rt liI/121!! Iii.
In view of the condition (3 < IT -+- I, we have that Ix f (1)1 ~-. 00 as ( ---+ Ttl' that is, the thermal wave engulfs the whole space in tinite time. Furthermore, it is not hard to deduce from (I) that in the I-IS-regime
From (4) we can also derive the analogous expression for the half-width of the spatial profile of the wave: IX,-I(I)I
(:10)
-+- I )fll
I).fl' N
= Ilk
(32)
mN)
which after the transformation
111;,(0:= lI1(1r
= -(IT -+-
0 for Il
>
l/!
It is equivalent to the problem
:= el)(fl),
where the function I
sequence Iflk I, such that /111-' ...... the problem
,dT)
(IBllTfJ')'(T))---T)
. 0
189
I Three lypes of self-similar hlow-up regimes in combustion
11 1/;1 / 11T/11Tf-11 1
-
-+-
(lr
-+- 1).fl
II/I I-' I1II
N x
1
nO') As in the proof of Lemma 2, we have from here that the assumption 1/1}1 > 0 in R+ for any fl > 011 leads, by the compactness theorem, to the existence of a
= (;.(T o " n lll
'
111/12111 III. () < 1<
1'1),
where ~, > () is a mot of the equation O\(.f) ,= Os(O)/2. By monotonicity of Os, /;. is unique_ It turns out that sufficiently general initial perturbations IIII(X) in the problem (0.1). (0,2) behave according to self-similar rules. For (3 < IT -+- I all unbounded solutions 11(1. x) satisfy (32) and are nOllocalized. The theorem concerning absence of localization for (3 <: IT -+- I is proved in 4. As an illustration. we show in Figure 41 the results of a numerical computation of the Cauchy problem (0.1), (0.2) for N = I. It is clearly seen that here, in distinction to the S-regime (see Figure 37), as f ...... 1'1). the thermal wave accelerates, engulting and heating to infinite temperature all the space 1-00 < x < 00 I·
~ I Three types of self-similar blOW-Up regimes in combustion
191
IV Nonlinear equation with a source
190
60
in the cases (3 = IT + I and (3 < tT + I it was shown that in the class of all solutions O(~; J.l) of the Cauchy problem for different J.l > O. there was always a family of strictly positive functions O. oscillating around the spatially homogeneous (homothermic) solution 0 == () /I (sec Figure 40). The osci \lations there were which ensured strict positivity of damped. and their amplitude decreased with the solutions. For (3 > IT + I. when 11/ > O. this. in general. is not the case. For example, for N = I the identity (29) ensures exactly the opposite. i.e .. if for some J.l > 0 there exists a solution 0 = ()(g; J.l) > O. which oscillates about () == (11/. and gl. 6 (~I < ~c) are any lwo maximum (minimum) points, then
___-_6
~
r
'i
4
20
J
•
o
2
_~ 20
40
m_~
.r
Fig. 41. Numerical manifestation of the I-IS-regime. The parameters arc: IT = 2. {3 = S('>. N = I; I: II == O. 2: 1~ == 1.63.3: 11= 2,030. 4: r~ ,= 2.320. S: I, == 2AIO. 6: II, = 2.S0S
4 Localization in the self-similar LS blOW-Up regime,
fJ >
IT
+1
In this subsection we consider self-similar solutions for (3 > IT + I. which illustrate even more clearly than in the S-regime the property of localization of processes of heat diffusion and combustion. Let us consider the boundary value problem (5)-(7) for (3 > IT + 1. It is not hard to show that unlike the cases (3 = IT + I (subsection 2) and (3 < IT + I (subsection 3), for (3 > tT + I there are no generalized solutions with compact support. and Os has the following asymptotics:
where Col = C,\(tT. (3. N) > 0 is a constant (see Remarks). The fact that there does not exist a point ~ = ~o > 0 such that Os(~o) = 0, WU/\)(~o) = 0 and Os(~) > 0 for 0 < ~ < ~o follows directly from a local analysis of equation (5) in a left half-neighbourhood of ~ = go. Below we shall prove existence of the simplest monotone solution of the problem (5)-(7). More complicated non-monotone solutions (so-called COII/hustion l'igenlilllction.\' the nonlinear lIIedil/lII) were studied in detail in 1349.391. 267. 268. 274. 90. I. 21· As usual, side by side with the boundary value problem (5)-(7). we consider the family of Cauchy problems for the same equation:
or
I mO' f; - -:-_._- 0 (3 - I O«)·''II)
= '-' II 0' (0' II) = O' ~ ''.
III
=
+ \0\13
III
= O. g >
(3 - (IT + I) -2«(3 - -- -I)- > 0 ,
0,
(36) We shall take into account the following easily established fact: if ()(~; J.l) -> S as g -> :x; (.I ::: 0). then 01 = 0 (to prove this. it suffices to analyse the equation locally in a neighbourhood of ~ = (X) (see [I. 2])). I Lilleari::,ation aroulld ()
== 011
A fairly precise picture of undamped oscillations for J.l sufficiently close i'b 011 is provided by solutions v(~) of the problem obtained by linearizing the original problem around the homogeneous solution 0 == Oil· Let us set
where E > 0 is a constant, which plays the role of a small parameter in the sequel. Then. after substitution of (37) into (34). (35) we obtain the following problem for v(~): I (N 1 I: > (...) (38) 11',',--~ - I v,), -II1Vl:;+V=E().(Vl.!> ! "
~N
I
-
11(0)
(39)
= I'. v' (0) = O.
Here (\>.(11) : C 2 -> C is a bounded quasi linear second order operator. Boundary values I' in (39) and J.l in (35) are related by II-
= Oil + Ell.
(37')
(34)
(35)
where J.l > 0 is a constant. Let us show that for some J.l the solution II = O(g; J.l) ::: 0 satisfies condition (7) at infinity. and thus defines the required function Os. Observe the following property of solutions of the problem (34). (35). Earlier.
From (38) it follows that by continuous dependence of the solution of the equation 011 a parameter, for sufficiently small E > 0 the solution v(~) of the problem (38), (39) is close to the solution of the corresponding linear problem:
O'lrf~-~ (I:Nl y ')'_ 1I1y'~ + .\' = O. ~N-I!>' .
I; > O.
(40)
= II :j;; 0, .1"(0) = O.
y(O)
(41)
Because of that, let us consider the problem (401, (41) ehange of the independent variable
more detail. The
III
reduces (40) to the degenerate hypergeometric equation
<,(c - TJ) -
a\'
= O.
TJ> D: -,,(0)
where c = N/2, a = -1/(2111) = -(fJ - 1)/lfJ boundary condition assumes the form
I TJ 1,1'1 '--",,(1)) ,,0= I
(rT
= 1'.
+ 1)1.
(43)
Then the second
() .
v(TJ)
= // (
a TJ a(a + I) TJ2 a(a + I )({/ + 2) 1)' ) I + - - + - - - - - + ---- + ... , c I' c(c+ I) 2' c(c+ l)(c+2) 3'
(44)
which defines the degenerate hypergeometric function
In general, the function (44), and thus the solution v(g) of the problem (40), (41), is non-monotone. In the cases when -- a
= (f3 - I )/[fJ -
((T
+ I) 1=
(45)
K,
where K > I is an intcger, the function y(T)) is a polynomial of degree K, since the series (44) terminates at the (K + I )-st term. Furthermore, it is known [351 that it has for 1) > 0 exactly K "zeros". Equality (45) holds if I
fJ = f3,,' : ; ; - --" + K - I
(for convenience we set fJl = 1'(1))
.
(0).
= Jl ( I -,
and therefore thc equation
y( TJ)
K
----·(rT
K - I
+
I), K
= 2. 3, ...
For example, if fJ = f32
4 TJ N
4 ') + ~---Tr N(N+2)
,T)
;:;c
2(T
> D,
+
(46)
I, then
1)1/2 >
I'~( • 1)0
y(~)
of the original
O.
The equality (45) determines the number of zeros (and thus the nature of nonmonotonicity, Of, we might say, the degree of complexity) of the function y for all values fJ > (T + l. Namely, for any fJ,,' \ I S fJ < fJ,,' (K = l. 2, ... ) the function -"(~) has exactly K~1zeros in g > 0 (sec 135J, where approximate formulae for computation of zeros and positions of extremum points of the function y(g) can be found). Combining all the cases considered above, we obtain a general formula for the number of zeros of the solution of the problem (40), (41): K
Therefore a suitable solution of (43) is one with a bounded derivative -,,;/(0). It can be written down in a convergent Kummer series [35, 3171:
.
to which there correspond the following zeros of the solution problem (40j, (41):
goci = 2'1r:::.2(2(T) (42)
TJY~" '.f-
193
§ I Three types of self-similar blow-up regimes in combuslion
IV Nonlinear equation with a source
192
= -[ai,
a
=-(fJ - 1)/lfJ --
(rr
+
1)1
<
D,
(47)
which is valid for any fJ > (T + l. By (47) K ::: 2 for all fJ > (T + l. that is, the solution y(g) will always be non-monotone 2 . Let us note that the oscillations around zero are undamped (this follows directly from the form of equation (40»). Returning to the original linearized problem (3X), (39), we sec that by continuous dependence on the parameter f > D in a neighbourhood of f = 0 of the solution l' on any compact set, there exists an f > 0 small enough, such that for all 1//1 :::: ) the function v(~) has for ~ > 0 at least K (K ::: 2) zeros. For the original problem (34), (35) it means that for any 0 < (III -- f < f-L < (III + f the solution fJ(g; f-L) has for g ::: 0 at least K extrema. In particular, if fill < f-L < (III + E, then there exist at least IK/2jminima and K -IK/2Imaxima, for which (36) holds. 2 Glo!Jai properlies of Ihe solulions
fJ(g: f-L)
Thus, we have determined the behaviour of (I(g: f-L) for all f-L close to 011. Let us show now that for large enough f-L the function (J(~: f-L) vanishes at some point. For that we could use the samc mcthod as in the case f3 S (T + I (though due to (36) some additional difficulties appear). However, for N = I this result can be obtained in a much simpler way. In the following we coniine ourselves to a fairly brief analysis of the case N = I, and at the samc will prcsent some results pertaining to the multi-dimensional case. Lemma 4. Ll'I N = I, f3 >
(T
_
f-L ::: f-L
*
+
I. The/1
Fir (//1
[ f3 + (T + I ] 1/1/3 = (rf=i")(;;-:;- 2)
II
( 4X )
== D has two positive ruots: 2Let us nole that in the case of a scmilinear equation (eT :;:;; 0), K~c conclusion plays an important part in § 7.
the solution 0 = O(~; f-L) of the prohlem (34), (35) vanishes at some point and has no extrema in l~ > 0 I f}(~; f-L) > 0,0 < ~ < ~) (that is. it is strictly decreasing). Proof Let us consider the identity (29), which for N
=
195
Three types of self-similar blow-up regimes in combustion 8(~jl'-)
flit }LJ
I has the form
80
(49)
l'-2 Ii,
Here the function
8H
has the same form as in the case f3 :s. (T + I (see Figure 39). Let us assume the opposite. For example, assuming that (48) is satisfied, let the solution {}(~; f-L) have a pDint of minimum at ~ = ~. < x, where. naturally. o < 0/1. Then, setting in (49) {; = ~,. in view of the condition m > 0, we obtain the inequality cl'(f-L), and therefore we must have that {}(~.: f-L) > f-L > 01/, which is impossible. In the same way it is proved that for f-L .::: JL' the function O(~; JL) cannot be a positive solution in R+ (i.e. the case t;. = 00 is also 0 impossible). In a similar fashion, we derive from (49) Corollary. Any soilltion of' the prohlem (5)-(7) sati.ljies j<J/' f3 > the estimate
, Os W < f-L
=
[
f3 + (T + I ] (f3 _ 1)( (T + 2)
1/(#
IJ
(T
+
I,
N
=
l.
tI
:=
tI(~:: IJ.}
for different IJ.. f3
> 'T
+
I
In Figure 42 we sketch the behaviour of the functions {)(~: JL) for different values of JL == 0(0: f-L): the thick line shows the solution Os(t;). which corresponds to JL = {}o = sup J't. Thus, we have proved the following
Theorem 3. Let f3 >
Let {; = t;. be a point of absolute extremum of the function O,dt;). Setting in (49) first t; = 00, and then t; = ~., and subtracting the second equality from 0 the first, we obtain (T + l, N I. Let us set .N' {JL > 0 11 1 there exists a compact set K I0, ~ 1\ I, such that O(~: f-L) > 0 on K and has at least Dne minimum point on K). Then .N' I: Vi (by
=
Fig. 42. The function
(50)
.
Pro(~l
=
O+----....-l;--~----
(T
+
I, N
I. '!1u'll the prohlem (5)-(7) has a strictly
111OIIotolle positil'C' sollllioll.
Some additional properties of the function (Js(~) will be mentioned in subsection 4.3, as well as in 6.
*
=
the analysis of the linearized equation) and .N' is bounded from above (Lemma 4). Therefore there exists (51 ) supN = 00 E (0/1. (0). It is not hard to see that by the choice of (}o the function Ott;: Ro), first, has no minima for {; > 0 (this follows from continuous dependence of (I(~; f-L) all f-L on any compact set, where (J > 0), and, second, cannot vanish (see Lemma :.'\). Therefore fJ(g; (lo) is a positive, strictly monotone solution O,l'(t;) of the problem (5)-(7) for f3 > (T + I, N = I.
Remark. As we already mentioned above, in Theorem 3 we determine the simplest self-similar solution: in fact the problem (5)··(7) has at least K' = K - I different solutions O.~. O~ . ... ,O~ I. Each one of these has one more extremum point than the preceding one. Among these there are at least IK'/2] solutions for which Os(U) < 0/1 (t; == 0 is a point of minimum) and at least K' - [K'/2\ solutions such that OslO) > 01/ (~ == 0 is a point of maximum). Proof of existence of these solutions follows the same lines: the set N I: Vi now including values of f-L for which the corresponding function O({;: JL) has a more complicated spatial profile than the desired function (}s(t;) (see [1, 2j).
~
IV Nonlinear e4uation with a source
196
3 The mulli-dimensional case, N > I
The method of proof of Lemma 2 can be used to show quite easily that whether or not a statement analogous to Lemma 4 holds for N > I depends on the properties of the solution of the stationary equation
f~
I
(t
Illl"l} + Il\fJ l(O)
= o.
I
1
g > 0;
(52)
= I. ('(0) = 0
= '-(IT + l)t lN r~
7]NI\I/I/,1 1fJ1 "t 111/1"tll /'l'd7] '
.In
where 1/11'(0) = I. 1/1;, (0) = is bounded in C: G(I/I/,) ::.:: m(lT
Theorem 4. LeI IT
0
and
G(I/'I')
+
I)(N
+ 1-'-1/3C(I/II')'
(53)
Remark. In the case f3 :::: (rr + I )(N + 2)/(N - 2); the solution l(g) of (52) is strictly positive in R+ (see Lemma I in 3) and the question of existcnce of Os(t) remains open. In the sequel we shall need the following surprising property of the self-similar function Os.
is the following integral operator. which
t
Ttl
[_1f3 -
.-mN]
I
1//1,1 '
I//x.(t)::.:: -(IT'+ l)t
r~ 7]NI'/I~
.In
I/''X,(O)
I <
f3 ::::
(IT
+ I )N/(N
- 2)+ ol/d leI ()s(t) be
llll
arbilrarv
(54)
,,/1"tll'/ll'd7].
which is not a contraction in a neighbourhood of 1/1/, == O. Unlike the case f3 :::: (T + I, for f3 > (T + I. a priori nothing can be said about boundedness of 1/1 1, and 1/';" on compact sets, sincc possible oscillations of O(g; 1-'-) around () == 0" are undamped (see (36)). Therefore we shall use a method based on continuous dependence of 1//1' on I-'- in a neighbourhood of I-'- :-..:: (Xl. For I-'- ::.:: (Xl, (53) becomes formally the equation l .N
+
solulion oFlhe problem (5)-(7). Then
+ I )~II/,/,1"/{a 111'/'1' +
.In
1 < f3 < (Xl for N = I or N = 2 and IT + I < f3 < - 2) for N :::: 3. Theil Ihe {lroblem (5)-(7) has ({ slricllv
pOsilil'e 1/1OIIolOne soluliol/.
Theorem 5. LeI IT
+(rr+ l)t lN
+
+ 2)/(N
*
Indeed. let us rewrite (30') in the form I/I;,(t)
Schauder fixed point theorem. Then by "smallness" of C(I{lJL) the derivative Ijl~(g) does not change signiflcantly and as a result 1/1 1, (g) will be zero if I-'- is sufflciently large (1-'-1-f3 is a small number). This fact allows us to prove the following result.
(IT N
197
I Three types of self-similar blow-up regimes in combustion
Inequality (54) displays some important properties of the unbounded self-similar solution (I). (4). For examplc. from it immediately follows
Corollary l. For IT+ I <
==
iI --udl . .1') ill'
1"llil/
l
"JII I{I'X,d7].t
f3 ::::
(IT+ I )N/(N - 2)
t
Ihe solwioll us(T . .1') is crilical.
Ihol is.
(Tn - I)' 1/(ji11·1
[I + f3 - ] //If/.(e)/;
.\
!.
-
·_·-Hd/;)
I . -
> O.
(55)
> 0; IE (0. Tn) . .I' ERN.
= 1.1/'~(O) = O.
Ulld there/IJI'(' .fiJI' allY I E (0. 1'11)
and its solution coincides with the function \ll"l(~). where l(i;) is the solution of the problem (52). In 3 we shall show that for any f3 < (rr + I)(N + 2)/(N - :2), the function .ng) of (52) vanishes at some point g ::.:: ~. > 0, and. moreover. I{I~(~.) < O. Therefore on every compact set K, = 10. f, - fJ. f > 0, on which I/I oc > 0 there is continuous dependence of 1/'1' (g) on I-'- in a neighbourhood of I-'- = (Xl, that is, I/'I'(~) is close to l/h,Jt) on K, for all sufficiently large I-'- > O. Let us fix a sufflciently small f > O. Then
*
lI'here C\ ::.:: es((T,
f3.
N) > 0 is Ihe cOllslull1 of Ih/' lIsymplolic /'xpollsio/l (33).
Integrating the inequality (54). we obtain the following estimate, which again demonstrates strict positivity of 0.0;(0 (it correctly reflects the asymptotic behaviour of the function Os as i; (Xl). -)0
I{lJL(~' - f)
-)0
f"JI({:. -
f).f/,;,ri;. - f)
= O(III/'JLII;lltrtli) I{IJL (~) from a point {: ,,: t, ._But
C(I/'I')
-)0
f
-)0
(f"II({:. -
E)}' < ()
as I-'-
-)0
00.
0 as 1I1/IJLlle O. Therefore in order to extend into a neighbourhood of {: = f, we can use the -)0
~ I Three lypes of self-similar blow-up regimes in combustion
IV Nonlinear equation with a source
19H
F({;I) .:::
')IT
((:III
N - I ,,'
'I' 'I + ---{,I' ., (;, +
H'"
(T-'-0l'
Os'
) + "I' llf3 ,
= /'. (d. . l:
(57)
D, OS({;I) > D, {;I ::: D. Then O~I({;I l < D and therefore
"(l: / . sJl
l:
== InSI .::: 1Il{;1
["
,
O,(~I)
.
" [ 0S({;I)
+
(3-(T+ I H'I.(E,I)] ",::: (3-«(T+I) {;I
+ -{;II
(N - I -
2cr),] 01·({;Jl •
(3-«(T+I)
'
as (f3-(T+ 1)/If3-((T+ 1)1::: N-I-2(T/[f3-(rT+ III forrT+ 1< f3::: «(T + I) N/ (N - 2) I . Then from (57) we have 0" (;: )
_.1_'_,,_I.
1/11;1
since
F' (E,.1 )
<
F(E, ) < D. ,1._
0', < _ I 1 Os 1/1 (f3 - I) E, I .
199
Hence we have thal in the LS-regime «(3 > (T + I) the half-width decreases with time and Ix"j(T() II == O. Thus intensive combustion takes place in an ever shrinking central region of the structure. As a result blOW-Up occurs only at one point; in the rest of the space the temperature is bounded from above uniformly in I by the limiting profile Ils(T(). x) (see (56)). Thus the self-similar solution is effectively localized. Strict localization, howN ever, cannot obtain here, since lIs(t. x) is strictly positive in (0. Tn) x R . Numerical computations show that self-similar estimates hold. and. moreover, testify to the occurrence of strict localization in the LS-regime (for a proof see § 4). An example of such a computation is presented in Figure 43. Here un(x) is a compactly supported (not self-similar) initial function. Up to the time I = II the initial perturbation spreads oul, then reaches its resonance lenglh (I = 12)' after which fast growth of the solution starts. It is clearly seen that as I ~ 1'(1 combustion occurs in an ever narrowing central region of the structure. During the process the front points of the solution lI(t. x) hardly move at all and heat is localized in the fundamental length L u ; of LS-regime. Wc emphasize that herc. unlike thc situation in thc S-regime. LIS dcpends on the initial perturbation L/n(~)· Figure 44 shows the LS-regime, which is close to the self-similar one as I ~ 1'1';, which develops from a spatially homogeneous initial perturbation lIn(.r) == I, due to instability of homothermic combustion with finite time blOW-Up,
From here it follows that F(I;I) < 0, so that the function F({;) is decreasing on an intcrval (E,I. 1;1 + (»), 8 > D, where Hlf ( l:) _s_s_F'(E,) < F({;) < O. III 1 ; '
.
Therefore Fi{;) < 0 for any {; > (;I. But by (57) it also mcans that
The last inequality, under the assumptions we have made, ensures that the function Os({;) vanishes at some point and, conscquently, is not a solution of the problem (5)-(7). This is especially simple to prove when N = 1,2, f3 > (T + I. Analysis of the casc N ::: 3 uscs the same mcthod as in the proof of Lemma I in subsection 4.1 in § 3. 0 4 Properties of Ihe
seU~sil/1il(/r
Vi-regime
Let us again write down the expression for the time dependence of the half-width of the self-similar thermal structure: IXI'I(tll == {;.(T o - I)I/J
-('f
f
111/121f3- Iii
0 < I < Tn.
Fig. 43. NUlllcrical manifcstation of the LS-regime. The parametcrs arc: iT '" 2. {3 = 5. N l; I: /1 1.14.2: 12 2,34.3: 'co 3.559, 4: I., 3.5712. 5: /s 3.5714
=
=
=
I.,
=
=
*2 Asymptotic behaviour of unbounded solutions
IV Nonlinear equation with a source
200
201
heating the spatia-temporal structure characteristic of the self-similar solution u (t,:z:) 180
(3)
6
m=I,B-(cT+ll!/12(,B-I)].
Difficulties of the analysis of asymptotic behaviour of unbounded solutions are related to the speed of evolution of the blow-up regime. which is not stable with respect to arbitrary. even infinitesimally small. perturbations of the initial function
140
lIoLr) .
Here we present the results of a qualitative analysis. Qualitative theory allows us to obtain a number of quite subtle results: for example. for ,B > IT + I + 21N we can find a family of global solutions of the problem (I). (2). whieh correspond to sufficiently small initial functions lIoLr). At the same time we show that for ,B ::: I + IT + 21 N there are no global solutions II '1= n. These results are justified in ~ 3. The idea of the averaging method consists of reducing the problem (I). (2) for a partial differential equation to a system of two ordinary differential equations with respect to certain parameters that characterize the evolution in time of the spatial profile of the thermal structure. As such parameters we can choose. for example. the amplitude and the half-width of the structure. or the amplitude and the position of the front of a radially symmetric structure which has compact support in x. The latter averaging. "amplitude-front position" allows us. in particular. to describe the localization of unbounded solutions in the S- and LS-regimes. and absence of localization in the HS-regime.
100·
60 -
20--
o
,.?
10
.7:
Fig. 44. NUlllerical manifestation of the LS-regillle. The parameters arc: N = I; I: /1 O.2S24. 2: /2 I, = O.2S3679. 6: /(, = O.2S-'6R I
O.2S.'')S. -':
/1
O.2S-'6'). 4:
I.l
2. f3
4. = O.2S-'674. '): (T
§ 2 Asymptotic behaviour of unbounded solutions. Qualitative theory of non-stationary averaging In this section we consider questions connected with asymptotic stability of selfsimilar solutions of the problem ;\((1)
u(O, x)
=
11 1 -- \7. (11"\7//) -
= //o(x):::, n.
//# =
O. I>
n,
x ERN,
x ERN; //0 E C(RN).//;;II E HI(I~/'\
I The non-stationary averaging "amplitude-half-width" Let the initial function 110 E L I (R N ) in (2) have compact support and be elementary in the sense that lIo(.r) has a unique maximum at x = n. We shall take lIoLr) to be close to a radially symmetric function. Then we should expect that the solution uU. x) will also be almost radially symmetric and that the half-width of the evolving thermal structure will be approximately the same in all directions. Taking this as our departure point. let us seek an approximate solution of the problem (I), (2) in the form lI(l. x)
We shall be interested in the following question: under what conditions docs the unhounded solution of the problem, 11(1. x), aequire in the domain of intense
where 1/1(/) and c/J(t) arc. respectively. the amplitude and the half-width of the structure. which depend on time. and ()(~) is some fixed function of compact support. which is monotone decreasing in all its arguments and such that R(O) = I. ()"+I E HI (R'\\
In its form (4) is the same as the self-similar solution (3), which was studied in § I, where the specitic form of the functions ~I(I), (/1(1) is determincd by substituting (4) into the original equation (I). Therefore the self-similar solution (3) satistles the problem (I). (2) for some specially chosen initial functions I/o(x). In our case I/o is. in general. an arbitrary function; therefore we do not require that the approximate solution satisfy equation ( I) in strict sense. Instead. we shall demand that (4) satisfy the two following equalities (conservation laws)': .
A(I/(I. xl) dx::;: O.
./.R'
I'
203
§ 2 Asymptotic behaviour of unbounded solutions
IV Nonlinear equation with a source
202
A(I/(I. x))I/(t. x) dx::;: O. , > O.
It is not hard to resolve the system (8). (9) with respect to derivatives: (II) (12)
while from (11). (12) we easily pass to the single equation
(5)
(\ 3)
./R'
After integration by parts, these equalities have the form where ~
d
d,III/(I)III I IR'
j::;:
(6)
1II/(I)IIf./J IR j· '
We shall take the condition I d
/.
_ I ,
,1/ . R·I
2 -') -, 1II/(I)III.'IR'1 ::;: -
IT
1'VIII
2
dx
~jl
+ 1III(I)IIf.!'·'IR·'j·
b ::;:
a::;: (/I, - /11)//12. 111
>
(/I, -
(\4)
21/ 1 )/1/2.
to hold. so that
2/11
(7)
>
II
In the tirst equality. which is the energy equation. there is no contribution from the diffusion operator, while in the second there arc contributions of both the source term and the diffusion operator. Substitution of (4) into (6), (7) gives us the following system of ordinary differential equations for the functions '/1(1), 41(1):
(15)
O. b> O.
#
The inequalities (15) are. generally speaking, necessary for (II), (\ 2) to admit blow-up regimes. Let us move on now to analyse the equation (13), which describes the dependence of the amplitude of the thermal structure on its half-width.
fJ ::;:
, S-regime.
(T
+I
In this case equation (\ 3) has an especially simple foml:
("II where
//1. /12. /I,
-, I ::;: ( Cfl
arc positive constants:
-N
I/J a
I' ~I > 0,
-I.'
1 )'!'- (/1
>
o.
(16)
It is easily integrated. and its general solution has the form 0 C.o ::;: /II"I/,.NI
(10)
I
/I -
-
2/1
1/'
1"N/'1/1211""2/"I!
V2
where Co ::: 0 is a constant determined by the initial conditions: if 1/1(0) and O. then
Here we arc assuming that the function make sense.
(I
(17)
-.1- - - 0 -
= ~Jo
> 0
in (4) is such that all these expressions
31nstead of these two laws we could take others; for example. instead of the second one we <.:Ould take the identity (A(II(r . .1')) • .1') =0: 0 and retain the first one as it is the simplest possible. This does not ailed the results of the analysis below.
where ¢~ i= /12/(1'1 - 2/11)' For (17) tll correspond to a blOW-Up regime. we have to demand that the inequality 11 3 > 2//( holds. that is. that both conditions (15) are satisfied.
~
IV Nonlinear equation with a source
204
205
2 Asymptotic behaviour of unbounded solutions
We are in a position to check how exact the averaging is, using the self-similar solution of § I for the case {:3 = IT + I, N :::: I: (.1'
11.1'(1, x) ::::
_ 1)-I/IT ( 0
{
2'
I 'IT
(IT
IT(IT
+ I) cos 2 7TX)' ,Ixi < L)2 + 2) r; - .\ '
0, Ixl > Ls 12; 0 <
I
(18)
< To.
where L s :::: 27T((T + I) 1/2 I (T is the fundamental length. Taking the precise structure of (18) into account, let us set
Igi < I, Igi :': I.
(19)
to which cOlTesponds the amplitude ~/(I)
2((T+ 1)]I/IT, =--.----(10 [ U(IT
+ 2}
- I)
--1/IT
Fig. 45. Phase plane of equallon
(20)
,
(](1)
(f3
:= (T
+- I. S-regime)
that is,
as well as the half-width I/J(I) ::::
For the function (19) the coefficients IT
+2
IJI :::;; - - . - - - - ,
2(IT + I}
[.l? :::::::
-
Ls12.
111.1'2,1/, 7T
If/(I)::::: (In/l) 1/"(1'0 - 1)-I/IT, I -->
(20') have the following form:
2
~-- ......_-.
IT(IT + 2}
IT
+4
1/_, ::::: - ' - -
(21 )
IT + 2
(let us stress that here I', > 2111, so that the inequalities (15) hold). Substitution of the functions (20), (20') into (16) leads to an identity. Therefore, the averaging method gives us an absolutely exact description of the self-similar solution (18). A clear understanding of the evolution of the thermal structure (the dependence of the amplitude on the half-width for different initial data) can be obtained from considering the behaviour of phase trajectories of equation (16), which are depicted schematically in Figure 45. The thick line denotes the trajectory
which is the same as the dependence on time of the amplitude of the self-similar structure (see § I). If for N :::: I we take the value of /'1 from (21), then we obtain precisely the expression (20) for the self-similar solution. In conclusion, let us remark that the rcsults of numerical computations agree well with the phasc plane picture of Figure 45. 2 HS-regil11e. (:3 < IT + I Let us first determine the gencral solution of cquation (13) for (:3 =f: rT + J. Let us set ~,rl--IIT + II(f/ = 2((f»). Then from (13) we obtain a separable equation:
I]
dz. [2-N({:3-if-I)-az -¢=z . dIp !Jz - I the general solution of which for {:3 =f:
which corresponds to the self-similar solution of the S-regime (it is also the isocline of intlnity of equation (16)); the dashed line denotes the nullcline (/) = (/ 1;2 < (f)s. That figure shows, in particular, that for (:3 :::: IT + I all solutions become in fact infinite in finite time. and as the amplitude Ifl(1) grows, all the trajectories converge to the self-similar one: 4)(1) --> 4).1', I --> Til' Using this conclusion, we immediately obtain from (II) for (:3 :::: IT + J If/ (t) ::::: If/'I 1(I ) I' I, I --> Til'
To,
(T
+ I + 21 N
(22)
has the form
wherc £I
= /'2 1'1
[11, _ 1'1
{:3 - (IT {:3 - ((T
+ I + 41N)] + I + 21N)
1
(:3-((T+I) ['/' f3-(IT+I+4 I N)] a=NIf3_«(T+I+2I N )j2 ~- f3-(IT+l+2I N )
Fig. 46. Evolution of trajcctories of cquation (13) in the HS-regimc «(3
< IT
+ I)
Thus. let (3 < (T + I. Using (23), it is not hard to draw the phase plane of equation (13). For convenience, we have shown in Figure 46 the nullcline (24) and the isocline of infinity III
= IIIcxo(I/I) = b-I/I13(fT11)Jc/J-21113- (fTj III
(25)
The thick line is used to show the separatrix III
= Ills(c{J) = dl/lli--ifT111!c/J-2/11i-ifT-tlll,
(26)
which is an exact solution of the equation. In (23) to this trajectory there corresponds the value Co = O. For (3 < (T + I we have the inequalities Illoo(c/J) < 11/.1'(1/1) < 1110(4/), which deline the nature of the evolution of trajectories in Figure 46. Thus. as can be seen from that figure. in the HS-regime all the trajectories converge as III -+ 00 to the separatrix III = IIIS(IP), that is, (27) Substituting this estimate into equations (II), (12), we deduce that at a fully developed stage of evolution 111(1) ~ (To - I)I/II! I), (1)(1) ~ (To - I)W- 1,,
- - - - - - - - - -rp
o
(28)
that is, as r -, To- unbounded solutions develop a spatio-temporal structure which is closed to the self-similar one.
l<'ig.47. Evolutioll of trajccturies fur f3
E (IT
+ I. IT
3 LS-regime, (3 > er
+
-1
I + 21N) (LS-regime)
I
As can be seen from (23), for (3 > (T + I the phase planes are different in the cases (T + I < (3 < (T + I + 21N, (3 = (T + I + 21N, and (3 > IT + I + 21N. The case IT + I < (3 < (T + I + 21 N: ul/boul/ded so/uriol/s. Figure 47 shows the phase portrait for (3 E «T + I, (T + I + 2IN). Here the separatrix VI r/Js(¢) (see (26» lies above the isoclines (24), (25). With time, all trajectories converge to the separatrix, that is the asymptotic equality (27) holds, so that as r -+ Til the estimates (28) are satisfied. Thus for (3 < IT -+ I + 21N all solutions of the problem are unhounded and as I -+ Til their evolution follows that of the self-similar
=
solution. The case (3 = (T + I + 21N: ul/boul/ded so/uliol/s. For (3 = (/ equation (22) assumes the form
dz
-41
dc{J
+ I + 21N
_2
= 2(b
II) .._.::...-.-.
bz -- I
Its general solution is determined from the algebraic relation
(29) For (3 = IT + I + 21 N the phase plane of equation (13) has no separatrix; this follows from (29). As (/ > 17, V/O(IPJ < r/J,dc/J), and the trajectories behave more or less as in Figure 47. except that in this case there is no special trajectory r/Js·
208
§
IV Nonlinear equation with a source
2 Asymptotic behaviour of unbounded solutions
209
Thus, for f3 = IT + I + 21N, as before, all the trajectories correspond to unbounded solutions of the problem. Moreover, as the amplitude 1//(1) grows, the half-width cP(t) decreases. Let us flnd out what is the asymptotic behaviour of 1//. IP as I - 4 To'. From (29) or directly from equation (13) it is easy to deduce that as the solution grows without bound, the relation between 1/1 and 1/) can be determined from the approximate equality (30)
that is, (30')
where 8 0 > 0 is a constant that depends on the initial conditions. The dependence (30') is not a self-similar one. whieh corresponds to the asymptotic equality (27). since here Nr:..>I) f3
-
2 (IT
+
IT
+ I + 21N
(31 )
.=N
(32)
where I
>
I)
(we remind the reader that (I > 11 by (14». The estimate (30') gives us the following asymptotic expressions for the amplitude and the half-width of the thermal structure as I --, To:
N(I a= [ -,;(f3.I) - I J
Fig. 48. LS-regime. {3
== [(I-,;(NIT+2).- I J
I
Let us compare (32) with the self-similar expressions (2X), The amplitude 111(1) is the self-similar one (there is a rigorous justification for that), while the halfwidth 4)(1) behaves as I -+ Til in a non-self-similar way, since in general a is different from the exponent 1f3 - (eT + I) 1/12(f3 - l)j == I I (N IT + 2), which appears in (2X). Thus there arises the question: what invariant or approximate self-similar solution describes the asymptotic (J -> To) stage of the process') Therefore for f3 :::: IT -+ I -+·21 N we can in principle expect unbounded solutions which evolve not according to self-similar laws at the asymptotic stage. Let us note that in (30') and (32), apart from the non-self-similar exponent. we also have signiflcant dependence on the initial conditions (through the constant 8 0 in (30'). which differs from one trajectory to another); we recall that for f3 < IT+ I +21N all the trajectories converged to the self-similar separatrix (26) with a lixed constant d determined from (23'), At this point it has to be stressed that the averaging theory considers solutions with compact support, u(I. ,) ELI (I~N) for any I < To. It is well known (see ~ I) that for f3 > IT + I self-similar solutions u.\ do not have compact support; however,
if iT+ I < f3 < eT+ I + 21N. the inclusion 11.1' ELI (R N ) still obtains, that is. liS have finite energy. This. apparently. can guarantee self-similar asymptotics of compactly supported solutions. For f3 = IT + I + 21N (and (I.!iJrliol'i for f3 > IT + I + 21N) the energy of 11.\ is infinite 4 and therefore liS docs not necessarily describe the asymptotics of a solution with compact support and flnite energy. The case f3 > IT + I + 21N: 1I/l!JO/I/lded .wlllliO/ls. In this case there exists a separatrix (26l. and in the phase plane it is placed so that '11.1'(11) < Illo(ljJ) < 111'X-(ijJ). This determines the behaviour of trajectories in Figure 4X. The asymptotics of unbounded solutions as I -> To here is the same as in the case f3 = IT + I + 21 N, that is. it is non-self-similar (see (30'), (32». The c(lse f3 > IT+ I +2IN: glo!Jalsolulio/ls. From Figure 4X it can be seen that for f3 > IT + I + 21 N there are initial conditions to which there correspond global solutions (that do not blow up in finite time), The corresponding global trajectories lie below the separatrix III = 111.1'(1/1), As I - , 00, the amplitude of the global solutions goes to zero. while the half-width grows without bound (extinction). A rigorous construction of the family of global solutions is presented in 3, Let us determine the dynamics of this extinction process, First of all let us note that to the separatrix III = 1/1\(I/J) there corresponds a thermal perturbation for which
*
(33)
,lThis follows from the nature or asymptotlCs of C's( - 2/1fJ·
~ 2. Asymptotic behaviour of lIl1bounded solutions,
IV Nonlinear equation with a source
In § 3 we shall construct a family of global self-similar solutions of equation (I) with the spatio-temporal structure of (33), so that the separatrix 1/1 = I/Js(c/J) is the image of some self-similar solution. What is the behaviour as I -+ 00 of the remaining global trajectories'? From (23) it is not hard to deduce that for them (34) where the constant Do. which depends on initial conditions, has the form
Do
= ICocr"]
_ NIIJ-ler+I+2/NI! 13-(,,+ II
an approximate solution of the same form: (38) where I/J(I) > 0 is the amplitude of the solution, and now g(l) > 0 is not the half-width. but the position of the front of the solution (in the symmetric case the front is the surface Ixl = g(l)). The function 8(~) :::: 0 is such that 8(~) > 0 for ~ E [0. I), (J(~) = 0 for all ~ :::: L (J(O) = I. (J'(O) = O. (Jer+1 E HI «0. I)). As the first equation for the functions 1/1. g. we choose the energy equation
!-,l
(35)
Substitution of (34) into the original system (\ I). (12) gives us the following asymptotics of global solutions:
[¢/(t)g,v (I)] ::::
om.
II,
= 'V. (lI
cr
'VlI) , I>
O.
X
E
R
N
(37)
(for a proof of this fact see § 3). Another indication of this is given by the asymptotics (36) (a self-similar solution of equation (37). which satisfies (36), is given in § 3, eh. \). Therefore for f3 > CT + I + 2/ N, the separatrix, to which there correspond selfsimilar solutions, is unstable in the class of both unbounded and of global solutions. Therefore it has to be expected that at the asymptotic stage the combustion process evolves according to different, non-self-similar rules. and. as shown by averaging. the form of the limiting thermal structure depends on initial data. These are the qualitative properties of the evolution of thermal structures initiated by an elementary perturbation with finite energy. As shown by numerical experiments, this method of "amplitude-half-width" averaging affords us quite a precise description of the behaviour of unbounded solutions on a large time interval. The question of the nature of the front motion. and thus the question of combustion localization. remain unanswered. To that end, we present below another method of averaging.
VI 1//' (t)gN (I). I
Let an elementary initial perturbation Uo ELI (R N ) be radially symmetric and have compact support. Then u = u(l, r), r = Ixl. for all I E (0, To). We shall look for
(39)
The second equation is obtained from the well-known expression for the motion of the front of a thermal wave (see Remarks): dg(t) - = -dl
.
illl(1. r)/ill
(40)
11m - - - - - . ,Ag (II
311(1. r)/ilr
Hence. using the original equation (I). we obtain dg(t)
.
rINJI(rN-'llIrTlI~)~
+ II fJ
- - - :::: - 11m .----.--------. dl
1I~
/1·.0
and. finally. resolving the indeterminacy in the right-hand side. using the known differentiability properties of the solution 1I(t. x). we arrive at the equality dg dl
= _ 1/--0 lim II cr - I II' ,.
(41)
In the derivation of this equality we assumed that the singularity of the solution close to the front has the algebraic form 11(1. x) ~, (g(l)- Ixl)l/cr. Therefore the presence of the source terlll u fJ has no bearing on the final form of (41). Substituting the approximate equality (38) into (41). we obtain the second equation:
dg(t)
I{/'(I)
dl
g(t)
_ _ :::: I l , \ - - ,
I> O.
(42)
where "'l = -UJ")'( I )/IT > n. The required system of el\uations for 1/1, g has been obtained. Solving (39) for ,//(1). we rewrite it in the form ¢/
= /l1¢,1J
-- NII4 1//'+l g .
g' :::: Il.l¢/'g I
2 Non-stationary averaging "amplitude-wave front position"
> O.
cI
(36) It is clear that the dependence in (34) corresponds to the energy conservation i.e .. as I -+ 00 the self-similar solutions are law (d/dt)(¢/(I)cpN(I)):::::: 0 (see close in a certain sense to the self-similar solutions of the nonlinear heat equation without a source term,
2.11
(43)
(44)
and then pass to the single equation (45)
*2 Asymptotic behaviour of unboundcd solutions
IV Nonlinear equation with a sourcc
212
213
which describes the evolution of the amplitude of the thermal structure as a function its front position. Here ;..t = vl/(NI}4) is a constant. Equation (45) is much simpler than the one obtained using the "amplitudehalf-width" averaging. Let us write down its general solution and discuss its main properties. Let us note that (45) is not applicable for the S-regime, when f3 = IT+ I. From the analysis of self-similar solutions (see I) it is known that the asymptotics close to the front in this case must have the form 1I ~ (g(1) - IXI)~", that is, in this case we should put «(/")' (I) = 0, 1'4 = O. As a result, we obtain from (42) g = const, while for f3 = IT + I (43) becomes the equation
*
= 1'1 1//,+'1(1),
,V(I)
I>
O.
o
From that we derive the self-similar dependence of the amplitude of the localized unbounded solution on time: (46)
For the HS- U' < IT + I) and LS- (f3 > IT + I) regimes equation (45) makes sense. In the case f3 ::f IT + I .+ 21N its general solution has the form
.'1
Fig. 49. Evolution of trajcctories of cquation (45) in thc HS-regilllc (f3 <
Co ~ 0 is a constant determined by the initial values gil. 1/111. In the case IT + I +. 21 N we have, instead, thc cxpression
1/12/N=lg2(Co-2;..tlng)]
I,
CII=const>O.
f3
=
'/lo(ljJ) IT
-+
I.
f3
'> IT
= ;..t1/liJ1"tll!g"/If!·I'"
+ I -+
! (!) I~ ! _ 'IS
11 -
-
II
I" I I) I
g .. "I/!·
III
I " I 1II
:=
/3 I"
j
1I
C' ~((i'i I+"/NI - II
•
that is, g(l) ~ g. as I ~ Til' This implies heat localization in the domain Ixi < g,. In this casc the amplitude grows according to the self-similar rule (51 )
21 N equation (45) has a special trajectory, the .._ II/ I{3
I)
(49)
Schematic behaviour of trajectories of equation (45) in the cases f3 < IT + I, CT + I < f3 < (T + I .+ 21N, f3 > IT + I + 21N is shown in Figures 49-51, respectively, The dashcd lines in all thesc ligures denotc thc non-trivial nullcline:
For f3 < separatrix
I
g ..
I
+-
and here g(l) ~ ex; as I -+ Til' that is, there is no heat localization in if f3 < IT+ I. For IT + I < f3 < IT + I + 21N (see Figure 50) each trajectory has its own vertical asymptote with coordinate
(47)
where
IT
(50)
which corresponds to Co ~-: 0 in (47). For IT + I < f3 < IT -+ I + liN wc have from (48) /0 SO, and therefore there is no separatrix. Let us indicate the main propertics of the solutions. For f3 < IT + I (sec Figure 49) all the trajectories evolve to the separatrix (50), which determines the asymptotic sclf-similar regimc
Thc same conclusions are true also for f3 = IT + I + 21 Nand f3 > IT + I + 21N, except that in the first case the expression for the wave pcnetration depth has the form g. = expIC II /(2;..t»), which follows from (49). In the case f3 > IT + I + 21N (sec Figure 51) therc exists the separatrix (50) in the phase plane. It separates unbounded trajectories from the family of global solutions. It follows from (47) that the latter evolve according to .1. f''lig ','I. 11(",)::::::
!
f''II = C' II /II
1/1" II
/11 '> () .
g -}
00.
From (43), (44) wc then obtain thc following asymptotic bounds for the global solutions:
which are typical or self-similar solutions or cquation (37) without a source term, which have finite cnergy: 1!1I(1, ,)II,IIR'1 == const.
where ?(t) = (1' - t)11J-1"+IIj/I~IIJ-III. A. a. l' are positive constants. The function blows up as I --> T- in a self-similar fashion. Let us find under what conditions this function will be an unbounded subsolution of equation (I). From Theorem 3. Ch. I. it follows that for this it is sufficient for II. to satisfy everywhere in (0. T) x R" apan from on the degeneracy surface (0. T) x 1lxl = a?(t)}. the inequality
II-
---'/'0
n Fig. 50. LS-regime. f3
E (IT
+ I. IT + I + 21N)
A(IL)
==
(IL)t -
":'1 (/,' 1(11)"(11.),) r -
(11 .. )#
:s O.
which reduces after simplifications to _I_(I:N.I{J"{J')' _ ~N-I!' _.
{3 - (u-+:~(J' I: 2({3-I)'!'
__ 1_ 0
(3-1
+ {J~
2: O.
j:
E
"
(O,a),
(5)
where (.)' = (d/d~)(·). The left-hand side of the resulting inequality contains the operator of the self-similar equation (1.5), which is not surprising, since II- and liS have the same spatio-temporal structure. Substituting here the function (J __ (~) of (4), we obtain after relatively simple computations the inequality
(6)
...
9
Fig. 51. LS-rcgimc. f3 >
IT
which is equivalent to
+ I + 21N
4A"
m=-,, +
+ 1 + 21N
(T~W
==
lit -
\7. (11"\711)
_lI fi
= 0,
, > O. x ERN.
= (I
{3 - «T + .I)' ({3 - I )(T
_.
(I)
'
I [ 1+2
(T
E (0. I].
(A") -, (N+':''))] . a(T
Let us determine the restrictions
on A, a. First of all, we must have the inequality
from which we obtain the restriction
11/
> O. that is
">
u+I-{3 {3 - I
4 A" The main approach is to construct and analyze suitable sub- and supersolutions of equation (I ).
e/a~)1
1/=-
Inequality (6) must be satisfied for all ~
*
In this section we justify some of the qualitative derivations. obtained in 2 and deal with conditions of global solvability and insolvability of the Cauchy problem for equations with power type nonlinearities, A(u)
Here ~
§ 3 Conditions for finite time blow-up. Globally existing
solutions for f3 > u
(5).
5Here. as usual. Cn+ "" max{O. fl·
(7)
216
Secondly, it is easy to see that for m > 0 inequality (6) holds for all ~ E (0, ~*), where ~. = min E (0, I), Hence it follows that (6) will be satisfied for all ~ E (0, I] if m .- n~ + A13 1~~fJh' . J II" ;:: 0, ~ E (~*, I), which is equivalent to the condition AfJ·
2 Non-existence of global solutions for 1 < Theorem 2.
LeI
E (I, IT + 1 + 21 N),
{3
fJl
A ;::
(
+ ';;
'(12
IT
u()(x»,·,.u(O,x)=T I/WI10.
At I -
eI
a2)
It'
and
( 1'113
1.11
It'+ 111/12113 I) I
)
(9)
, .tERN ,
1', a, A are posilive constanls, Ihe
IH'O
lasl
1',
An elementary analysis of the subsolution () following result.
Corollary.
LeI
I < {3 <
IT
+ I,
uo(x)
1= 0,
for {3 <
IT
+
I leads to the
Then Ihe sollllion of Ihe problem is
unbounded.
Prol~f. Since
V'. (1/'V'v), I > O. x E R N
(10)
*3, Ch. I)
.
"
vs(l . .\)
=:
(71
+ n -N/INa 1·21 .j' (1]),
_
TJ -
(1'1
Ixl + nI/INt'-f2I'
UIl1=0,
there exists a ball Ix E RNllx - xill < pl, P > 0, in which uo(x) ;:: E > O. Then, choosing in (3). (4), l' so large that the inequalities AT"I/W·II < E, aTlfJlajlll/121fJ,III < P hold, we have that uoLt) ;:: u (0, x~ xo), Therefore by Theorelll 1 the solution is unhounded and blows up at To ::: 1'., where
where j
,
2
(TJ)
= B(TJ(J
211/f
-
TJ)j
=
,B
[
]I//r
IT 2(NIT + 2)
o
(12)
Here 1'1, TJo are arbitrary positive constants, Let us show that in the case {3 < + I + 21N for any Uo 1= 0, after a finite time the solution u(l, x) of problem (I), (2) would have to satisfy condition (9) of Theorem I, and thus is unbounded. The stage at which the initial perturbation spreads, when the amplitude of the spatial profile decreases, will he described using the solution (II) of equation (10), in which production of energy due to combustion is not taken into account (for most of the spreading stage it is negligible). Without loss of generality, let lIo(O) > 0 and uoLr) ;:: E > 0 in a ball Ix E RNllxl < 81, Let us choose the number TJo = TJII(7'I), such that lIoer) ;:: vs(O, x) in RN , For this it is sufficient that IT
2It'7' N/IN,,121 < B TJo IE,
TJII
TI/INt"cl <
I
-
8
(13)
(here 1'1 can be arbitrary). Then by the comparison theorem (see Ch. I) lI(I, x) ;::
1.'\(1, x),
I
([4)
> 0, x ERN.
Let us show that for I < {3 < IT + I + 21 N there exists I I, such that for some T I the function vs(l I, x) satisfies condition (l). Then by (14) this condition will also N hold for the solution 11(1 I. x). The inequality vs(l I, x) ;:: II (0, x) in R will hold if
+ II) N/IN"+2IBTJ~/" ;:: l' Il\Ij.IIA, ' + I )I/IN,,!2) "> l'I/JltnIH/121/J III TJo (7 I I :.... a (1'1
A stronger result will be obtained below,
(II)
IT a-
ones being relaled hy (7), (8). Then Ihe solulion oj" Ihe Cauchy prohlel1l (I), (2) is unbounded and exiSIS al nlOsI j(n lime
V, !=:
(8)
(I
LeI
=:
Then Ihe sollllion I~j" Ihe Cauchy
In the N ,dimensional case this solution has the form (see
The system of inequalities (7), (8) has a solution (a, A) for all IT > 0, {3 > 1. Indeed, for {3 < IT + I condition (7) imposes a restriction on the ratio A" I a 2 . Then, by increasing A and {/ in such a way that the ratio N' I a 2 does not decrease, we can always cause (8) to be satisfied. If {3 ;:: IT+ I, everything is much easier, since then (7) is not taken into account. Therefore we have established the following assertion.
where II (~)
+ 1 + 2/N
}lfJ+""I)la
'~_=ic.: +~ + ~ At: {3'o, I
Theorem 1.
U
proceeds by comparing the solution u(l, x) with a known self-similar solution of the Cauchy problem for the equation without a source,
{I + 2(N + 3.) ~\;
-)
{3- I
1= O.
<
prohlem (I), (2) is unbounded.
Therefore the second inequality, which, together with (7), guarantees that (5) holds, has the form
l I N A"
uo(x)
f1
Proof It
(n - m)~:(f:i+"'I)/",
1 ;::
217
§ 3 Conditions for finitc limc blow,up
IV Nonlinear equation with a source
(here A, II is an arbitrary solution of the inequalities
Let us show that the system (15), (16) is always solvable with respect to if {3 < (T + I + 21N. Suppose that equality is attained in (15), that is
II,
T
219
Conditions for finite time blow-up
Theorem I (if 110 E V. then 1I(t, x) is unbounded). The set "V contains not only resonant initial perturbations, solutions through which start growing immediately and blow up in finite time (these are displayed in Theorem I). but also 110(.\"), to which there correspond unbounded solutions with initially decreasing amplitude.
( 17)
Here T I is fixed and T is sufficiently large. It remains to check, whether for sufficiently large T inequality (16) is satisfied. It then has the form
This will be obtained by constructing bounded supersolutions u+. which. as in subsection 2. are sought in the self-similar form
or, which amounts to the same, T.",h;If3·urll ~ l/NII <
~
- a
It is clear that in the case {3 the prooL
E
(I.
3 Conditions of global solvability of the Cauchy problem for (3>(T+l+2/N
(T
(!!.) . A
+ I + 21N)
I/N
T/2/N,Tt I.
(18)
0
it holds for large T, which concludes 0
Remark. In the course of the proof we have in fact showed that for I < {3 < (T + I + 21N the blOW-Up time is composed of two parts: To ::: II + T, where II is the time of spreading out of the initial perturbation practically with almost no energy production. T is the time of rapid growth of the resonant solution towards finite time blow-up, which was determined in Theorem I. For {3 > (T + I + 21N the spreading out stage can take infinitely long time, so that non-trivial global solutions are possible. Justification of this conclusion is the subject matter of subsection 3. Let us use inequalities (13), (18) to analyze the case of the "critical" value {3 = (T + I + 21 N . Corollary. Let f3 = (T + I + 21N alld leI Ihe illilial.limclioll iJe such thar ill {lx[ < 81. f > 0.8 > O. whl'l'e
where O+t§) == At 1- fl( 2)1,!"; A. T. a> 0 are constants. The choice of the function (20) is suggested by the form of the global selfsimilar solution of equation (I), which is considered in subsection 4. Taking into consideration the fact that II., (t. xl has a continuous derivative 'VlI~+ I. we concliude that (20) will be a supersolution if A(u+) :::: 0 in R+ x R N \{§ = aj, which gives us the inequality I (3-((T+I) I fJ --ttV 10" 0' )' + ------0' ~ + ---OH + 0 . < O. ~ 7'= a I;N-I++ 2({3-1) 1{3-1 I ~-
(compare with (5)). which is equivalent to the inequality
where
110 ( x) :::: f
ii == m, ==
(19)
\vhl'l'e a, A salisI.'''' Ihe illeqllalily (8).
Theil Ihe solulion or prohlelll (1). (2) is
I/Ilh/Jl/nded.
Since the product f8 N characterizes the amount of energy of the initial perturbation. condition (19) means that for {3 = IT + I + 2/ N unbounded solutions are all solutions with sufficiently large energy. Let us recall that the qualitative results of 2 indicate lhat in this case all non-trivial solutions are unbounded. This can be proved; see Remarks. This is also attested to by the analogy with the results of 7 concerning semilincar ((T == 0) equations. To conclude this subsection. let us note that in the case f3 > (r+ I +2IN, analysis of inequalities (15), (16) allows us to enlarge substantially the IInstaiJle sel '11' of
(21)
(I -
eI a
2
4.':~ _IJ.- (a~!l.II, = ~
(T-W
(T(f3 -- I)
(T
)+ .
[1_ 2~~ (N+ 7:..)]. a'
(T
Since the function F"fJ is convex (F:~r3 :::: 0). to satisfy (22) it is sufficient to have F"j3(() ::: 0 and F"IJ( I) ::: n. From that we obtain the required restrictions on the numbers A, a: m, ::: o. Ill, + II, + Af3"! ::: O. or 4 A"
f3 - (cT
+
I)
---.:; ._, < --'-_..'~--. a- a(T(f3 - I)
*
J I
AI-
*
III
2N A" 1 < --- ---IT a 2 {3 - I
(23)
(24)
Let us show that the system of inequalities (23), (24) has a solution only the case {3 > IT + I + 21N (for f3 < (r + I + 21N by Theorem 2 there
~
IV Nonlinear equation with a source
220
cannot be any solutions). From (24) it follows the necessity of the restriction 2NA"/(ITa 2 ) > 1/({3-I), which, combined with (23), gives us 2
4 A"
~
{3 _. (IT+- I)
...
IT({3 - I)
221
:1 Conditions for finite time blOW-Up
4 Global self-similar solutions for {3 >
(T
+ 1 + 2/N.
A lemma concerning
stationary solutions
(25)
This subsection is wholly devoted to the study of one particular case of global solutions of equation (I) of the form
The quantity AIT la 2 , which satisfies (25), exists if 2/INIT({3- 1)1 < [{3 - (IT + I) ]/[IT({3 - I) I, which is equivalent to the inequality [{3 - (IT + 1 + 21 N) III IT({3 I) I > O. Hence arises the restriction {3 > IT + I + 21 N. Then, varying the numbers A and a so that the ratio A" I 0 2 remains constant and within the bounds of (25 l, we can ensure that (24) always holds by decreasing A. Thus we have proved
(28)
----<--" 2
NIT({3 - I l
Theorem 3. Let (3 > IT T > 0 the inequalily
IT 2 0
+ I + 21 N,
uo(.\.).< ::: u+ (0') ,.\ --
.
and leI the .lilllction unCI) satisf" jin' some
u(l,x):::: (1
+1)
= .1'/(1' + 1)/II. m =
(IT
+ I )]/l2({3 -
where l' > 0 is an arbitrary constant. After substitution of (28) into (I), we obtain for equation:
Is :::
1)1·
0 the following elliptic
I
7'1/1/3-110
+-
(
TlfJ
1.1'1
ltf + 11I/121;3··111
)
.
x ERN,
1/lfJ II.
0-1
(
1.1'1
(T+l)lfJIITtIII/12[fJ111
) . R III
+ X
RN
.
-
(27)
+ n,·I/l/!
II, I>
N ,
(29)
f.\W
0,
->
I§I
-> 00.
which differs from the equation which cOlTesponds to self-similar blOW-Up regimes only in the signs of the second and the third terms. This, however, significantly alters the properties of the solution Is as compared with Os of I. At this stage we confine ourselves to the study of radially symmetric solutions
*
= f.d§)
= 1.1'1/(1' + 1)/1/ E R , .
::: 0, §
(30)
which. as follows from (29). satisfy the equation
Remark 1. From this follows, in particular, the estimate supu(t, x) :::: A(T
?;ER
(76)
Is
O.
I
~N1(§
\
Furthermore, using (27) we can estimate the diameter d(t) of the support of a generalized solution: d(t) :::: 2a(T + I)IIJ' IIT11 1I/121/3-I)I. Naturally, these estimates coincide with the self-similar ones. Combining the results of Theorems I, 3, we arrive at the following statement: for (3 > IT + I + 21N jiJr all large illitial functiollS the prohlem (I), (2) is glol}(/IIY insolvahle, while for slIjjiciently small Uo there exists a glohal solulioll. Remark 2. For (3 > IT + I + 21N, we distinguished in Theorem 3 the stable set 'W' of the Cauchy problem (I), (2), such that the inclusion lin EW entails global solvability of the problem. The set 'W = lun ::: 0 I 3 l' > 0: un(.\):::: (0, x) in RN } contains only functions with compact support, and its "boundary" consists of a one-parameter family (with parameter l' > 0) of also compactly supported functions. This does not mean that only compactly supported solutions can be global. In subsection 4 we construct a non-compactly supported stable set in the case f3 > IT + I + 21N, the boundary of which consists of non-compactly supported global self-similar solutions of equation (I).
"+
1{3 -
V~.(r~Vds)+mVds·§+ (3_lfs+I~=O,
where 0 1 (0 = A(l - r;z /( 2 ) I/IT alld Ihe COlIslanls A, a > 0 salisfy illequalilies (23), (24). Then the Cauchy prohlem (I), (2) has a glohal sollllioll alld "'
§
N
I 'IT .,
,
.,
.
I
·IJ
Ids) +mls§+/f'~-ils+ls=0, §>O,
(31 )
and the boundary conditions (,(0)
= O. Is(OO) = 0
(32)
(fs(O) > 0).
The generalil.ed solution fs must have continuous heat flux, that is, if Is is a function with compact support. then §o
(F~")' (~o) = 0
at the point of degeneracy
= meas
supp Is. Let us consider a family of Cauchy problems for the same equation:
_'_ (t;N
~N - I '
-I
f'" f")'
.'
+ mi"/' + ._1f + f'/i = n, . f3 - I . .
(0) =
~
o. flO)
= /-L > 0,
?; > 0,
(33)
(34)
and choose /-L so that I = I(§; /-L) satisfies (32). Before stating the main theorem, let us note two properties of the solution Is which follow directly from the form of equation (31).
+ I + 21N. Indeed, there were the function fs ::: 0 to exist, (28) would have been the global solution Us t= 0, which, as we showed above, does not exist. Let us note a peculiarity of this argument. Here, in order to study an ordinary differential equation, we usc results from an analysis of much more complex partial differential equations. Advantages of this approach in this case are not significant, since (a) admits another, simple proof. However, in the sequel (in the proof of (C» this approach leads to a noticeable simplification. The same result can be obtained by a different method. By (35) for f3 < IT + I + 21N equation (31) can be integrated over (0. (0) with the weight function ~N 1 As a result. after integration by parts we obtain the equality
(T
First of all, Is(~) is monotone, since (31) does not admit points of minimum ~ = ~III' such that IS(~III) > 0, tlj(~III) = 0, n(~III) ::: O. Therefore for all !J. > 0 any monotone classical solution of the problem (33), (34), defined for small ~ ::: o. can be extended either onto the whole axis ~ E R+ (then it is the required solution Is), or till it becomes zero. Local solvability of (33), (34) is demonstrated by analyzing the equivalent integral equation using the Banach contraction mapping theorem. Secondly, analysis of (31 ) for small Is reveals the possible forms of asymptotic behaviour of the solution as Is -> O. First is the asymptotics of a non-compactly supported solution:
I
C'-.
I .f-li( \ 1] ) 1] ,V··I 11]= -
.0
where C > 0 is a constant. Second is the asymptotics of a solution with compact support
thell the prohlem (31), (32) hilS 110 positi"e solwiolls lilly !J. > 0 the .fimctioll f(~;!J.) IJecomes ::.ero at .10/111' poillt ~ = ~* IIlId <
u"tl)~(~,;!J.) (b) for all
IT
#
2.)
+ I + - - f3 IC'-.
Is(1])1]'v ,··1 d1].
(37)
N.o
which for f3 < IT + I + 21 N cannot be satisfied, as in the right hand-side we have a negative quantity (to derive (37) we need an estimate of tlj(~) as ~ --> 00. which is easily obtained from the equation). (b), (c). Proofs of these assertions are based on the properties of a family of stationary solutions of the original equation (I), which are established below.
00,
where wen -> 0 as ~ -> ~ij. The asymptotics (35), (36) make sense for f3 > (r+ I. Let us note that a solution with compact support (36) formally corresponds to the value C = 0 in (35), that is, (35) becomes (36) for C = O. Properties of various solutions of the problem (31). (32) depend on the relation among the parameters f3, IT and the dimension of the space N.
Theorem 4. Let
N ( IT 2(f3-1)
0):
f3 > (T + I + 21N, if N = 1,2. or .lill' (T + I + 21N < f3 < (IT + I)(N + 2)/(N - 2), if N ::: 3. the problem (31), (32) hilS IItlellst 0111' solwioll Is with compllct support alld 1111 illfillite IlLImber (If strictly positive solutiolls: (c) ({ f3 ::: «(T + I)(N + 2)/(N - 2). N ::: 3. thell the problem (31), (32) hilS 110 solutiolls with CO/llPIICt support. For lilly !J. > 0 the solutioll I~r the Clluch\' prohle/ll (33), (34) is strictlv positil'e alld sati4ies cOllditioll (32) at i/(fillity. '
*
By analogy with results obtained for the case IT = 0 (see 7), we can expect that for f3 = IT + I + 21N all nOll-trivial solutions of the problem (I), (2) are unbounded, and therefore a function II ::: 0 does not exist in this case. This is true for IT > 0, see Remarks. prol~r Assertion (a) follows immediately from Theorem 2 concerning nonexistence of non-trivial global solutions of the problem (I). (2) for I < f3 <
I A lemma cOllcemillg statiollary solutiolls
Let us consider the stationary equation
+ Vf3 = 0
\7 . W"\7V)
(38)
for arbitrary values of parameters IT ::: 0, f3 > O. For our purposes it suffices to analyze the family (V ::: 01 of radially symmetric solutions, which satisfy the equation (38') Let us set V" + 1 = V and make the change or variable denote by V A the solution of the following problem: I
v
_ _ (1',1 V~)'
rN,·1
VA(O)
+ V',\'
= A.
= O.
V~«()
I'
= Ixl
I' ._."
>
1'( (r
+
I) In. Let us
O.
= O.
(39) (40)
where A > 0 is a constant (which parametrizes the family (V AI), a :::: f3/(IT
+ I)
> 0,
Lemma 1. Let a > O. Theil: I) For lIIl," a > O. ({ N = 1,2. or 0 < a < (N + 2)/(N - 2). i{ N ::: 3. the problem (39), (40) Illis no solutillllS V A ::: 0 ill R., (that is. .!ill' llll." A > 0 the .filllctioll V A becomes ::.ero at sOllie .finite point. where 1 # 0):
V:
~
IV Nonlinear equation with a source
224
2) if cy :::: (N
+ 2)/(N
225
3 Conditions for finite time blow-up
- 2) .li)r N :::: 3. then for any A > 0 the solutiO/l of the R t-. VA(r) -'> 0 as I' -'> 00.
proiJIem is dl~fi/led and strictly positil'c in
Local solvability of the problem is proved by reducing (39), (40) to an equivalent integral equation. I) Let us first consider the case N ~ 2. From (39) we have
Proof
and therefore (if we assume that V A > 0 everywhere), for each
!
bi
o
I' ::::
r
Fig. 52. Solutions of the problem (39). (40) for different A > 0 in the case a < I.
Hence VA(r)
~
VA( I) - ('\
r Tl
I N - d7]. I'
./1
from which we cannot draw a conclusion concerning extension of V A(r) > 0 into the domain of large r. From here it follows that > I.
VA(r) <
that is, VA(r) ~ V,I(I) - ('\(1' YAir)
~ VA(I)·-
I)
('llnr
= I). (N = 2);
(N
I'
> I.
rN
'1
T}N.I dT} .0 /
II. I'
(43')
> O.
Let us assume that I) does not hold. and that V A > 0 in R t for some a < + 2)/(N - 2). N :::: 3. Then, using the identities we derive below, we shall N 1 arrive at a contradiction. To derive the first of these, we multiply (39) by r - V A and integrate the resulting equality over the interval (0,1'). As a result we have (N
This means that for N ~ 2 V A becomes zero at a point; furthermore. by that point V~(r) =F 0 (that is, the heat flux cannot be continuous). Now let N :::: 3. Then by monotonicity of VA we have from (41) rN .. IV~(r) < -V~(r)
IN )1/\,,-11 2 ---r- /(". ( a-I
= -V~(r)-.
(41)
at
-rN1V:\(r)VA(r)+
(42)
N
Since
V~(r) <
r
./0
7]N
IV~2(7])dT}=
7]N'IV~II(7])d7].
1'>0.
0 and (by assumption) VA > O. we have from here
ri.T}N-IV~2(7])d7] ~
Integrating this inequality we derive the following estimate: for a <
r
./0
./0
{'"
./0
T}NIV~II(7])dT} <
(44)
00,
where convergence of the integral in the right-hand side is ensured by the estimate (43).
and therefore VA (I') is defined on an interval of length not exceeding
I' A
= 12N/ ( I -
Now let us multiply (39) by a different equality:
rNV:I(r)
and integrate over (0, rJ, This results in
a)II/2A(I""1/2.
Let us note that for a < I the function VA depends in a monotone way on the boundary value A = VA(0) (Figure 52), I h f ' V' .. I' I ((l '::'N' (II) . III () ' W Ilere::' N > Por a = t e unctIon A IS pOSItIve on t le mterva is the first root of the Bessel function J IN-21j2· If, on the other hand, tY > I. then from (42) follows the "non-compact" estimate (43)
N J -V' '(I') ,\ "2
=
--.!'l_ tt
"N I + a__ (I') = + I \1"+ A
r
+ 1./0
T}N-I
V~ II (7]) tiT}
-
N
:.;1 -
r
(45) 7]N
1
I
2
V A (7]) dT},
I'
> 0,
./0
It is not hard to see that p(r) == rNV~ 2(1')/2 + rNV~11 (r)/(a + I) -'> 0 as I' ~ 00, Indeed. from (45). since the integrals converge, it follows that p(r) ~ Po as I' ~ 00, while from convergence of the integral 1;;" Ip(T})/T}1 dT}. which follows
which, combined with (44), gives us the inequality (N - 2)/2:::: N/(a + I). i.e., a:::: (N + 2)/(N - 2)+, which leads to a contradiction. 2) Assume the opposite. that is, that there exists a :::: (N + 2)/(N - 2)+ (the case of the cquality will be considered separately), such that for somc A > 0 the solution V A vanishcs at sOllie point r,\ > O. Thcn V A =1= 0 is a solution of the boundary valuc problcm (39) on the interval (0. rA), satisfying the boundary conditions
Fig. 53. Functions VA(r) for I <
a <
(N
+ 2)/(N
- 2)+
(46)
However, as we shall show, thc problem (39), (46) has no solutions. For that. N 1 as in the proof of I). we first take the scalar product of (39) with N KNr - V A (r) and then with NKNrNV~(r), whcre KN is thc volume of the unit sphere in RN As a result, aftcr integrating by parts and taking into account the conditions (46). we obtain liV~II;!::::: liVAII~',;.\ (L"::::: L"({lxl < rA})). (47)
V/.
Substituting into (48) the cxprcssion for II V~ II;., from (47), we obtain the cquality
t·
Fig, 54. Functions VA(r) for a .:: (N
+ 2)/(N
-- 2)+
(49)
which, of course, cannot be satisfied for (l' > (N + 2) I (N - 2)+. Finally, for the critical value a = (N + 2)/(N - 2) f the problem (39), (40) has for all A > 0 the positive solution (50)
Insolvability of the boundary valuc problem in this case also follows from (49), since here V:\(r.1l i- 0 by (41). 0
Remark. Returning to equation (3X') we obtain that for all f3 > 0, N = 1.2 or for 0 < f3 < «(r + I)(N + 2)/(N - 2), N :::: 3, there are no stationary solutions in R~. On the other hand, for (3 :::: «(T + l)(N + 2)/(N - 2), all its solutions are strictly positivc.
In Figures 53, 54 we sketch the behaviour of the functions V A(r) for different A > 0 in thc compactly supported (Figurc 53) and non-compactly supported (Figure 54) cases. For a > 1 (unlike a < I) there is no monotone dependence of VA(r) on A if a < I + 4/(N - 4 - 2JN=1), N :::: II [227.378]. In conclusion. let us note that a statement similar to the one proved above, is valid in the case of equation (39) with a fairly general nonlinear term q(V) in the place of V" (see I. eh. VII).
*
2 Proof' or (lssertion (h) or Theorem 4
Let us first establish a simple claim, which is relevant to assertions (b), (c). Lemma 2. LeI (3 >
(T
+ I + 21N.
Then for (1/1 (51 )
*3 Conditions for finite time blow-up
IV Nonlinear equation with a source
228
229
f(~;p)
the solution ()lproblem (33), (34) is strictly positive ill R.j (and, consequentlY, has the asymptotic behaviour (35)). Pro(}j: Let us rewrite the equality obtained by integrating equation (33) multiplied
by
tN. I over the interval (0. tJ. as follows:
(52)
By (51) and monotonicity of f the right-hand side is strictly positive for ~ > 0, Let us assume that f(Y]) vanishes at ~ =~. E R I , Then r'nt.) ::: 0, f(f,) = o. so that the left-hand side of (52) is non-positive for t = t .. which leads to a contradiction. 0 Thus we have established the second part of assertion (b), Its proof is completed by appealing to the following lemma,
f3 > (T + I + 21N it' N = I. 2, or (T + I + 21N < f3 < (cT + I)(N + 2)/(N - 2) ir N ::: 3, 711ell there exists JL > 0, such that the sollition or the problem (33), (34) becol//cs zero. Lemma 3. Lct
(IT
+
I)(N
+ 2)/(N
N
JL > {
Let us first note that strict positivity of all radially symmetric solutions of the stationary equation (38) for f3 ~ «(T+ I )(N +2)/(N -2), N ::: 3 (Lemma 4), indicates. in principle. that assertion (c) is true. A proof which proceeds by analyzing the ordinary differential equation (31) encounters various difficulties. Therefore our
I(O). the case
IT
+ I + 21N
<
[(
2(f3--=--1) f3 -
2 )] } I/I/!
(T
+ I + IV
II
Then, as in the proof of (b), we conclude that there exists a non-trivial solution f,(~) ::: 0 of the problem (31 J. (32). with compact support and having the asymptotic behaviour (36), As we show below, this conclusion leads to a contradiction. We established that for f3 ::: «(T + I)(N + 2)/(N - 2).\_ the Cauchy problem for equation (I) has the self-similar solution u=lIs(t.x)=(T+t)
I/II! 110 _ (
.\
Ixl
)
(T+t)liJ- 1".1111/12(JJ-liI·
'i3 C.)
which has finite energy
.
l:(I)
3 Proof' ot' assertioll (e) ot'Thcorcl/1 4
=;
- 2)1
proof will be based on a curious property of solutions of the corresponding partial differential equation (I). Thus, let us assume the opposite: that in the conditions of (c) there exists JL > 0, such that I(~: JL) vanishes at some point. Observe that by Lemma 2
*
Proof of the lemma follows the lines of the proof of Theorem 4 in I (see also the analysis of the case f3 < (T + I in subsection 3, I). In the final count, the assumption contrary to Lemma 3, that is, that for all JL > 0 the solution of problem (33), (34) is strictly positive in R I , leads, after passing to the limit. to a conclusion that for A = I there exists a strictly positive solution of the stationary problem (39), (40), which is impossible by Lemma I, Next, denoting by Atf. the set of all JL o > O. such that I(~: JLl > 0 in R+ for all 0 < JL < JL o, we have that ,AA :f. VJ (see Lemma 2) and that AA is bounded from above (see Lemma 3). Therefore there exists JL, = sup ,M < 00, and, using standard methods, we can show that the function Is = I(t: JL,), which corresponds to JL = JL" is a solution of the problem (31), (32) and has the asymptotics (36), In Figure 55 we show curves I(t:JL) for different values of JL > n. The thick line shows the compactly supported solution,
*
Fig. 55. Solutions of the problem (33). (34) for different fL (3 <
I "t I 2 + I II 1#1" II = ::;----I-IIV'II II t.'IR'1 - -f3 L«(T + ) +-(T-+- I u!tJ/'''''iR''I' . (T
.rl·
'i4
(.
)
Indeed, Is has compact support, and tf E C(R+): it is not hard to check that V'1I"tl(l) E [}(RNl. 1I(t) E l/!l"tl(R N ). so that \£(t)1 < 00 for all t > O. Energy functionals of the form (54) are an important attribute of solutions of the problem. and we shall frequently use them in the following (see 2, Ch. VII).
lime. Proof Under the above assumptions we can take (J(I) > 0.1111",1(1)111.' =j.:
For convenience, let us introduce the functional
II VII IT + I (1)111.'
0, and <: 0,
=j.: 0 for all I > O. Then by (55), (56) we have (J'(I) > O. E'(I)
and, furthermore, the inequality (57) is strict: _, «(T (J (I) > -
Then
=
(J' (I)
«(T
+ 2) . (11.'" I~
= -(iT + 2) Since (3 >
(J
II" 1 III,
(3+(T+ (T + I
+ I,
=
dx
I[I (3 +
iT
+ 2) /.'
«(T
. 11."
+ I II VII
lilT
+I
(_I_LiIl",1 + II fJ ) iT
+I
(T+ (T
(1112
1I /.'III.,v) - (3 +
I
dx
=
+ 2)((3 + (T
=(T
+ 1)<' ',,1#
X { 2«(T
4«(T
+I
(T
+ I) E'(1),1>. ()
(57)
I
+ I)
1"11
IIV
/"",.1
(. S
Ix I:
(d
112
U -
(T
(3 +
+ (T
(58) I
11/'
+ I .
S
(l:
~)
11{i+ff+l} 0"""
.
Let us show that for (3 > «(T + I)(N + 2)/(N - 2), N ;:,: 3, a global solution cannot have such energy. For the critical case (3 = «(T + I )( N + 2) / (N - 2) this is obvious. From (58) it follows that E(I) = const, that is, E'(I) == 0, which contradicts (55), as II, ~ O. Assume now that (3 > «(T + I)(N + 2)/(N - 2)'1,' Then from (58) we have that for the energy E to be strictly decreasing (see (55)), we must have E(O) :s O. Then E(f) <: 0 for all I > 0 (if 1:"(0) > 0, then by (58) E'(I) > O. which contradicts (55)).
-(J(I)E(I)
+ I)
> - - - - , (II «(T + 2)-
All the above transformations arc justified for the function (53). Let us see now what energy corresponds to a global solution (53). It is easily computed that for it £:'(1)
,
/H'l!-l]
rr+1
1+'1/2
= 4«(T+1) IIII I_ «(T + 2)2
. (II
" (iT ( J(I);:,:-
(T
+ I).
.--£(1). I >
(60)
O.
Using the Cauchy-Schwarz inequality, we derive from (55), (56), following estimate:
it follows that the function GU) and that there exists
==
1I111+(f/2U)117.~ cannot be bounded for aliI> 0,
Therefore the solution 11((. sup
lI(I.
x)
x)
s:
233
is globally bounded: fs(O)(T
+ t)
I/I/!
I)~, 0, 1-> 00.
\ER\
o such that GU)
~
ex) as I
-+
T(I' i.e .. the solution
11(1, x)
is unbounded.
0
Therefore for (3 :::: (IT + I)(N + 2l/(N - 2)., the compactly supported solution liS in (53) cannot be a global one. so that in this case all solutions (:n). (34) are strictly positive, which concludes the proof of assertion (c) of Theorem 4. 0 This analysis establishes a certain similarity of solutions of equation (31) and of the stationary equation (38'), which is quite obvious for sufficiently large values of /-L = f(O)· There arc also significant differences between them: for small /-L > 0 lower order (linear) terms in equation (31) are of importance. A consequence of this is existence for a range of (3 of the solution fs(~) with compact support. In the case of the stationary equation (38') (see subsection 4.1) a solution with the required asymptotic behaviour exists only for the one value (3 = (IT + I)(N + 2)/(N - 2),.
5 A non-compactly supported stable set
*
+ I + 21N, I :3 fs
+ 1 + 2/ N
(T
Let us lind out the relation of the self-similar solutions constructed in subsection 4 to the asymptotic behaviour of arbitrary global solutions of tbe Cauchy problem. Is it true that the particular solutions (28) describe for large I the amplitude and spatial prof1le of decaying thermal structures, which exist for (3 > IT + I + 21N? Below we present an analysis of asymptotic stability of symmetric in x selfsimilar solutions (2X). First we shall show that the question posed above can be answered in the affirmative if f.dl~l) has the power law asymptotics (35). In particular. if we denote by f,U, {l the similarity representation of the solution of the Cauchy problem (I), (2),
Theorem 6. LeI (3 >
IT
+ I + 21N.
I/l1d lellhe sel(silllilar/illlclioll Is ill (30) he
sllch Ihal
-r'i=
2) + (3(/\'{O))' .
N . ( IT+I+--(3 2((3 - I) N
'J
. '
I
< O.
(63)
Theil IIII' sel(silllill/r sollllion (2X) is I/s\'nl/JIOlically .I'll/hie in 1. 1 (R N l ill Ihe fol· lowing seils/,: if/ill' sOllie I' > 0
Using the results of subsection 4, it is not hard to determine the stable set IV of the problem (I), (2) for (3 > IT + I + 21N, which consists of non-compactly supported functions. The boundary ofW' consists of non-compactly supported global self· similar solutions (28). Here we shall assume that the solution 11(1, x) obeys the Maximum Principle and depends monotonically on the initial function (see 3. Ch. I). Let us denote by :ij; the set of non-compactly supported solutions f.d~) of the problem (31), (32); to :9; belong, for example all f(t; /-L) of Lemma 2 (']0 #- Vl for (3 > IT + I + 21N), The set 'W is determined in the following statement. Theorem 5. LeI (3 > IT
6 Asymptotic behaviour of global solutions for
Ihell
E :'F. I' > 0
Ilf,(I,·) -
= const :
(65)
f\(·)II"i1f'l :::" OU .I) ..+ 0, 1·-> 00.
(62) 110(.1')
:.'S
Till!! III\(IxlITlfJ itr , lilI12Ii!!
Prol~j:
By the Maximum Principle the restriction provides us with the bound for the solution: 11(1, x)
s:
(I'
+ I)
IlijJ Ilfs(lxl/(T
(62)
+ 1)"'),
III)}.
on the initial function
I> 0, x
ERN.
Remark. The inequality (6X) provides the following restriction on the sIze of /s{O) = sup I\(ltIl: [( N 'dO) < { (3 - IT 2((3 - 1)(3
From Lemma 2 (see subsection 4.2) it follows that then Is(lt"I) > () in R • and therefore the theorem deals with asymptotic stability of a non-compactly supported N self-similar solution Us. Let us note that by (35) lIs(f, .: T) if. L 1(R ) for any I > O. Formally. the argument is the same as in ~ 13. eh. 11. Let: = liS - U E N for each I :::: n. By (64) U ::: US. i.e., : :::: 0 in R+ x R . From the parabolic equation for the function: it follows that
Prol)j:
L1(RN)
£I cI
-, 11:(f)III.'IR'j ::: (:(1), O(lI, 11.1». I >
O.
where
unstable as I -+ 00. In this case the asymptotic stage of combustion with extinction is described by self-similar solutions of the equation without source: N
a(II,
u,d
= 13
t
.10
(7]11.1'
+ (I
1
_7])lI)I3· £17]::: 13(/.1'(0))13. (1'
i.e .. for large times combustion is negligible compared with diffusion. A similar situation was already encountered in § 13. Ch. II. Therefore we shall not attempt an exhaustive analysis. and will study only the most interesting (and hardest where proofs are concerned) case of a compactly supported initial function I/o E LI(R N ). It will be shown that in this case the global solution of the Cauchy problem evolves as I -+ 00 according to the rules determined by the spatio-temporal structure of the self-similar solution of equation (70) (see § 3. Ch.
where l' > 0 is an arbitrary constant, gS(7]: B-x.("I) .~,
However
==
IT
V,/, ("I,V,/"I) ,~. ,~,
a) ::::
a),
(71 )
0 satisfies in R N the equatiog N~
I
+ -Nlr - -+ V2, / ".~.I " '/ + Nlr--\-2 = ()
(72)
rI
(67) and has the form and the estimate (65) follows from (66). (67): by (63) it implies the stabilization IT -+ Is as , -+ 00 in the norm of L I (R N ). 0 Theorem 6 demonstrates asymptotic stability of non-compactly supported selfsimilar solutions in the class of initial functions (sec (64')) 110(.1') ~ ClxI2/lf311T-tlli. 1.1'1 -+ 00.
(68)
lr ] [ 2(N(T + 2)
lilT
(73)
Here a > 0 is a constant. To prove the above-mentioned fact. let us introduce. corresponding to (71). the similarity representation of the solution 11(1, x) of the problem (I). (2): (74)
Thus if uo(x) satisfies (68) (then 110 if. L1(R N )). and to a given initial function there corresponds a global solution of the Cauchy problem. then the amplitude and the half-width arc estimated asymptotically exactly as , -+ 00 by
and write down the equation it satisfies with the new time iJg
sup
u(f, x)::::: ,,1/1 13 11.1.1'<,/(1)1 ~ ,lfi "",111/121{i III.
(69)
\cR'\'
What will happen if (68) is not satisfied. for example. in the case of an initial perturbation with compact support Uo ELI (RN)'I It follows from Theorem 4 that for lr-\- 1+2/N < 13 < (lr-\- I)(N -\-2)/(N -2)+ there exists a compactly supported self-similar solution Us of 'the form (28). which. it would seem. should describe the asymptotic behaviour of such solutions, However. this is not the case. Unlike the non-compactly supported solutions in Theorem 6. this self-similar solution is
-:-j IT
= BT(g) ==
T :::;
In(7' -\- I):
V,/ . (gITV"g) +
(75)
I V -\N g -\-., /'T 13 I 7' -\- Nfr+2"g'7] NlT-\-2 (' g ,T>To= n • ryE
RN
•
(76) In (75) we denoted by
II
the constant
1/ .::::
Nlf3 .... (IT -\- I
13 > IT -\- I -\- 2/N it is positive, which is important.
+ 2/N)I/(NfT -\- 2).
For
*3 Conditions for tinite time blow-up
IV Nonlinear equation with a source
236
Let us prove f1rst that in the case of a function uo(.r) with compact support the behaviour of global solutions of the problem (I), (2) as t ~ 00 "obeys" (71). First of aiL it is clear that the function (73) is a subsolution of equation (75), since agS
-'J-:=
()
<7
:s B
( T
g,1'):= e
I'T
/3 gs·
Thus, if uo(x) is a function with compact support and there exist constants T > 0, af > 0, such that
where
h == (T A{:' I (a: )1 13-1 I/IT I Il,
Therefore g(7, 1]) ?:: g.I(1]: a.) in R N for all admissible 7 > 70· [t remains to construct a similar supersolution of equation (75). Naturally, the functions gs cannot be used to that end. However, they can be easily modifled to give a supersolution. Lemma 5. Let
f3
>
(T
+ I + 21N
and (78)
where b
==
(TAr;
CXJ)
X
R N the estimate
:s g j.(7, 1])
(82) D
Combining the above results, we arrive at the following conclusion. Theorem 7. Lei f3 > (T + I + 21 N, and let the ./illletioll uo(x) sat i.I:/)' the conditions (77), (81). 71lCn, iF g(CXJ, 1]) exislS. (83) This result shows that for f3 > (T + I + 21N the evolution of global solutions of the Cauchy problem (I), (2) is described by the self-similar solutions (71) of equation (70). The estimates (83) mean, in particular. that
I
(79)
(70,
(and, actually, problem (75), (76) will have a global solution).
sup
Then the ./imction
1 (a 2 )I/J.·.II/1T Ill,
then we have in
g(7, TI)
Thus if uo(O) > 0, there exist constants T > 0, a. > 0, such that (77)
237
u(t, x) ~ t- N / IN "t.2I, Ix,'I(t)1 ~ 11/INlTt21, t ->
CXJ.
clt'\
Recall that the same conclusions were obtained earlier using the qualitative nonstationary averaging theory (see 2). As far as asymptotic stability of the self-similar solutions (71) is concerned, we shall coniine ourselves here to proving one simple assertion.
*
is a supersollllion of equation (75).
Proof We shall seek a supersolution in the form g.t == 1j;(7)gs(1]I
(80)
Theorem 8. Let N == I, f3 > (T + I + 21N := IT + 3 and /el l/o(x) have compact support. Then under the conditions of Theorem 7 we call ./i/l(/ a E Ia._ , a + j, such that g(t, 1]) -> gs(1]: a), t -> CXJ, almost el'e/l'll'hcre in It
where d == (l/2 - ( ) I E (O,1I 2 1· Let
Proof First of all. by (83) the Cauchy problem (75), (76) is equivalent to a boundary value problem in some bounded domain n ::J Ia ,al], g == 0 on R+ x an. In addition, we can derive the estimates
by taking scalar products in Setting here ,//'(7) == I supersolution if h?:: (TA[;-I
These estimates show that the w-limit set, w(Ro) = (R'(17) I 3T" -+ 00 : g'HI(T",.) -+ [R,(.)!",I in L 2 (il)}, consists of "stationary" solutions of equation (75) for T = 00, that is, Boo(R,) = D. Indeed, using the estimate (83) and (84) in passing to the limit in equation (75), we have that, given a monotone sequence T" -+ 00, g(T" + .1',') -+ h(s,') in L~J(TI, 00) x il), where h(s,') is a weak solution of the limit equation hoi' = B",,(h) for .\' > D, h(O) E w(Ro). It follows from (84) that uniformly in .I' E [0, II (Q' = I + ITI2) IIg"(T"
+ .1')
.-
g"(T,,)II;-'{(ll ::c
I .IT"~T" I.:) g"(T)\ 1!T11 'T
II,
= (II"U,), + 1/3 • 1/(0. x)
•
::c
= 1I0(.r)
where the initial perturbation connected support:
1.·11li
::cj'T"I'I\;gO(T)\\2 dT
where, applying this method, we shall solve the question of describing asymptotic spatio-temporal structure of unbounded solutions for times close to the blOW-Up time. We shall consider the Cauchy problem in the one-dimensional case,
2
1.1
T"
239
~ 4 Proof of localization of unbounded solutions
IV Nonlinear equation with a source
238
dT-+O
w(O)
I.-'{(II
as T" -+ 00, that is, h docs not depend on .I' and is a weak stationary solution. Boo(h) = O. By (83) .II' are functions with compact support, and therefore w(go) <; Lt:s(17; lI), II E [II ,Ill]}' Finally, independence of the limit function .11'(17) of the choice of the sequence T" -+ 00 follows from "monotonicity" of the solution of the problem (75), (76):
-00 <
II
>
+
O.
x E
R:
E
R;
IT>
O.
11",1 E
f3
>
I.
c' (R).
(I)
(2)
t 0 is a compactly supported function with a Rlllo(x) > O}
= (h
h.(O) < hl.(O) <
(O).h,(O)).
(3)
00.
= SUppll(t, x) = (h
(t), ht·(t)).
(4)
0 of 0
Remark. It is not hard to show by the Bernstein technique that the derivative (g"l·1 ),/ is uniformly bounded in (TO + I, 00) x R. so that stabilization g'" I(T, .) -+ g'f"'(·; lI) as T -+ 00 is uniform in R.
§ 4 Proof of localization of unbounded solutions for f3 2: IT absence of localization in the case 1 < f3 < 0' 1
E
Then, for all t > 0 for which the solution exists, the support of the generalized solution 1I{t, x) will also be bounded and connected. w(t)
(-llg(T)IIL'(lll is a Liapunov function), and also strict monotonicity in the expression Il.lis« lI)II/.'llll'
O. x
:::
1I0(X)
= SUppliO == {x
I>
+ 1;
Results of preceding sections give us quite a good overall picture of the main properties of unbounded solutions of the Cauchy problem we are considering. The aim of the present section is to prove localization of unbounded compactly supported solutions of the S- (f3 = IT + I) and LS- (f3 > IT + I) regimes, as well as absence of localization in the HS-regime (I < (3 < IT + 1). At the same time we shall obtain a number of important estimates of the size of the support of unbounded solutions. The method of proof presented here will be used in 5,
*
The functions IL.(t) and h,(t), which determine at each moment of time the position of the (left and right, respectively) front points of the generalized solution, are, respectively, non-increasing and non-decreasing, so that the length of the support meas w( I) == h, (I) .- h . . (I) is non-decreasing with time. It is not hard to show by comparison with travelling wave solutions that hl: E C([O, To))· Let I = To(l/O) < 00 be the blOW-Up time of the problem (I), (2). First of all we shall be interested in the behaviour of the functions ":f:(t) as I -+ Til' It turns out that for (3 ::: IT + I the functions het (t) are bounded on (0, To) and Ihl:(T(I)1 < 00. which. as can be easily seen. is equivalent to localization of the unbounded solution. Conversely. it will be shown that in the case f3 E (I. IT + I) the functions he(t) are unbounded. and h .. (tl .... ±oo as t -+ T(j (there is no localization) . In I we studied in detail self-similar unbounded solutions. which explicitly illustrated various interesting properties of blow-up regimes. To prove that these properties are shared by a wide class of solutions of equation (I), we shall use the method of intersection comparison of the solutions being considered with exact self-similar solutions having the same blOW-Up time. We start by presenting the main ideas of this comparison theory, which is especially suited to analyze the spatial structure of unbounded solution close to the finite blOW-Up time.
*
~
IV Nonlinear equation with a source
240
1 The number of intersections of different unbounded solutions having the same blOW-Up time (main comparison theorems) Let us observe at the outset that Proposition I below, concerning the non-increase of the number of spatial intersections of two different solutions 11(1. x) and v(l. x), is an immediate corollary of the Maximum Principle for linear parabolic equations. Certain technical difficulties. which crop up also in the definition of the intersection used here, have to do with the fact that (I) is a degenerate equation, which admits generalized solutions. Therefore we shall not present this result in its maximal generality, or in all the possible detail. We shall mainly emphasize the parts of comparison theorems that make essential use of unboundedness of solutions under consideration. Thus, let v(l, x) be a generalized solution of equation (I) with a bounded nonnegative initial function v(O. x)
::=
voCr) ::: O..\ E R: vg
We shall assume that the function
v(l. x)
fl
E
l
C (R).
is defined for [0.
To)
x R.
Definition. For a fixed to E 10. To) the interval la\. a" I c R is called an intersec(or an inlersection poil/I if a\ ::= a") of functions 11(10. x) and 1'(10. x) if the difference ll'(lo. x) := 11((0. x) v(lo. x) is such that 11'(10. x) = 0 for all x E lal' a:d: for any sufficiently small <5 > 0 the function w(lo. xl does not have c'5. a" + c'5] and 11'(10. x) ji 0 for all x E (al - <5. a\) and the same sign in [a\ x E (a:>, a) + <5). 11", on the other hand, w(lo. x) ::= 0 on lal. (I21. w(lo. x) has constant sign on la\ .... <5. a2 + 6[ for any sufficiently small 6 > 0 and 111(10. x) ji 0 for all x E «(/1 - 6,a\) and x E (a". a:> + 6), then we call1a\,a21 a /(II/gcnc.'" inlerval (or poilll if al ::= a:» of the functions 11(10, xl and v(lo. x). We shall denote the number of spatial intersections in R (of intervals or points of interscction) of the solutions 11(10. x) and v(lo. x) by N(lo), and we shall always assume that N({)) < 00. It is clear that N(lo) is precisely the number of sign changes in R of the difference ll'(lo. x).
tion interval
n'
'n
Proposition 1.
The .limctio/l N (I) is non-increasing lvilh lilll£'. a/ld in !wrliclIiar N(I)
:5:
N(O)
li)r all
(6)
IE (0. To).
Proof As we already mentioned above, technical difficulties arise in the analysis of gcneralized solutions 11(1. x) and v(l. x) having compact support. In this case. in accordance with the method of construction of generalized solutions (see ~ 3. eh. 1) we shall first consider the functions 11,(1. x) and v,(I. x). classical strictly positive solutions of equation (I) with initial functions
11,(0.
x) := lIoCr)
+ E.
v,(O. x)
:=
voCr)
+ E.
X
E R.
(7)
4 Proof of localization of unhounded solutions
241
where E > 0 is a fixed sufficiently small positive constant. By the comparison theorem 11, ::: IE, v, ::: E everywhere in the domain of existence of the solutions, and therefore equation (I) is uniformly parabolic on these solutions. Let us denote the number of spatial intersections of the functions II, (I, x) and v, (t, x) by N, (t). It is easily seen that by construction N, (0) == N(O) for every E > 0. The difference 11'(1. x) = 11,(1. x) - v, (I. x) satisfies the linear parabolic equation lI',
= (a(l.
X)lI',),
+ h(l. x)ll',
(8)
where the coefficients of the equations (/ (I. x)
=
r' 177
./0
II,
+ (I
- 77) v,
I" d 77. /J (I. xl
::=
{3
t
./0
177 II,
+ (I
- 77) v, ]{1- I d 77 .
arc sufficiently smooth. and a(l, x) ::: E". Equation (8) is uniformly parabolic in the domain under consideration. Therefore Proposition I is valid for the function N,(I), which is a consequence of the Maximum Principle. A simple short proof of this fact is presented, for example, in 13551; for similar statements see 113, 316, 303. 315, 171. 175, 2631. The original general ideas of such a comparison go back to C. Sturm, 1836 13681· Thus, N,(I) is non-increasing and N,(I) :5: N,(O). Using now the fact that 11,(1. x) --->- 11(1. x). v,U. x) --->- 1'(1. x) as IE --->- 0 uniformly on every compact set in 10, To) x R, after the necessary elementary consideration of the possible configurations of the intersections of the generalized solutions 11(1. xl and v(t. x) and the corresponding regularized solutions II, (I, x) and 1', (I. x). we arrive at the desired result for the function N(I): see 1140. 1751. 0 Let us note that for II :::: E the coefficients of equation ( I ) are analytic. Therefore its solutions 11,(1, x) and v,(I, x) for I > 0 are analytic in x functions (see, e.g. 1100, 249, 256\). Therefore each of their intersections for I > 0 is an isolated point. However, as we pass to the limit IE --->- O. an intersection point of LI, (I, x) and 1',(1. x) can be transformed into an interval of intersection of the gcneralized solutions 11(1. x) and v(l. x). Nonetheless, any intersection in the domain of strict positivity of the generalized solutions 11(1. x) and fJ(I, x), where the equation (8) for their difference lI' ::= 11- V is uniformly parabolic. and the solutions are classical and as smooth as is allowed by the coeflicients (therefore we can also claim that the solutions are analytic in x in the domain of their strict positivity), is an isolated point. Hence an intersection interval can only arise when it contains end-points of the supports of the functions under consideration. Thus the number of spatial intersections is non-increasing in timc. As we already observed. this fact is true for a wide range of parabolic equations, and we always have the upper bound (6) for the number of spatial intersections. However, to be able to use intersection comparison of solutions we also need, roughly
speaking, a lower bound of the number of intersections. Indeed, for example, the fact that the number of intersections cannot decrease to zero during the evolution of solutions would mean that there exists a certain relation between spatial profiles of the solutions on the whole interval of their existence. Unfortunately, it does not appear possible to obtain such a lower bound for the number of intersections in the general case; for that, solutions must share some common properties. In this case such a shared property will be equality of blOW-Up times. Thus we shall assume that 11(1, x) and v(t. x) have the same blOW-Up time:
rrm SUpll(r. x):=:
I+~.. T{I ,\ER
lim supv(l.
243
§ 4 Proof or localization of unbounded solutions
IV Nonlinear equation with a source
242
(9)
x):=: 00.
which, obviously, contradicts (9), sincc the function in the right-hand side is uni: formly bounded in R. 0 Proposition 2 can be described (with certain caveats) as providing a lower bound for the number of intersections: under the above assumptions N(I) > 0 for all I E [0. To). Without any caveats, one such result is presented below. Corollary. LeI
11(1. x) and v(r. x) hm'e Ihe same iJ/o\\,-IIP lillle t :=:
To: assume
Ihat lIo( x) has collllWCI support alld Vo (,t) > 0 in R. Then Ihe soluliolls u(l, x) alld 1'(1. x) inlersect/in aliI E
[0. To). FlIrthermore.
f,~~TII .\FR
N(r) :::
(II)
2 ill 10, To)·
Let us now state the main intersection comparison theorem. Proposition 2. LeI
11(1. x) alld v(l, x) IUII'(' Ihe same h/oll'-llp tillle I :=:
'W' ==
(t E
10, To) I u(l.
To. Theil
x) ::: v(l. x) in Rand (10)
suppv(l. x) C suppu(l. x)} Pro(lt: Let us assume the contrary: let 'W"
:=:
0.
i= 0 and there exists
that 11(1 •• x) ::: v(l •. x)
I. E 10. To),
such
in R.
supp v(l •. x) C suppu(l.,
x).
Then, first of all, hy the Strong Maximum Principle. applied to equation (X) in any suhdomain where it is uniformly paraholic, in which 11(1. x) and v(l. x) are separated uniformly from zero, and also using continuity of solutions and of the boundaries of their support, there exists a sufficiently small time r I > 0, such that 1I(1.+rl,x) >
v(t.+rl.x)insuppv(r.+rl.x),
suppv(l.+rl.x)C suppll(l.+rl.x).
+ rl. x)
::: v(l.
+ r, + r".
x)
In the sequel we shall have to compare a solution of the problem (I )'A (2) with exact solutions, which are not defined everywhere in [D. To) x R. Below we formulate an intersection comparison theorem for a solution u(r, x) of the Cauchy problem and a solution V(I. x) of a boundary value problem for equation (I). Thus, now we shall assume that a generalized solution v(t. x) is defined in some domain of the form [0. To) x tYJI (r). YJ"(I)), where YJI (I) < YJ"(t), vet, YJi(t)) are continuous functions in 10. To), and it is unbounded in the sense that lim / "~'~ Til \ (~(
sup TIl (f
l!(I. x) C:~OO.
l. TL (I))
In this context we shall denote hy N (10) the number of spatial intersections of the solutions 11(10. x) and v(lo. x) in the domain tYJI (10). YJ"(lo)). We shall take N(O) < 00.
Proposition 3. I) For all."
Using again continuity of the solution v(l. x) and of the houndaries of its support, we conclude that there exists a sufficiently small r" > D. such that 1/(1.
Proof Since under these assumptions the solution 11(1. x) has compact support, and v(l. x) > 0 in 10. To) x R. the estimate (II) follows immediately from (10) if we replace u(l. x) by v(r. x). 0
10 E
[D. To) Ihe nlUllher N(ro) does not exceed Ihe
numiJer of chwlRes (If siRIl 0( Ihe di/.li!rel1cc w(t. x)
==
v(l' x) on Ihe
11(1. x) -
paraholic h01/lldarr oFlIle domaill 10. To) x (TII(I). YJ"(r)). 2) LeI the SO/lIIiOIlS v(l' x) alld 11(1. x) exisl.!ill' IIle same lime
To. Theil
in R. ''V.
:=:
{to E (D. To) III(to. xl::: v(lo.
x) ill (YJ,
(to). TI2(10)),
Therefore hy the usual comparison theorem 11(1. x) ::: v(l
Setting here
I :=:
+ r2.
x)
in
(12) (I,
+ r"
To) x R.
To - r2, we arrive at the estimate veTo. x) :::: H(T o - r". x)
in R,
sup v(l. fE(ln.T(I)
1],(r))::::
.
inf 1I(I.1]i(l))jin
i:=:
1.2}
:=:
0.
tetro,/ll)
PrO(~r The first statement is a corollary of the Maximum Principle and, as Proposition I, is proved by first regularizing.
~
IV Nonlinear equation with a source
244
The proof of statement 2) is also similar to Proposition 2. Indeed, if there exists 10 E 'V t • 10 < To. then. after "small translations in time" of size II and 12, justified by the Strong Maximum Principle and boundary data comparison theorem, we conclude that the solutions u(l. x) and v(t. x) have different blow-up times (that of v(t. x) is somewhat larger). Observe that Proposition 2 is a direct corollary of this more general assertion. 0
Remark. Since proof of statement 2) is entirely hased on the Strong Maximum Principle and boundary data comparison theorem, it will still hold if v(t. x) is a generalized subsolution of equation (I) in the domain ID. To) x (7)1 (1).7)2(1)) with blOW-Up time To. We shall start our applications of the intersection comparison theory by analyzing the S blow-up regime.
2 Localization for {J
+
where L s
= 27T(lT + I) 1;2 / IT
h + (I) ::: h + (0)
h(l) 2: h (0)
meas w(D)
+ 1. .1'. h
(I)
2: h
((») --
Ls.
( 13")
Proof' of' Thcorem I'. In ~ I we presented an example of a simple localized selfsimilar unbounded solution for f3 = IT + I:
= (To -
t)
1;"OsCr). 0 < I
<:
To, x E R.
~
Ixl
I III/holl!lded sollllio!l \\'1' //(/\'1'
(14)
Ls /2
(13)
+ L."
( 13')
is Ihe .Iimdllmenllli lenglh 01' Ihl' ,'i·regime.
Below we shall also prove other theorems. which descrihe more precisely the penetration depth of the thermal wave for specific initial perturbations. Actually, the word "unhounded" in the statement of the theorem is superf1uous. since, as we showed in ~ 3, for all f3 E (I. IT + 3), N = I. to any initial function 110 ¢ 0 of the Cauchy problem there corresponds a solution that exists only for a tinite time. The estimates ( 13) mean that in the S-regime the front of a thermal wave can advance a distance which docs not exceed half of the fundamental length L s. We stress that the estimates (13) and (13') are independent of the spatial struclUre and amplitude of the initial perturbation I/o( x): therefore the length L s is indeed a fundamental (independent of I/o) characteristic of the nonlinear medium. Proof of Theorem 1 is based on intersection comparison of the solution 11(1. x) with a family of exact non-self-similar solutions II. (I. x) presented in Example 14
<:
L,/2,
( 15)
Ixl 2: Ls /2.
jilr all
IIlld, ill pllrlicllillr, :':
Theorem I'. III Ihe condilions IIf'Theorem I, .IiJl· all I E (0. To)
11,,(1. x)
(0. To) Ihe eslimales
meas (viTo)
Proof of Theorem I proceeds in two stages. First we shall prove a weaker result. which will serve as a simple and illustrative example of the power of the intersection comparison method applied to an exact self-similar solution.
where
of'lhe Callch.\' pmblem (I), (2) is lo('(/ked. and if' (3) holds.
+ Ls /2.
I Comparisoll \\'ilh lm exacl se/f~silllilar sollllio"
(T
Theorem 1 (localization in the S-regime). For f3 = IT
h , (I):,: hi (OJ
245
in ~ 3, Ch. I. Since during the time of existence thermal fronts of such an exact solution travel a distance exactly equal to Ls /2. for arbitrary compactly supported solutions the estimates ( 13) are optimal.
= + 1 (S-regime)
The main localization result is the following claim.
I E
4 Proof of localization of unbounded solutiolls
The support of this solution is constant in time. and its length is Ls == meas supp Os. We shall prove the estimate (13") by intersection comparison with the above selfsimilar solution existing for the same length of time. Since (14) is a solution in separated variables and has a very simple spatio-temporal structure. this allows us to give an exhaustive graphical illustration of the proof. Let us denote by 11.1'(1. x; xo. Til) the self-similar solution (14). symmetric with respect to the point x = Xo ( 14) is symmetric with respect to x = 0). Let us prove the first inequality (13"); the second one is established in a similar manner. Let us set XII = h-t (0) + L,/2. and in addition to 11(1. x) let us consider a different solution v(l. x) = 11.1'(1. X;Xo. Til), localized in the domain IIx - XIII < Ls /21, having the same blow-up time I = To. The interrelation of graphs of the corresponding initial functions lIoLr) and v(O. x) == 11.,(0. x: xo. To) =: I'll I;" 0.1'(.1 - xo) is shown in Figure 56. It is clear that they intersect only at the point x = hi (0). so that N (0) = I. Then by Proposition I N(I) ::: N(O)
=
I for aliI
E (0.
To)·
(16)
Let us prove that h! (I) ::: hi (O)+L s . i.e .. that the thermal wave cannot advance heyond the right front point of the solution v(l. x). Let LIS assume the contrary. Let f' = supll > D I hi (I) ::: hi (0) + L.d < To, that is. 1I(t. x t ) > 0 for alii E (f', To)
I; initial functions have a unique intersection at the point
.I
==
I I I
h+(O)
I
I I
I I
0
Fig. 58.
N(I2)
_.........-1. h,IO) X o
x
x.
h.(t?)
== 0; such a sillJation is prohibited by Proposition 2
Thus, we have considered the two cases: N (I') = I (which leads to Figure 57) and N (1*) == 0 (see Figure 5X). By (16) there are no other possibilities, Therefore t' = To, which concludes the proof of Theorem I'. 0 Fig. 57.
N(I I)
= 2; such a situation contradicts Proposition
at the point x, == h f (0) + L\,. Then there exist two ways the front point x == h ,(I) can move past the right front point x == x* of the self-similar solution LiS· The first of these corresponds to the case N (1*) = I. Then, since, as is well known, the moving front of the solution 11(1, x) cannot stop, fur any arbitrarily small T > 0 there exists tiE (I', t' + T) such that N (11) :::: 2. This situation is shown in Figure 57, where N(lI) = 2. Indeed, by continuity of the solutions, under a small shift in time the intersection that existed for t == t' persists, and at least one "new" one is created to the left of the point x == x, due to the motion of the right front of the solution 11(1, x). Hence N(I I) :::: 2. In other words, during the evolution the number of intersections of u(l, x) and v(l, x) increases, which is forbidden by (16) (and contradicts the Maximum Principle). In accordance with ( 16), the only other way of violating the first bound in (13") is by having N (I') == O. Then by Proposition I we have N (I) ::: N (1*) = 0 for all t E (1*, To). and therefore by the usual comparison theorem with respect to initial functions, at any moment of time t = t2 E (r', To) we must have the situation as in Figure 5R. (The case of 11.1'(12. x; XO. To) being tangent "from inside" to the spatial profile of u(l2, x) is ruled out by the Strong Maximum Principle applied to equation (X) for the difference of these solutions in the domain of uniform parabolicity. where the solutions are uniformly separated from zero.) Clearly, such a configuration of the graphs of the functions u(l2. x) and Lls(l2, x: xo. To) contradicts the condition of them having the same blow-up time.
Next we are going to exploit similar ideas of the intersection comparison method applied to the self-similar solution ( 14), (15) to study in more detail the dependence of the character of the motion of the front of a thermal wave in the S-regime on the spatial structure of the initial perturbation lIoCr). 2 Conditioll o( time-independence of the sllpport o{ WI //nbo//nded sollllion
The support (localization domain) of the self-similar solution (\4), (15) does not change during the time of existence of the solution t E (0, Tn). Let us show that in addition there are many other (non-self-similar) solutions, that are localized in the domain supp lIoCr) of their positivity at the initial moment of time. Theorem 2. Ass//me that f3 = IT + I. conditio/lS (3) are sati5:fied alld meas (supp //0) > L s . Let the initial functioll lIoCr) sati.I:/Y the jill/owing conditioll: there exists a cOllstant Ao > 0, s//ch tlzat
= lz+(O) - Ls /2, while the jimctiolls lio(x) alld IIS(O, x; xo, A) have exactly olle intersectioll {JOilltjill' all A E (0, Ao) IIs(O. X;Xo, Ao)::: lIo(x) ill R, where Xo
(this situatioll is shown in Figure 59). Theil
248
*4 Proof of localization of unbounded solutions
IV Nonlinear equation with a source
249
is quite large and contains functions other than the initial functions corresponding to the self-similar solutions (14). 3 COlldilioll FJI' localiz.arioll
011
Ihe IlIlldal11ellllll lellgth L s
Let us show that under certain conditions an initial perturbation with a small support (meas(supp 110) < Ls } cannot propagate beyond the domain {Ixl < L s /2} in tinite time of existence of the solution. Here we shall assullle that in addition to (3) we also have the conditions lIo(-x)
Fig. 59.
Pro{)t: Let us denote by N (t; A) the number of spatial intersections of the solutions 11(1. x) and 11.1'(1. x; xo. A). From the condition of the theorem it follows that To s: Ao. If To = Ao, that is, if N(O; Ao) = O. then the hypothesis concerning the motion
of the right front leads to a contradiction with Proposition 2 (here obtains the situation of Figure 58). Therefore we have to consider the case To < Ao. Let us tix A = To and consider the solution IIs(l. x; xo. Aj, which has by construction blOW-Up time To. Then for I = 0 there is a unique intersection point of the initial functions I/o(x) and IIs(O. x; Xo. A} == A·1/"OsCr - xo), i.e., N(O; A) = I. Arguing now as in the proof of Theorem I', we conclude that there are two possible scenarios for the motion of the right front of the solution 11(1. x). According to the first of these (N(I') = I). we arrive at the contiguration of spatial protlles as in Figure 57 (which contradicts Proposition I). The second scenario (N (I') = O)leads to Figure 58. that is, to a contradiction with Proposition 2. Hence the front of the solution 11(1. x} must be perfectly immobile. which concludes the proof. 0
Remark. It is of interest that for the right front point of the solution to be immobile throughout all the time of its existence. we need a non-local condition on the behaviour of the initial function 1I0Cr} in an Ls-neighbourhood (h+ (0) - L,\' , h + (0)) of the front point x = hi (0). If the condition of Theorem 2 holds, the behaviour of 110(.1') in the rest of the space. (x s: h.,.(O) Lsl. has no influence on the immobility of the right front of the solution. Formally, the initial function can go to any large value as x - 4 -00 (as long as a local (in time) solution exists). This again emphasizes the universality of the fundamental length L s characteristic of a nonlinear medium, which here plays the part of a kind of effective radius of influence of thermal perturbations. Imposing similar conditions on the behaviour of 110(.1') in a neighbourhood of the left front point. we obtain a set of initial perturbations 1I0(.r), which generate unbounded solutions with a constant support. It is easy to show that this set (1I 0)
= lIoCr}. x
E
R; lIoCr) is non-increasing for x> O.
(17)
Under these conditions, due to uniqueness of the solution and the Maximum Principle we have that II = 1I(1.lxl}. 11,(1. x} s: 0 for x E 10. h ,.(I}} and SUP,II(1. x} == 11(1. OJ. Theorem 3. Assllme Ih(/I f3 {T + l. cOlldiliollS (~), meas(supp 110) < Ls. Lei lIo(.r} also s(/Iis/)' Ihe cOlldilioll
(17)
Ihere exists Ao > O. such Ihal IIs(O. x; 0, Au} ~ lIu(x} ill while Ihe .limcliolls lIu(x) (/Ild 11.1'(0. x; O. A) illiersect precisely (1/ IWO [Joillls for (/11 A > Ao (.1'1'1'
hold (/Ild
R.
Fig II 1'1' (0). Theil Ih,(I}1
s:
L s /2/(I/' (/111
E
(/Ild ill [Jar/iclIlar
meas
(v(
To )
:s:
Ls ·
(0. To) ( 18)
By Proposition 2 To > Ao. Setting A = To and denoting by N+ (I; A) the number of spatial intersections of the solutions 11(1. x) and IIs(l. x;O. A) in the domain (x > OJ, we obtain N +(0; A) = I. Then by Proposition I. in view or the condition II = 11(1. Ixl), we have that N I (I; A) s: I for all I E (0. To). Therefore we can now use the method or proof of previous theorems. 0
ProOt:
Thus in the conditions of Theorem 3 the thermal wave can move in any direction. but the total distancc covered by thc thermal perturbations up to the blOW-Up time cannot exceed ( 19) L s - mcas (v(O) < L s . 4 COlllpariso/1 with
l/
.I(I/11ily o( ('.1'(/1'1 /1{J/l-s('/f~silllilar soll/tio/1s. Pro{)( o( Theorelll I
We move on now to prove the optimal bounds ( 13). As we already mentioned. the proof is based on intersection comparison with the more complex solution 1I. (I, x)
facets of the intersection comparison method by comparing not with a single fixed solution, but in fact with a continuously parametrized family of exact solutions. all having the same blOW-Up time. For a fixed E > 0, 0 E R, let us denote by v(t. x) == v(l, x; E. 8) the function tI.(I + E. x - (h+(() + 8»). where Co = Co(T o + E). Then v(l. x) is an unbounded solution of the Cauchy problem (I), (2) with initial function tlo = U.(E. X (h+(0)+8)) and support supp v(l. x) == llx- (h+(O)+8)1 < g(l+E)I. Let us note that the function 11,(1 + E. X - (h+(O) + 8» is continuous in E. 8 in [0. To) x R. By construction, for all 0 E R the solutions tI(I. x) and v(t. x; E. 8) have the same blow-up time To. For any I E [0. To) let us denote by N(I: E. 8) the number of spatial intersections in R of the functions tI(t. x) and v(t. x). It is not hard to check that for any E > 0 and 8 ~ 8. = g(E) supports of the initial functions tlo(.r) and v(O. x) do not intersect. so that N((); E. 0) = I. Then by Proposition 1 we have that
Fig. 60.
N(r;E.O)::,: I for all I E (0. To).
of equation (I). presented in § 3. Ch. I. It has the form 11.(1.
for
= {(/>(I) [Ip(l) + COS(27TXILs)]t }1/lT
(-L s I2. Ls/2) and 11.(1. x) = 0 for is determined from the equation
x E
1//(1)
1//
and
x)
c/>(I)
= (r(IT + 1)-ICo [I
= Coll __ I/, 2 (1)ll Co
tr
'2)/2.
= Co(T o) =
-
x E
~0
(20)
R\(-L I /2. L s I2). The function
if/2]~lT/2. I>
0: Ip(O) = -I.
+ I )IT ITo
IS(
1+ (r/2. 112).
(24)
Thus. let us fix sufficiently small E > 0 and 8 = 20.; then supp v(r. x; E. 20.)i = = h+(O) + 28•. Obviously, I. -+ ',+(0) as E -+ O. Let us show that h+(t) ::': '. + g(t + E) in 10. To). Assume that this is not the case and I. = sup{t E 10. To) I h+(t') ::': I. + g(t' + E) (Ix - 1.1 < g(l + E)l. where I.
(25)
(21)
where
(IT
251
(22)
Then 1/1(1) is defined on (0. Toj. and 1//(1'0) = 1 (hence c/>(To) = +00). so that (20) is an unbounded solution with blow-up time To. The generalized solution u.(t. x), which is symmetric with respect to x = O. has compact support. and its right front is located at the point (23)
Clearly. h~(t) is a strictly increasing function and h~(I) < L s l2 for all IE (0. To). so that the unbounded solution (20) is localized in the domain {Ixl < L sI2\. In § 3. Ch. I we showed that II. (t. x) satisfies the singular initial condition LI. (0. x) = Eoo(x). Eo > 0 is a constant. and supp u. (t. x) -+ {Ol as I -+ O. Therefore for intersection comparison we shall in the following take functions of the form II. (I + E. x) (the constant E > 0 is taken to be sufficiently small). to which correspond regular bounded initial functions II.(E. x). In view of the fact that the support of this exact solution (unlike the self-similar one) varies with time. we shall modify somewhat the proof. We shall exhibit new
for all I'
E
10. III
< To.
Clearly, hj(l,) = I. + g(l. + E) == x •. By Proposition I. two cases are possible. Case 1: N(I,:E. 28.) = I. Here we shall arrive at a contradiction similar to the one in Figure 57, where IIS(') should be replaced by the solution 11,(,), In this case the difference w(l,. x) == 11(1 •. x) - 1'(1 •• x) changes sign in R exactly once. Then we can find --00 < X2 < XI < x, ;;;:: h t (I.). such that either
or. on the contrary.
If (26) is satisfied. then choosing 01 E (15..215.). 01 ~- 202, we have that by continuity of the function v(l. x; E. 0) in 0, inequalities (26) would still hold if the function v(r •. x; f. 20,) is replaced by V(I •. x; f. 8 1 ) and. furthermore v(r •. x': E. 01) = 0 < 11(1,. x') at the point x';;;:: Ix. ~... (20. - 01)] E (XI. x.). Therefore N(I,:E. 01) ~ 2. which contradicts Proposition I. If. on the other hand, we have (27). then the same contradiction is obtained by comparing the functions v(t •• x; E. 0 d and II, (r. x) for 8 1 > 20•. with 01 - 215. sufficiently small.
*4 Proof of localization of unbounded solutions
IV Nonlinear equation with a source
252
=
Case 2: N(I.:E, 21\) O. We shall show that this case leads to the situation of Figure 58 (lI.d·) is replaced by u.(-)). Then either u(l •• x) :: v(t •. x) in R (but then by the usual comparison theorem u(t' x):: v((. x), suppu((. x) <;;: suppv((. x) in [t •• To) x R, which contradicts (25)), or u(( •. x) :::: v((., x) in R. In the latter case by (25) there exists 12 E ((n Tol. such that h+(l2) "> I, + g(l2 + E) and in addition obviously U(t2, x) :::: V((l. x) in R. But then we can find 8 1 E [0. h+ ((2)(If + g(l2 + E))), such that supp v(l2. x; E. 8 1 ) <;;: supp U((2. x) and v(ll, x; E. 8,) :: u(12, x) for all x E R. This contradicts Proposition 2. Thus, h l (!) :: I, + g(t + E) in (0. To) for any arbitrarily small E "> O. Passing in this inequality to the limit E ~ 0 (then If ~ h+(O). g(l+E) -> g(l), we obtain the first of the bounds (13), which completes the proof of Theorem I. 0 Let us note that the above argument proves a sharper optimal time-dependent upper bound for the motion of the front.
Corollary. III Ihe cOlldiliollS or Theorem I h,(I):: hl(O)
+ (L s j27T)[7T/2 + arcsinl/f(!)].
IE
10. To).
(28)
Clearly, by (23). for the solution u,(I, x) instead of the inequality we have in (28) an exact equality. Since the initial function for the solution u.(I. x) is singular, the estimate (28) describes, in particular, the maximal speed of motion of the thermal front for small I "> O. It is not hard to compute from (28) that I
'1
hi (I)~~ h, (() + bltl
I·
...
Remark. For /3 = (T + I we have m = 0, and as will be seen in the following, ~. = L s , i.e., in the case /3 = IT + 1 this theorem becomes Theorem I'. Let us stress that unlike the S-regime (Theorem I) the "fundamental" length of the LS-regime L tl = meas Iu( Til) depends, via its dependence on To, on the initial function. An upper bound on the blOW-Up time To = To(uo) in this problem was obtained in § 3. / Conslruclion or II self-simi/ar suhsollllion In § I it was shown that for /3 "> (T + I equation (I) has no localized self-similar solutions. All the self-similar solutions constructed there are strictly positive and are only effectively localized (us(l. x) grow without bound as I -> T(l only at the point x = 0 and remain uniformly in I bounded in R\(O}). However, it is easy to verify that Proposition 2 still holds, if as the second solution v(t, x) we t'lke some unbounded subsolution of equation (I). We shall construct such a subsolution for the LS blow-up regime, and. since it is not defined in (0. Tn) x R, we will in fact be using as our main intersection comparison theorem Proposition 3, which is specifically suited to deal with this case. We shall seek the self-similar localized subsolution in the usual form:
1+
o(
I) as
1->
(0::::
0 satisfies almost everywhere in R the equation
(fJ'r 0 " ) ~ I/If) I
..
~
Lemma l. For allY ,8 (32)
3 Localization for
f3
>
(T
+t
-
(IT
+
) ::
measw(O)
I) 1/[2(,8 - I) I
">
+ 2CT;;'
0 alld ~.
">
"> ()
< 00,
(T
+
(30)
t~ -
I Ii /3-1· + 0 .. = o.
-~-{J.
(32)
I Ihere exisls a /loll-Irivial solutio/l (J_ (~) suri.II)'illg > O. us wel/ as Ihe cOlldiliolls
(-f'. 0), C (1(0)
Theorem 4 (localization in the LS-regime). LeI /3 "> (T + I. Theil all /11lho/lluled solutioll of'the pmhlem (I), (2) having !J/ow-up lime To = 1'0(110) < 00 is locllli:ed alld (29) measlu(T 11
O/l 1111 illll'l'\'(" b
(l~S-regime)
The main assertion concerning localization in the case of the LS blow-up regimes has the following form:
i.e.,
(31 )
O.
where
where m = 1/3 ollly on IT, /3.
I/II!~II{) (g), ~=x/(To-I)III.
us(t.x)=(Tn-1)
where the function ()
1
II" t _I(
253
= O. (0" (j'
)(0)
= 0,
(33)
From (33) it follows that the function (J (~), which is the same as the function of Lemma I for g E (--g'. OJ and zero for ~ "> 0, is a generalized solution of equation (32) on (-g', (0), Then (31) is a generalized unbounded solution of equation (I) in the domain (0. Tn) x (x.(t), (0) with a moving left boundary x,(1) = -~'(T() - 1)'", on which 11.:;(1. x.(t)) = O. Hence we have that (31) is a localized solution: even though u~ (I. x) grows without bound in any left neighbourhood of the point x = 0 as I ~ 1'11' the front of the solution x/(t) == () is immobile, and perturbations do not penetrate into the domain x"> 0 (see Figure 61).
is a COlls{{/nl Ihl/I depends ('Here the constant C
ccc
l;' ((T. f3) is the same as in the statement of Theorem 4.
(0. To) x {R\{x = Xo _. (.'(1'0 - I)'"}l. All this allows us to compare the solution II(T. x) with III~s as was done in the previous subsection. In brief. it goes like this.
Fig.
(It.
Localized subsolution
n I)
for
f3
>
(T
+
If now we set (L(t) == 0 for f. :s -t', then u~ subsolution of equation (I) in (0. To) x R. Proof' of
1.£'11//1/(/
I at difrcrcnt timcs
(I, x)
will be an unbounded
I. It is not hard to demonstrate local solvability of the problem
(32), (33) for small ItI by reducing it to an equivalent integral equation and using the Schauder fixed point theorem. The property of the solution extended into the domain of f. < 0, alluded to in the lemma. follows immediately from the results of I (see (29)) for f3 > IT + I, N == ). 0
*
Let us prove localization on the right. Set Xo = h+(O) + (.'7'(;'. x,(T) = .toC(T o - t)1II (To is the blOW-Up time of 11(1. x)). Then 1I0(X) and IILS(O, X;Xo· To} do not intersect in (x,(O). (0), and, clearly. in the course of the evolution there can be only one intersection of 11(1. x} and IILs(l, x; xo. To) in (x.(I), (0) (if II >PilLS for x == x,(I)). Therefore 11(1. x} will become larger than v == lIi:s(l. x; xo, To} at the point x == Xo (there lJ == O) only aner it becomes at least as large as v(t. x) for all x < Xo. However. by Proposition 3 it contradicts the fact that the solution II and the subsolution v have the same blOW-Up time. Therefore h.,. (I) :s xo, which is the same as (29). 0 In conclusion, we present a result for the LS-regime, which is similar to Theorem 2 in statement and method of proof. 3 COlldilioll of'imll/obilily of/rolll poillls of ,III IInbOllnded soilltion
Theorem 5. Assllme th(/I
7Notc that this fact is useful in dcriving conditions for global insolvahility of boundary valuc prohlcl11s for ( I).
>
IT
+
I (/nd leI
1I0(X) (/Iso slIIi.\j)' the conditioll
therc exisTs Ao > O. slIch th(/I lIoLd ::: lIi,'s(O. x; h, (0), Ao) ill R, whilc the/ill/Clions 1I0Lt) (/nd II;:S(O, x; h+ (0), A)
2 Proof of' Thcorcm ./ Let us denote by II 1.1'( I, x; Xo, To) the function which coincides for (0. To) x {-f.'(T o - 1)'11 < X - Xo < 01 with (31) (fL(f.) ::: 0 is as in Lemma I and 111:.1 == 0 outside that domain). As we already mentioned 11 1..1 is an unbounded subsolution of the Cauchy problem 7 in (0, To) x R, i.e .. if 1I0(.r} ::: 111:.1(0, x; xo, To) in R then 11(1, x) ::: 11 1..1(1, x; .to, To) in R for all admissible I > O. Therefore Proposition 3 remains valid if as the function v(I,x) we take 1I;.s(l,X;xo, To} (or any other subsolution of a similar form). Let us note that at the same time the function 11 1.5 satisfies the equation in a generalized sense in the domain
f3
illtersccl precisely at olle poilll .lin' (/11 0 < A < Ao. Then
where To < 00 is Ihe blOW-lip time of Ihe solwion 11(1, x).
A graphical interpretation of the condition of the theorem is presented in Figure 62. In the LS-regime the length of the part of the support supp uo(x), which. in accordance with the condition of the theorem, influences immobility of the front Unlike the S-regime case. this length depends point x = h t (t) == h, (0) is
en;'.
§ 5 Asymptotic stability of unbounded self-similar solutions
IV Nonlinear equation with a source
256
on the behaviour of the initial perturbation uo(.\") (via whole space.
1'0(110))
on practically the
4 Non-localized unbounded solutions of the HS-regime, 1 <
f3 < u + 1
Absence of localization of solutions of the Cauchy problem in this case (all nontrivial solutions are unbounded; see 3) is easily proved by the method of stationary states, which is presented in Ch. VII. II is used there to study the localization phenomenon in arbitrary nonlinear media. However, for equation (I) sharper results can be obtained by comparing the solution 1I(t, x) with a self-similar solution of the HS-regime. which was constructed in § I.
*
IT + l, and let condition (3) be sati.l:fied. Then the l/nbollnded solution (if the Callchy problem ( I), (2) is not localized. lind if t = To is the blOW-lip time. 11'1' hal'e the estimates
Theorem 6 (absence of localiz.ation in the HS-regime). Let I < f3 <
h.(O)
+ ~olCTo -
I)'" -
T;;'J,
IL(t) :::: hj(O) _. ~ol(To - I)'" -
1';;'1,
hf(t)::::.
Inequalities (34) mean that as t meas w(l) == h+ (I)
- IL (I) ::::.
-+
lilt (1)1
00
--'>
as t
-+
+ 2~01 (To -
I)'" -
1';;'1
-+ 00.
(35)
*
I)'" E
R.
~o(To -
1)111
== h t
(0) -
~ol(To -
1)111 -
1';;'1.
(37)
cannot overtake the left front point x = h. (I) of the solution 11(1. x). Were that to happen, then either at some moment of time t = t 1 we would have two intersections of 11(1. x) and IIl/s(I. x; xo. To), which contradicts the Maximum Principle. or for some 12 E (0, To) we would have a situation precluded by Proposition 2. Therefore h . . (I) :::: XI (I). Taking into account (37), we obtain the second bound of (34), which concludes the proof. 0
(38)
for example. with the proof of Theorem I. Let us write down the unbounded self-similar solution of equation (I) for f3 < (T + I (see I):
= (To _. t ) 1/({J'I)f)s(~), ~ = x/(T o -
= Xo -
TIl'
Proof It is similar to the proofs of previous theorems; there is a direct connection,
1I,\(t • .I)
x/(I)
Therefore for any compactly supported initial perturbation. the fronts of the thermal wave in the HS-regime move as t -+ Til (110) not slower than at the selfsimilar rate
Ttl
meas lV(O)
different due to the nature of the bounds in (34) (they are lower, not upper bounds as in all the other theorems). With these positions of the supports of 110 and III/S(O. x; xo. To) (supP 110 n supp LlI/S(O, x; xo, To) = 0), the number of their intersections in R is equal to one. As the solutions evolve. this number cannot increase. Therefore the number of intersections of different solutions 11(1, x} and 111/.1'(1, x; xo, To) having the same blow-up time t = Trj{lIo) < 00 docs not exceed one for all t E 10. To}· This means that the left front of the self-similar solution III/S, which for all 1 E 10. To) is at the point
(34)
t E (0, To),
where m = 1f3 - (IT + I) 1/12(f3 - I) I < 0 and there.f{lre The constant ~o > 0 in (34) depends only on (r, f3.
257
In the next section we shall show that as t -+ To the motion of the front points x = h±(I) approaches asymptotically the self-similar one. that is, in addition to the inequality (38) we also have the reverse one. As far as equivalence of conditions (h±(I) -+ 00, t -+ To I and (u(t. x) -+ 00 in R. t -+ 1'1') I for I < f3 < (T + I is concerned. we shall prove assertions of that sort in I, Ch. VII. For example, it is especially easy to prove
*
Theorem 7. Let I < f3 < IT + lund U.I.llIlI/e thut conditions (3), (17) hold. Then lilt. x) -+ 00 in R us t -+ T(l (110)·
(36)
The function f)s(~) has compact support: meas supp ()s(~) = 2~o < 00. where the constant ~o is as in the right-hand sides of (34). As before, let us denote by IIflS(t, x; xo, To) the self-similar solution (36) symmetric in x with respect to the point x = xo. Let us. for example, sketch the proof of the second bound in (34). Let us place 11/1.1'(0, x; XI), To) relative to the initial function 1I0(X) as in Figure 56 (replace Us in that figure by 11/1.1')' For that we have to set Xo = h+(O) + ~oT;;'. The relative position of these two functions is, in principle, the same as in the proof of Theorem I'. However, the general structure of the proof is somewhat
§ 5 Asymptotic stability of unbounded self-similar solutions We have already discussed above certain essential difficulties in the analysis of the spatio-temporal structure of unbounded (singular in time) solutions, which always arise when solutions are unstable with respect to small perturbation of the initial function. Therefore in this section we shall not strive for maximal generality of presentation. which would entail exerting a great amount of effort in overcoming
* 5 Asymptotic stability of unbounded self-similar solutions
IV Nonlinear equation with a source
258
complications that are not of any particular importance. We shall use the example of the Cauchy problem in N = I to present the crucial stages of the proof. First, to simplify the presentation below, we state in a compact form the methods of comparison of unbounded solutions we used in ~ 4 in the study of the Cauchy problem (I) II, = (11"11\)\ + IIi!, I> 0, x E R: (T> O. {3 > I. 11(0, x)
:::
0, x
E
R;
110 E
Self-similar solutions of equation IIS(t,
x)
= (To -
have the form
(I)
t)-I/IIJIIHs(~). ~
= x/(T o -
1)'" E
R.
(7)
where the function Hs(~) ::: 0 satisfies the ordinary differential equation W:H'I'/ , . -
(2)
C(R),
I/IH'I~ . -
_I_HI' {3 - I
+ HI.I' = O. ~
E
(8)
R.
h _I (I»).
I A lower bound for the amplitude of the unbounded solution
supported initial perturbation:
= (h
(0). hi (O)),
u;;
is Lipschitz continuous in
R:
supp 11(1.
x) =
uo(-x)
= 1I0(X}, x
This is the simplest corollary of the comparison theorem. 110(.1)
Theorem l.
LeI
(T
:::
O. {3 > I.
lIl/d leI u(l, x) be
(1/1
IIl/boul/ded solllliol/ (If Ihe
ClIuchy prohlem (I), (2). Thel/
supu(l,x) > HltCTo-1)
I/Ii! II
IE
10. To); OJI
= ({3-lr 1/ 1i!11
(3)
t.?'
~.
2 Similarity transformation. Restrictions on the form of initial functions
In ~ I we found that for all (T > O. (3 > I it has an even solution Hs(~), which is non-increasing for ~ ::: O. We shall study asymptotic stability of precisely these solutions (for {3 ::: (T + I there are other, non-monotone solutions Hs(~)}' Consequently. we shall introduce the following restrictions on the compactly
where supp Uo (/L (I).
= 110(.1)
259
where To
= 1'0(110)
Pro(~t: The estimate (3) follows from the corollary to Proposition 2 in ~ 4, if we
take as v(l, x) a spatially homogeneous solution of equation blow-up time:
= {l//(T o -
(I)
with the same
R; meas
SUppll()
= 21()
<:
x,
(9)
(10)
is non-increasing for x > O.
Then u(l. x), an unbounded solution of problem (I), (2) is even in x, non-increastflg in x for x ::: O. and sup, u(l, x) = 11(1.0) for any I E (0. 1'0(110))' Corresponding to (7), let us introduce the similarity representation (J(t.~) of the solution of problem ( I), (2):
H(t,~) is Ihe !Jlow-up lime.
E
= (To - 1)I/IIj·IIII(I. ~(T() - I)/Il), IE (0, To)· (; E R. (II) «(1 + I) J/12({3 - I) I. Similarity transformation of the solution (7)
where m = 1{3 gives us exactly the function Hs(~), We shall be interested in the behaviour of O(t.~) as stability of the self-similar solution (7) means that
I
-'>
Til'
Asymptotic
(4)
(12)
Then we have that 11(1, x) and v(l) must intersect for each I E 10, To). Moreover, each intersection point in the domain of strict positivity of both solutions is an isolated point. For the number of intersections we have
for a sufficiently large set of initial functions 110· Let us note that under the assumptions (9), (\0) the limiting function is necessarily even and non-increasing for ~ > O. Therefore we are analyl.ing asymptotic stability of the most elementary (in its spatial "architecture") self-similar solution. Many of the results stated below extend to the multi-dimensional case (see ~ 6).
v(l)
N(I) :::
I)
1/1i!II, IE
2 for aliI
E
[0, To},
10, To).
(5)
Obviously, this means that SUPII(t, x) > v(t), IE 'EU
10, To),
from which (3) follows. Let us note that this estimate can be obtained directly from equation (I).
(6)
3 Asymptotic stability of the self-similar solution for
p = IT + I (S-regime)
If (9), (10) are satisfied. the only "candidate" for a stable self-similar solution is
o
the following one
(~
I):
IIs(l. x)
= (To -
I)
I/"Hs(xJ. 0 < I < To, x E
R.
(13)
~
IV Nonlincar cquation with a source
260
A sharper estimate than (19), which is "distributed" over R. can be derived using the method of stationary states (see I, Ch. VII).
where the function 2(CT+I) el (. 7) _ ( '.I' :::: { ( CT cT
+
17TX)I/1T
cos- -r-: LS
0,\.1.'1::: L s /2
1\
,x <
L/' .I'
k,
*
(14)
= 7T(CT + 1)1/2/(T,
Proof The estimate (18) was obtained in the course of proving Theorem 1 in §
4. Inequality (19) is none other than (3) for {3
satisfies the ordinary differential equation ( 15)
OSSlIlIIe Ihol Ihe com/ilion\' (9), (10) are salisjied, (/nd
leI To < 00 be Ihe blOW-lip lillie for Ihe IInbolinded sollilion of Ihe problem (I),
(2l. Then H(I. x)
R.
-')0
(Js(x), 1·--+ Til'
(17)
where (ls( x) is Ihe .limclion (14).
The main obstacle that arises in the proof of (17) is the derivation of bounds in LOO(R) for the similarity representation, which arc uniform in I E (0, To)· Upper bounds guarantee global boundedness of O(r, x). while a lower bound is needed in order that the limiting function (J(T I )', x) in (17) be non-trivial. These arc the two most crucial stages of the proof. The point is that the function (J :=: (Js(x) is an unstable stationary solution of the parabolic equation satisfied by the similarity representation OU, x) (for a similar example see § II, Ch. II). Therefore *in the course of derivation of (17), we single out in the space of initial functions (0(0, x) I the attracting set of an unstable stationary solution. We emphasize that similar problems of asymptotic stability of stationary solutions arise precisely in the analysis of singular solutions of evolution problems, which have a singularity in time. / Auxiliary rcmlls
sUpU(I,X) >
1~lo
-- Ls , 10
0(0) :=: 0"
(T
+ Lsi:
2 Proof of 711eorCI/1 2. a) EqlWlioll Fir Ihc .limclio/l
'EI{
(TI/"(To-/f,I/,,;
(To
< 0,(1'0 - I ) 1/"
(20)
-I)Or
= ({)"O,),
" ) 0 ,.\) f(
(19)
Setting in equation x)
OU, x)
problem
( 18)
1/", such Iltal
sup 11(1,
0'(0) :::: O.
has a solution 0 = 0(.1), which vanishes at a point x :=: x,({J.) > O. This is always possible, as can be immediately seen from equation (15), which can be integrated in quadratures (see Lemma 2 in § I). Let us set O(-x) = O(x) in (-x" 0). Then v(l, x) = (To - 1)- 1/ITO(x) is an unbounded solution in (0, To) x (-x •. x,). It is easily checked that .1.',(0.) -> 7T/12(cT + 1)1/21 and (H"),(X;) -> -00 as 0, --+ 00. Since II;; is uniformly Lipschitz continuous, we can pick 0, > 0 so large. that if (9), (10) hold, the initial function 110 either docs not intersect v(O, x) at all in (- x" x,) (i.e .. N (0) = 0), or intersects it exactly at two points, which arc symmetric with respect to x = 0 (N(O) = 2). Then by the comparison theorem N(I) ::: 2 for aliI E (0, To). Let us show that 11(1,0) := sup, II ::: sup, v := v(l. 0) (this immediately results in the estimate (20)). If 11(11.0) > 11(11.0) for sorne II E (0, To), then 11(11, x) > 11(11,.1) in (-T.,X,):= suppv. Indeed, if this is not the case, then 11(1, . .1.') = 0 for x = ±x, (since N(lI) ::: 2). and this equality holds for I E 10, 'II· Therefore N(O) = 0, and by assertion I) of Proposition 3 NUl) = 0, which is impossible. Thus, 11(11 . .1) > V(lI,X) in (-x"x,). If sUPpv(lI'X) C SUppll(ll' x), we obtain a contradiction to Proposition 2. § 4. If. on the other hand, supp 11(11, x) = (-x,. x,). then, by slightly increasing the value of 0.. and thus decreasing 2x, = meas supp (J, we arc back at the previous case. 0
,,,R
Ihere exisls O. >
*
It is not hard to check that the similarity representation (16) satisfies the Cauchy
Lemma 1. /11 Ihe condilion,\' o(Theor(,1/1 2, .!iiI' (/111 E 10. To) \I'e ItUl'e Iltc eslil/1(1/cs suppu(l. x) C
= (T + I.
Inequality (20) follows from Proposition 3 in 4. Let us pick the value O. > Oil = (T I/IT large enough, so that, first, H,T;~ 1/" > 110(0), and, second, that the Cauchy problem for equation (15) for x > 0 with the conditions
In the S-regime the similarity transformation has an especially simple form:
IIni(ormly in
261
5 Asymptotic stability of unbounded sclf-similar solutions
The initial condition (22) remains the same. Comparing (23) with the "self-similar" ordinary differential equation (15), we see that in the new notation the study of asymptotic stability of the unbounded self-similar solution of the S-regime is equivalent to the analysis of asymptotic stability of the non-trivial stationary solution (14) of equation (23). It is important to note that for arbitrary initial perturbations fJ o(.<), when the quantity To in (22) has been chosen "inclln"ectly". and does not equal the blow-up time of the solution 11(1, x), the problem (22), (23) can have both unbounded (sup, fJ(T. x) -'> 00 as T -'> T ) < (0) and global solutions. which stabilize to the trivial stationary solution I fJ == 0 as T -'> 00. In other words, the stationary solution (14) is unstable with respect to arbitrarily small perturbations. A proof of this fact is presented in ~ II. Ch. II. h) Eslill1ales ot' rhe .limclioll fJ(I. x). In the conditions of the theorem (with a "correct" choice or To = 1'0(110) in (22)), the Cauchy problem (23), (22) always has a global solution, which stabilizes to the function (14). The proof is based on the estimates (18)-(20), which assume the following form in the new notation: Corollary of Lemma l. III rhe cOlldiriolls ot'Theorell1 2 Stipp fJ(T, x) C 1-/0-- I- s . /0
sup (I(T.
+ 1-.1' I, 1-.1' = 27T((r + I) 1/2 I(T;
x)
>
(r
By (26) the right-hand side of (30) is bounded from above, from which we obtain the estimates (27)-(29) (for details see ~ 2 in Ch. VII). 0 c) Passage 10 Ihe /ill/il T -'> ex). Thus, the Cauchy problem equivalent to the boundary value problem with the condition O(T. x)
(24)
lifT,
V(/J)(T)
(25) (26)
fJ(T.X) < fJ,
./fIr all T :=:: 0, x E R.
(9), (10) hold. alld Ihal To ==
he a dOll/aill ill
R.
+ L s ) en. L ""(R,; L'(n»,
slich Ihar (-/0 - L s , /0 ()11"i2 E
((I11"!2), E (I,,'I E
Pm()( By (25) ()
of equation
(23)
== 0
with
Oil
r
4((T + I) (lr 2)2 ./0
+
(T
+I
an
WIT,I)T
+
Theil (27)
L 2 (O)).
(28)
L""(R+: HI11(n».
(29)
"
(1)11
2 dl' l'I!I)'
I -1I1)( )11211T11I 2 T I.'"'' "Illi
+
+
I 2l1T III --_111001l/'''''''Il!I+
2(lr
I
+ I)
I 2(lr
(r
(
+ +
I)
IIW
IT
2
+
'L
+
T 11
( )
I'llll
11(0"+1) 11 2 0
I{
T :=::
O. x E
an = o\n.
I ( fJ ,,+ I)' + · (T + I ------ - f )",0-
+
2((T
which is non-increasing in tions, we have
T
d -V({I)(T) dT
n
for any T :=:: O. Taking the scalar product in L'(n) and integrating over T, we obtain the equality
11({lIIITI2)
+ -----1I0(T)II'T'2 _ (T((T 2) I.''' 'ill) -
L'(R,;
1'0(110). LeI
==
. 0
From that we immediately have AI'SlIlI1e Ihm cOlldiliollS
= 0,
(23). (22)
is
(22')
and the estimate (26) ensures its global solvability. Stabilization of OfT. x) for T = T, -'> 00 to a stationary solution in the weak senses follows from the estimates (27)--(29), which ensure boundedness of the sequence O;T+I(T.X) = (J'T+I(T+Tj.X), 11 == 1.2.... in HI((O.I) x n) (see § 2, Ch. VII). By compactness of the embedding HI C L 2 and the estimate (28), this allows us to choose from any sequence T, -'> 00 a subsequence (which we also 1 -(r+ I . , denote by T,), such that O;T' (T, x) -'> 0 (x) as T, -> 00 in 1-"(0. I) x 0). See subsection 6, ~ 3. Passage to the limit in equation (23) is also effected by using the estimate (29), as well as the ract that (22), (22') admits the Liapunov function
\cR
Lemma 2.
263
~ 5 Asymptotic stability of unbounded self-similar solutions
IV Nonlinear equation with a source
262
I)
\
(T((r
+ 2)
o} (x. I
I ",n -- -fJ2
on any solution of the problem. By formal computa-
=
4 ( ( r + l )OUT+21!2 / r ' ( ) ' dx::: ((T+2)-.0 '
-"---0
O.
Stabilization in ('( n) follows rrom stronger estimates; using the method of Bernstein, it is not hard to show thal 1(0" + 1(T. x) L 1::: const < 00 everywhere in R I X R. By (26) this means that the trajectory WT II (T. x) IT> OJ is compact in C(n). Thus, (I(T, x) -'> H(x) for T == T, -'> 00, where H is some stationary solution of equation (23), and 0 E Co(n). Then, lirst of all, (25) means that "jj ¢ 0, and, secondly, by (9), (10) HCr) is an even function, which is non-increasing in x > O. Now, since 7} is a function with compact support (by (24) supp 0 C 1-/0 -- L s , /0 + LsD, from the uniqueness of the stationary solution Os ¢ 0 (see subsection 2, ~ I), we have 0(.1') == Os(x). Stabilization of O(r, x) to Os(x) as T -> 00 (that is. on any sequence T,-> (0) also follows from uniqueness of the stationary solution H = Os(x) with the required properties. This concludes the
~oor
\ I.'illl-
I ".1' ") IIHoll/"'-'Il!I,T>O. .
0
Let us present a corollary of ( 17). ..._ - - - -
(r (T +~)
(30)
-_
SSee examples or analysis of degcneratc equations in [20. 30X. 3591·
,~,'
.\
.
~.\.:
~
IV Nonlinear equation with a source
264
Corollary. In Ihe condilions of Theorem 2 U(I, x) , 4 00 as I ~ Til al any poinl (Iflhe domain x E ( L s /2, Ls /2). Thus, inside the "fundamental" localization domain Ilxl < L s /2), a non-trivial solution of the problem (I), (2), where f3 = (T + I and Uo satisfies (9), (\ 0), grows without bound as I -+ Til' Here condition (17) does not preclude unbounded growth of the solution outside the localization domain: this growth has to be at a rate o((T o - I)I//T), i.e at a slower than the self-similar rate. In numerical computations iJ more striking phenomenon was observed: for practically all non-monotone initial perturbations uoCr) in the process of evolution. a thermal structure was formed in a neighbourhood of an extremum point of UoCt). which developed as I ~..> Ti;(uo) < 00 as the self-similar solution (13), (\4); furthermore. outside the localization domain the solution was bounded from above uniformly in lEW. To) (see, for example, Figure 37). It is of interest that an optimal result of this sort can be obtained by combining Theorem 2 and Theorem 3 of § 4. Theorem 3. LeI f3 = (T + I, meas supp Uo < L s , a/l(l assllme Ihal condilions (9), (10) and Ihe cOlldilion of Theorem 3 in § 4 hold. Theil 11(1, x) -+ 00 as I -+ T('I (110) < 00 aI all poinls (If Ihe localiz.mioll domaill w(T 1', ) = 11.11 < Ls/21 and 11(1, x) == 0 el'ervwhere in 10, To) x {Ixl ::: Ls /21· AI all poinls x E R Ihe solulion approaches Ihe selfsimilar one: (31 )
From Theorem 3 in § 4 it follows that II == 0 in 10, To) x {Ixl ::: L s /2), while from Theorem 2 (sec (17) follows (31), and therefore the fact that 11(1,.1) ~ 00. t'l -+ Til in {Ixl < L s /21. 0 Pro(~l
Lemma 3. Under Ihe ahm!e asslllnplions foT;;' ,.
fo(To'~
1)/1',1 0
+ {joT;;' + fo(T o -
1)11/1.
°
WsW> 01 <
00.
m = 1f3 -
supu(l.x) > (f3-I) 1/({3II(To _1)
((T
+ I )1/12(f3 -
1)\ < 0;
I/W II.
rr:R
(33)
Ihere exisIs a conSllI1II O.
such Ihal
> Oil,
supu(t.
x) <
0,(1'0 - I)
(34)
1/1/1 II
lER
Proof Estimate (33) was obtained in Theorem I. inequality (34) is derived by the method used in Lemma I to analyze the S-regime. Existence of the unbounded subsolution v(l. x) appropriate in this case was established in § I. The magnitude of O. here can always be chosen such that lIoCr) and v(O. x) do not intersect. The estimate (32) of the length of the SUpp0l1 follows from Propositions I and 2 of § 4 (using an argument as in the proof of Theorem I' in § 4). 0
Remark. In the course of proof of Lemma 3 we established a stronger result: FIr I < f3 < (T + I and lIny inilial jill/Clioll lIoCt) of compliCI slIpporl 1\'1' !lave Ihe eSlimatl's meas sUPPJ 1I(r,
x)
= ~o(To - 1)'"
+ 0(1 l.
I ~
Til'
(32')
I\'here suPP+ 1I(t. x) = Ix > 0 lu(l. x) > 0). supp 11(1. x) = Ix < 0111(1. x) > O}. Let us show that (32') implies the following important claim (which could not be proved in § I by analyzing an ordinary differential equation): Corollary. LeI I < f3 < (8) is even and unique.
(T
+ I.
Thl'lI Ihl' compactly sllpporled solllliO/l of equalio/l
Proof: If Os ¥ is some compactly supported solution of equation (8), then the corresponding self-similar solution 11.1 (see (7)) satisfies the condition
We shall consider the Cauchy problem (I), (2) for I < f3 < (T + I with initial function satisfying conditions (9), (10). Asymptotic stability of the self-similar solution means that the similarity representation (II) satisfies (12), where Os(f) ¢ 0 is the unique non-trivial compactly supported solution of the ordinary differential equation (8). Existence of the compactly supported function Os has been established in Theorem 2 in subsection 3 of § I; uniqueness will be proved below. We start with some auxiliary estimates.
-
where {;o =meas {{j >
°
4 On asymptotic stability of self-similar solutions of the HS-regime, I <{l
SUppll(l. x) C 1-/0
5 Asymptotic stability of unbounded self-similar solutions
265
(32)
meas sUPPills(l.
x)
== (To
-I)II/
meas supPJOd{j);
so that by (32') sUPPIO.1 = supp .Os = ~:o. Equation (X) is invariant with respect to the transformation {; -, -{j, and, as can be easily verified by a local analysis, admits a unique nontrivial extension from the point ~ = {jo into the domain ({j < fol with (l~+I(~II)
= (ot I)' (fo)
= O. Therefore 0s(?;) is an even solution. Now.
if there exist two different solutions with compact support O~ and {)~.. then (32') ensures that their supports arc the same and therefore O~ == O~" which completes the proo f. 0
*.5 Asymptotic stability of unbounded self-similar solutions
IV Nonlinear equation with a source
266
From Lemma 3 we deduce the following estimates of the similarity representation (II): supp O( 7, t)
c
[-to -
+ ~oT;;') To
(10
,1/
exp(1Il7}, ~o
+ (10 + toT;;') Til
11/
exp(1Il7} I, (35)
sup
(j(7,
(J > 0 11
:=
(f3 - I)
(36)
1M3 II,
~€R
sup 0(7. ~
t)
< 0,:
7:= -In( 1- tlT o ) E 10,
X).
c It
Let us note that from (35) and from the estimates of Theorem 6 in that meas [SUppO(7, t)\(-to. to)\-+ o. 7 -+ 00.
(37)
so large that n c (-~!J.,I;!J.) and 0 0 (1;) ::: O(II;I:/-L) in n. Then by the Maximum Principle H(7, 0 ::: 0(11;1: /-L) in R-t x n, i.e., the problem (38H40) is globally solvable and estimate (37) holds. Thus we have proved the following general statement: Proposition 1. Let I < f3 < (T + I, let lIo(x) he (lfl arhitrary compactly sllpported .!imctioll alld To = T o(uo) < X he the tilllc or existencc of the solution of t!le Callchy pmhlem (I). (2). Thcn.!c}r al! t E 10, To) \I'e ha\'e the estimates sup
*4, it follows (35')
267
1I(t. x) >
(f3 ..- I )-I/I{J-II(T o - t),I/l/3-t,:
\(R
there exists O. > 0, sllch that sUP,,'R 1I(t. x) < 0.(1'0'- t)
meas suppu(l.
x)
I/I{JII;
= 21;0(1'0 - nlfJ ·,,,+111/12I!3-111 + O( I). 1-+
T(I'
Let us consider flOW the equivalent boundary value problem:
5 On stability of the self-similar LS-regime, 0(0,
t) := OIM) == 0(7,
T:t/J-1lIlIMT;;').
t) := 0,
7>
O. ~
E
an.
tEn.
(39)
(40)
Here n is a bounded domain in R. such that supp O( 7. ~) c n for any 7 ::: 0 (such exists in view of (35». The estimate (37) ensures global solvability of the problem, while (36), by force of an easily derived uniform in 11,00) x R bound for (0"1 I )~, precludes stabilization as 7 -+ 00 to the trivial stationary solution
For f3 > (T + I, the self-similar functions Os(~) (let us note, that in general, there is more than one) are strictly positive in R (see § I). The similarity representation (11) has compact support in 1;, but as t -+ To the size of the support goes to infinity:
n
H == 0.
Therefore, since we have shown that the admissible non-trivial compactly supported solution of equation (38) is unique, uniform in R stabilization to it as 7 -+ 00 would follow from existence of a Liapunov function with appropriate properties for the prohlem (38)-,(40). Such a function can be formally constructed using the general approach of [42, 3831. However, it cannot be written down explicitly and admits a representation in terms of a two-parameter family of solutions of the ordinary dilTerential equation (8). This makes verilication of the necessary properties of such a Liapunov function difficult, and therefore we do not consider this problem here; see Remarks. Remark. Conditions (9), (10) were not used in the derivation of (32), (33). Let us show that (34) (or, which amounts to the same, (37» for I < f3 < (T + I also holds without these restrictions. Indeed, let us consider a solution O(~; /-L) of the stationary equation (38) for t > O. satisfying the conditions 0(0; /-L) := /-L > 0, O~(O; /-L) := O. From the analysis contained in the proof of Theorem 2 in subsection 3 of I it follows that there are sufficiently large /-L > O. such that H(/;: /-L) vanishes at a point /; := /;1" We also have that~11 -+ 00 as /-L -+ 00. Lei us choose /-L > 0
*
f1 > u + 1
meas suppO(l.l;l ~ (To -
tj"!fi-IlTllil/121{3wIJI -+ 00,
t
-+
I'll'
It is not particularly hard to show that under the assumptions (9), (10), 0(1, g) is uniformly bounded; for f3 > (T + I we have the estimate (34) and, therefore (37). This is done as in the case f3 ::: if + I, using the results of § I (see Lemma 4).
By Theorem I we also have the lower bound (36), so that if 0(7, g) -+ 8(1;) as 7 -+ 00, then 8 ¢ O. However, the following difficulty arises in the analysis of the behaviour of 0(7,1;) as 7 := -In(1 - tlT o ) -+ 00. Unlike the cases of HS- and S-regimes nothing so far stops 0(7,1;) from stabilizing to the spatially homogeneous solution~ of equation (38), 0 == (f3 - I I/I {3-I). That would mean that the asymptotic behaviour of the blOW-Up process does not follow a self-similar pattern. Sufflcient conditions of non-triviality of the limiting function 0 (8 ¢ (f3 - I)wIM3 .. II) are given by Theorem 4, where we have denoted by Os(l;) the elementary solution of equation (8) constructed in Theorem .3 of I.
r-
*
(T + \, cOlJditiolJs (9), (10) are satisfied, alJd To := To(uo) < is the hioll'-llp time or Wl unbounded solution ()r t!le problem (I), (2). Furthermore, leI Ihe ilJitilll.!illletiolJ un(.r) he such thar TI1/I{J- 'luo(I;T;;'J inlersects the
Theorem 4. LeI f3 > 00
9This occurs. for example, for
IT
= tl (sec Remarks).
IV Nonlinear equation with a source
268
269
§ 6 Asymptotics ncar singular point
shall show that under certain restrictions on 1I0(X) and f3. the combustion process in the LS blOW-Up regime leads to unbounded growth of the temperature as I ~ To at one singular point only, that is meas WI == meas Ix
111'0
. illliI POIllIS
7· 01! 1/3
Ii 1I0() ()
I > 0.1'( () ).•I" zen 1I'e Iw\'e Ill'
_~_
r'\ ... 1
eslimale 11(1,0) >
R N I II(T I) , x)
= 001 = 0
(see Figures 43, 44). Let us recall that this is a property of unbounded selfsimilar solutions of the LS-regime. existence of which was established in ~ I for all IT + I < f3 < (IT + I j(N + 2)/ (N - 2).;. Here we shall consider non-self-similar solutions. Let us introduce a class of functions 1I0(\X\) with compact support, for which we shall show that meas WI. = O. As 110 we shall take functions U(\xl; Uo). which satisfy the stationary equation
Fig. 63. II ( ~.) exael I y at . 11.1' .!.IlIlellOn
E
Os(O)(To - t)~I/lfJli,
IE [0.
To).
(41)
(rN . 1UaU')' + Ui i = 0. r = Ixi
U;.(O; Uo) = O. U((); Uo)
= Uo.
> 0.
(1)
Uo = const > 0.
f3 < (IT + I)(N + 2)/(N - 2)+. U(lxl;Uo) vanishes at some point r = ro(Uo) > 0. Let us set U(!xl;Uo) == 0 for Ixl ::: ro(Uo). Let IT + I < f3 < (IT+ I j(N + 2)/(N - 2) ;. Let us consider the Cauchy problem
It was shown in subsection 4.1. ~ 3, that for
Proof In the situation of Figure 63, (41) follows from Proposition 2 in ~ 4. Indeed,
considering the unbounded solutions 11(1, x) and v(t, x) == ".1'(1. x) we have that N(O) = 2. Since N(I) > always. and the functions II, v are even in x, then N(t) = 2 for all I E 10, To). This means that
°
sup II ==
,
11(1,0) >
supv ==
,
(2)
11.1'(1,0),
from which (41) follows.
(3)
o
In the conditions of Theorem 4 ()( T, 0) > 0.1'(0) > (f3 - I) 1/lfJ- II for all T ::: 0. and therefore H(~) '¥= 01/. In ~ 6 we shall obtain a pointwise estimate, which precludes stabilization of OtT, ~) to a spatially homogeneous solution. Therefore one of the main difficulties which arise in the proof of stabilization of 0(7, .) to (}sU as T ~ 00. has to do with lack of a uniqueness theorem for a non-trivial self-similar function Os of the simplest form. Another difficulty, mentioned in subsection 4, is of constructing a good enough Lyapunov function. See Comments.
Then II = 11(1, Ixl) is monotone decreasing in Ixl and is unbounded: 11(1, 0) ~ 00 as ( ~ I'll (see Theorem 3 in ~ 6. Ch. V. and the Remark following it). Moreover, the solution, which has compact support in x. is critical: 11,(1, x):::
(I. x) E (0,
To) x {x ERN
111(1, x) >
see ~ 2. Ch. V. This means, in particular. that for each x (tinite or infinite) limit II(TI;·x) = li'l1 l1 (1,x). ,---1 0
E
OJ;
(4)
R N there exists a
Our goal is to prove that 11(1'0 ,x) < 00 in RN\IOj. A lower hound for II(T;;. x) is proved relatively easily using the method of stationary states. The following assertion will be proved in ~ I. Ch. VII for quite general 110 = lIo(lxl). Theorem 1. Le(
IT :::
§ 6 Asymptotics of unbounded solutions of L's·regime in a neighbourhood of the singular point This whole section is devoted to the proof of effective localization of unbounded self-similar solutions of the Cauchy problem (0.1). (0.2) for f3 > IT + I. Below we
Derivation of an upper bound for lIe T(i ' x), which proves validity of the equality meas WI. :::: 0 and the fact of effective localization itself. is accomplished by comparing u(l, x) with the self-similar solution lI.I'(I, x) :::: (To - I) 1/1{3II(}S«(j, ~
= Ixl/(To -
1)111,
(6)
where 11/:::: 1(3 - (IT + 1)1/[2«(3 - 1)1 (existence of the function (}s(~) ::: 0 has been established in Theorem 4 of I). Solution (6) is effectively localized; in particular, for IT + 1< (3:::: (IT + I)N/(N - 2)+ (Theorem 5. I)
*
*
illls(l, x)
..
- - - - . > 0, IE (-00, To) x
RN.
,
ill
(8)
IT :::
O.
IT
+
inequalities of the form of (10) are derived in I, eh. VII. Below we shall briefly discuss the main idea behind the proof. N Let us fix A E (0, lIS(O. 0) I. The self-similar solution (6) is defined in R for all I E (-00. To), such that, moreover, liS ~ 0, (11.1'), ~ 0 as I ~ -00 uniformly on every compact set in R N Therefore there exists 10 :::: 0, such that 1I~(lo, Ixl) intersects U(r; A) only at one point for I' > O. However, 11.1' and U(r; A) are classical solutions of equation (2) in (10, To) x Ilxl < ro(All and 11.1' > U = 0 for r = ro( A). Therefore the number of intersections of 11.1' and U cannot increase in I, and thus at time 1= I. :::: 0, when 11.1'(1.,0) :::: U(O;A), we must have the inequality ".1'(1., 1') > U (r: A), r > O. By (7), (10) follows from that. Thus, U(O; U o) > 11.1'(0.0) and the functions U(r; U o ) and 11.1'(0, r) intersect. We shall show that there is precisely one intersection point. Assume that this is false, and that there are several inersections. Let us consider the family of stationary solutions IU(r; A)l. For all A < lIs(O, O) the functions U(r; A) and IIS(O, r) do not intersect (see (10». Obviously, for sufficiently large A > 0 there is only one intersection (this follows from well-known properties of the functions U(r; A) for (3 < (IT + I)(N + 2)/(N 2) \; see 3). Therefore by continuous depende~ce of U(r;A) on A there exists A :::: A. > 0, such that the curves 11 :::: U(r;A.) and 11 = 11.1'(0,1') in the (II, r) plane are tangent at some point I' :::: r. > 0, and at the tangency point we have 11.1 = U, lI~ = U;. II~~ ::: U~,. But then (il/ill)lIs(O, r.) :::: 0, and that contradicts (7). 0
*
Let us state the main result. Theorem 2, LeT
271
I < (3 ::::
(IT
+
I )N/(N - 2),. Then Ihe sollilion
of
The prolJlem (2), (3) sll1i.l/ies Ihe eSTill/{/le 11(1, x) :::: lI(To ' x) < Cs!xI2/l/j·
(a
1 I II
(9)
The following lemma is a direct corollary of Lemma Lemma 2. Under Ihe condiTions oI Lell/II/{/ I. Remark. From (5), (9) we immediately obtain the estimate C.I > C., where C s is the constant in the asymptotic expansion of the similarity function (}s(~) ~ Csf2/lfJ l'rllll, ~ ~ 00 (see subsection 4, I), Let us prove lirst some auxiliary claims.
Let us assume that at some point is violated. Then by (4) 1I(1,r.)::: (\1',2/ 1/3 1"1111, T,
The functions lIo(r) and 11.1'(0, r) have to intersect, since the corresponding unbounded solutions have the same blOW-Up times, and 110 is a function with compact support (see Proposition 2 in 4). Let us prove now that
Proof
*
U(r; A), r > 0;
°
< A :::: 11.1'(0,0).
( 10)
< I < To.
11.1'(1.1')
To,
I'
= r.
> 0,
will have different blOW-Up times. More general
(II)
Let 11.1'(1, x) blow up at the same time as 11(1. x). Let us compare these functions, considering them as solutions of boundary value problems for (2) in the domain (I,. To) X iLl" where ltI, :::: (Ixl < r.}, illLl, is the boundary of ltI,. From (8), (II) we have 11.1'(1, x) < CslxI2/I/H",11I _ 11(1, x) in (I., To) x ilw.. (12) From Lemma 2 it follows immediately that
It is clear that the condition U(O; Uo ) > IIs(O, 0) will follow from that, since in the
opposite case 11(1, r) and
I :::: I, <
lI.dO, r) inlersecT (in r = Ixl!
exaclly (II one poinl.
IIS(O, r) >
*4.
> 0, lind Ihere/clre
Proof oj" Theorell/ 2.
Lemma 1. LeT
and Proposition 2,
11.1'(1 •• x) < u(l" x), x E iLl •.
(13)
~
IV Nonlinear equation with a source
272
But then u, Us have different blow-up times (see Proposition 3 in ~ 4). Indeed. from (12), (13) it follows that there exists T E (0, To--t.), such that IIS(t,+T. x) ::: u(t., x) in w•. By (12) and the Maximum Principle it means that IIS(t + T. x) ::: lI(t, x) in (t., To - T) x (I) •. Passing in this inequality to the limit as t -,> (To - T) -. we obtain the inequality IIs(1'(). x) ::: II(T o - T. x) in (I)., which is impossible. since IIs(T
o' 0) =
00, II(T o - T.O) < 00.
0
Here we would like to stress again that all the assertions of the intersection comparison theorems of ~ 4 are applicable to comparison of radially symmetric solutions of equation (2) with the same interval of existence. As an example of a fairly general application we shall now derive an exact "self-similar" upper bound. We shall consider the Cauchy problem for equation (2) with a radially symmetric initial function II(D. x)
From Theorems I, 2 we immediately have
Theorem 3. I. Let IT
f3
+ I + 2/N :::
Theil for (/Ily.fixed p ?: 1f3 (3) sllti,~fies the condition
-
(IT
0: (IT
+ I )(N + 2)/(N - 2)
+ I) [N/2
?: l,
E
lIu((,·)II/I'III'I' 00,
2. Let
[f3 -
(IT
j
-
> D, solution of the problem (2).
t
-,>
lIu(t, ')11'-"111'1'<11 <
[ 2r~(N/2)
]1/1'
c.\
< p <
x
( 15) III'
N _.
X
E NI I' 1 /11!-IIf,liI 0: 00.
f3 -
(
(IT
+
I)
)
Let us note the two main requirements on 110 = 1111 (I xl ), for which estimate (9) holds. First of all Uo is a critical function, that is, U t ?: D almost everywhere in (0, To) x R N • and, secondly, 1I11(lxl) intersects us(O.lxl) (lIs(t, Ixl) has the same blOW-Up time t = To < 00) only at a single point I' = Ixl > D. As far as the first requiremcnt is concerned, no special problems arise here. The family of critical 110 (I xl) includes, in addition to functions V(lxl, VII) with compact support, for example, smooth functions of the form III1(1xl)
«(T
= A(a" + Ixl")
1/10
1 "111 1.
x ERN: A> 0, (/"> O.
It is casily verified that \l. (1I~~\luo) + IIf; ?: D in R N if AI! I'HI! ?: 2N/If3 + I) I (this is sufficient for criticality of the classical solution: see ~ I, Ch. V).
The functions
1I0Cr) ;;;;;
A exp( -alxl") are also critical, if
Aljmllflll ?:
.., ~a N
exp
{1f3 -
(IT
N } + 1)1----2(IT + I)
?: D, x
E R
IV
(16)
,
where the compactly supported function 1111 ( 1') is non-increasing in I' = \xl ?: 0, sup 1(11(;)'1 < 00. Let To be the finite blow-up time of the solution. From the Maximum Principle, which can be applied to the parabolic equation for the derivative 1I,(t. 1'), we conclude that u(t. 1') is non-increasing in r. First of all, let us note that the elementary intersection comparison with the spatially homogeneous solution u(t) = 8//(1'11 - I) I/II!-Il leads to the self-similar lower bound: SUpll(t. 1')::= 1I(t.0) > 8//(T II - I)
+ 1+ 2/N 0: f3 ::: (IT + I)N/(N - 2)j. Then for allY I) IN /2, E > D, and all t E (0, Til) we have the estimate 1T N!.e"
= 1I11(lxl)
(14)
T(;.
iT
+
273
6 Asymptotics near singular point
. a> O.
However, for critical initial functions uo(x) different from V(lxl. U o ), the question concerning the number of intersections in Ixl of the function 11((. IXI) and the self-similar solution 11.1'((' Ixl) with the same blOW-Up time is a more diftlcult onc.
1/1#
II.
t
E [0. To).
(17)
r ;·0
Indeed, as in the one-dimensional case. solutions u(t. 1') and v(l) must intersect for each t E ID. Til), otherwise by Proposition 2. ~ 4 (its proof obviously holds for radially symmetric solutions of the multi-dimensional equation), they will have different blOW-Up times. The upper bound is proved by comparison with less trivial self-similar solutions. We shall state the most general result. which holds not only for the LS-. but also for the HS- and S-regimes of evolution of unbounded solutions.
Theorem 4. For 8, > all. so that
0:
f3
< (IT
+
I)(N
1I(t.O) o:O,(To-t)
+ 2)/(N-m I/II!
2) 1 there exists a COI1.1't([l/t
II. tEIO.T II ).
( 18)
Proof Proofs of all the three cases, f3 < IT + I, f3 = IT + I, and f3 > IT + I are similar: sec proof of Lemmas I and 3 in ~ 5 for f3 :::: IT + I. Here we shall consider the case f3 > IT + I. As solution l'(t. 1'), having the same blow-up time To as u((. 1'), let us take v(t. 1')
= (1'0-- t)
I/llj'IIII(~:p.), ~
= ,./(1'0- I)"'.
( 19)
whcre the functionll(~:p.) (solution of problem (4), (17) in ~ I) vanishes for all sufficiently large }-L > all at some point ~ ;;;;; ~I( > 0 (see subsection 4.3 in ~ I). Therefore (19) is an unbounded solution of equation (2) in the domain (0. To) x {I' :::: ~I«TII - t)1II}. As shown in subsection 4.3, ~ I, for all f3 < (iT + I )(N + 2)/(N - 2) I ' ~Il - , D and (OIf){(~ll: p.) -> -00 as p. -, 00. Let
N iJ.(t) be the number of intersections of the solutions lI(t. 1') and v(t,r) in the domain II' ::: ~iJ.(Tn - t)1Il}. Then under the above restrictions on lit), we conclude that N iJ.(0) = I for any fixed sufficiently large JL. At the same time the following
condition clearly holds for the support of the solution: supp v(t. 1') C supp lI(t, 1') for I E [0, To). Therefore by Proposition 3, § 4 (it is easy to check that it is valid in this context), it immediately follows that N !Jo(n == I (if N!Jo (t') = 0 at some time I = I', then solutions 1I(t. 1') and l'(t. r) would have different blOW-Up times). from 0 which we obtain (18) with O. = JL. Therefore the estimates (17) and (18) show that in the subcritical case I < {3 < + I)(N + 2)/(N - 2), the spatial amplitude of radially symmetric solutions grows according to a self-similar law. (IT
Remark. From the method of the proof it is easy to see that Theorem 4 is valid not only for the Cauchy problem, but also for the boundary value problem in (0. To) x B u, B u = Ilxl < R} (R = const > 0) with the boundary condition 1I(t. R) = 0 for I > 0; the initial function satisfies the same assumptions.
occurrence of unbounded solutions and, secondly, we shall establish conditions for their localization (or lack thereof). The main results will be obtained by applying methods, many of which are titled to the analysis of semilinear equations of the form (I). This has to do with being able to invert the operator (il/ill - u), as a result of which the problem (I), (2) is reduced to an integral equation of a sufficiently simple form.
1 A general result of non-existence of global solutions We shall start thc slUdy by deriving conditions for unboundedness of solutions of the problem (I), (2) with a general source term Q(II). A great advantage of the semi linear equation (I) in comparison with quasi linear ones is, in particular. the fact that the solution of the corresponding equation without a source term, v,
= ~v. I>
O. x
ERN;
I ~ = --1i0 (47T1) ,- ,
2
exp
1(\
In this section we study unbounded. as well as some classes of global. solutions of the Cauchy problem for semi linear parabolic equations of the form
= Ull + Q(u).
I >
N
O. x E R .
11(0, x)
= lIo(x)
:::
0,
R
x E
; 110 E
C(R
N
),
sup 110 <
(I) 00.
{1\'1 - -'-- } 41
IIO(.\'
+ y) d.\'.
(5)
with the solution of the original problem (I), (2). Let E(p. T) be a sufficiently smooth function. which is monotone increasing in p::: 0, E(/I. T) ::: 0 for all admissible p ::: 0, T ::: 0; £(0. T) = 0 and £(p, 0) == p. The function E has been introduced for an operator (functional) comparison of solutions of equations (I) and (4). By thc change of variables 1I(t, x)
N
(4)
It lUrns out that one can effectively compare the solution of the problem (4)
equations with a source
lit
= 1I0(.r). X ERN,
v(O. x)
can be wrillen down in terms of a heat potential v(t. x)
§ 7 Blow-up regimes, effective localization for semilinear
275
= £(U(t, x), I).
(6)
(2) in terms of the new function U, equation ( I) takes the form
which describe combustion processes in a medium with a constant heat conductivity coefficient k(lI) == I. It is assumed that Q(II) > 0 for II > 0 and that for all s > 0 F(s)
r'-
=I
dry Q(ry) <
00,
and that, furthermore, F(O) = 00 (this property is necessary for uniqueness of solutions of the Cauchy problem; see § 2, Ch, I). Equation (I) with a source term describes processes with an infinite speed of propagation of perturbations, and if 110 'I- D. then 1I(t. x) > 0 wherever the solution is defined. Therefore heat localization in strict sense is impossible here. unlike the case of §§ I, 4. and we have to use the concept of effective /oca!i:::.al;OIl. There will be two directions of inquiry: first of all, we shall clarify the conditions for
(7)
and U(O. x) = lin in R N by the identity E(p. 0) "" p. Then, comparing (7) with the linear equation (4) (see § I. Ch. I), by the Maximum Principle we have that in order to be able to compare their solutions, that is. in order that we have the inequality UU,
x) ::: v(t, x), I;>
it is sufficient for the function E(p.
T)
O.
x ERN,
to satisfy the conditions (8)
*7 Blow-up regimes for semilinear equations
IV Nonlinear equation with a source
276
Lemma 1. LeI Q(u) he a convex function in R+, Ihal is,
(9)
Q" (u) ::: 0, u > 0,
Proof First of all let us note that F- 1 (0) = 00. Therefore it follows immediately fro~ (12) that 1I(t, x) is unbounded, if we can lind I. > 0 and x" ERN, such that F(v(t •. x,,)) - I. ::.: O. or. equivalently.
Then E(p. T) ::::, F
where F-
I
is Ihe fitnelio/l i/l\'(:r,I'e
III (3),
I (F(p)
is a sollllio/l o{lhe s\'slem of i/lequalilies
(16)
F(v(t •. x.))/I, ::.: I.
( 10)
- T).
277
Let us set x.
= O.
(8),
From (5) and the assumption
Uo
N
ELI (R )
we obtain
v(t, 0) :::::: (41T1)" N!2111101IL'IR"I' 1-> 00,
Proof The function (10) transforms the second inequality in (8) into an identity. Let us write the first one in an equivalent form:
and therefore. resolving the indeterminacy in the expression we have nl'(t.O))
lim - - - It is satisfied for T = O. Therefore it holds for all of Q and monotonicity of F.
Operator (10) is thc identity for
T
(-'''C
T
> 0
(T
< F(p)) by convexity 0
= 0,
Lemma 2. LeI Q(II) he a CO!l\'ex fll/lclion, The/l j{JI' Ihe pro/J!el/l Ihe IOll'er hOlllld u(l, x) ::: FIIF(v(I, x)) -lj.1 >
O.
x ERN.
(I). (2) lI'e
III!,
f3
v
'
I
slt2/N
11111 - -
\ .0'
Q(s)
1-> 00,
2iN
= 21TNll u olI{I(R\ V. , I
Therefore by (15) for this class of initial functions (16) holds for x" = 0 and some sufficiently large I,. which entails unboundedness of solutions of the Cauchy 0 problem (I), (2),
Corollary. LeI l' = 0 ill (14), Theil fiJI' allY (2) is IIl1bollllded,
(II)
In the case II > 0 Theorem I defines a certain minimal initial energy needed for the occurrence of finitc timc blOW-Up: Ell/III = (21TNI/)NI2, In fact. for II E R j in many cases all non-trivial solutions of thc problem are unbounded. In the sequel this will be demonstrated for the example of a power type source term. Q(II) = up. Let us note that using the inequality (16), we could establish conditions of global insolvability also for II = 00 (in which case conclusions of the theorem to some extent indicate the possibility of existence of a class of global solutions),
Since F is a decreasing function, inequality (II) is equivalent to the following one: (12) F(V(I, x)) - F(u(t. x)) ::: I. I > 0, x ERN. ::=
0
= 21TNlluolI{I-/R'
as
haFe
where v(t, x) is delermilledfrom (4),
In the case of a sourcc term of power type. Q(u) - I) and (12) assumes the form
I
F(v(t. 0))/1
> I. we have n,l)
=
110
¥-
0, sollllio/l o{lhe problem (I),
,I,I-I! 1(f3
( 13)
These incqualities comc in handy for dctermining conditions of global insolvability of thc Cauchy problem (ll. (2),
Theorem 1. LeI
Q"(II) :::
n filr
II
> 0, alld assume Ihlll Ihe limil 1,lt2/N
lim :.....__ w._ \.0'
= I'
<
00
( 14)
Q(s)
( 15) (I). (2)
has
110
glohal solulions.
III
=
all
+ Il ll
In this subscction we present a detailed analysis of unbounded and global solutions of the Cauchy problem for an equation with a power type source term: A(II)
Some of the results are the analogues of those obtained in 3 for quasi linear equations; therefore they are stated without proofs. Observe that from Thcorem 1 we immediately have that all solutions u(l, x) ¥- 0 are unbounded for f3 E (I. I + 21N), Therefore globally existing solutions are possible only for f3 ::: 1+ 21N (in fact they do nOl exist for f3 = I + 21N either},
We shall start the study of the problem (17) by constructing unbounded suband global supersolutions. These provide explicit conditions of local or global solvability. I Conditions ot' global illSoll'ahilit." ot'rhe prohlelll
Construction of unbounded subsolutions of problem (17) allows us to derive a sharp upper bound on the time of existence of a solution, which due to the technique of the proof was not obtained in Theorem I. Let us consider in (0. T) x R N the function ( 18)
Let 0 E C 2 (\ 0. (0)), 0' (0) = O. For the function (18) to be a subsolution of N equation (17), it is enough to satisfy the inequality A( u ) 5c 0 in T) x R . Substitution of (18) into (17) gives us the following condition:
ro.
I -;:--l;N.
I
(c~
N
.
I () ' ) '
We shall seek the function 0
I , E, - - -I- 0 - ··0 2 . f3 - I
in the form
f)
(E,)
+ 0f>J O. 'l:, .
(19)
> 0.
= A exp{ -crE,2).
where A > 0,
Then from (19) we obtain the inequality
+
1)(
+ AfJ I exp/a( 1 -
mel::': 2aN + I 1(f3 -
Theorem 3. Lei I < f3 < I + 21 N. unbounded.
I).
(20)
It is easy to sec that it is satisfied for any
. - . •
::=
*
Theorem 4. Lei f3 global solution.
I
::=
+ 21 N,
Uo
E 10,
that is, f3 = I + 21N = 2. With slight modifications the same argument works for any N. First of all let us note that for any initial function Uo '# 0 we can always find 2 constants to. Ao, ao, such that u(lo, x) ::': Ao exp( -ao!xI 2 1 in R Therefore by the comparison theorem, it is sufllcient to prove the claim for functions of the fonn
u(t, x)
::=
P(u)
== (47T1)
I
E,.I, it is sufllcient that the
147T(t-r)r
lJ1(t. x)
= P(O);
U"f
2, f3
exp { -
t
= 2, is equivalent to the following
~/{} uo(y) dy +
/.' exp{_lt-yj2}(U(r,y))2dYdr. ' . R' 4(1 - r)
(23)
in (17) he such rhar
where T, fr, A lire positive cOllstwlI.I·, (flld fr, A sati.l.h' inequlllit\, (21). Theil rhe soilltio/l lit' problem (17) exists for rime /lot exceedi/lg T.
1(1.
x) :::::
P(l},,(I,
x)). n::::: I.
2, ....
From (23) it follows immediately that for any n 11(1. x) ::':
For convenience, we state an immediate corollary of Theorem I.
L..
::=
Let us form the recutTent sequence of functions
be satisfied. Ito
= 2,
The Cauchy problem ( 17), (22) for N integral equation:
and in order that (20) holds for the remaining E, inequality
'# 0. Then the solution of problem (\ 7) is
It is not hard to obtain this result by reasoning as in 3. subsection 2, that is. by comparing the family of subsolutions (18) with an arbitrarily small fundamental solution of the heat equation. In the case f3 < I + 21N this procedure is comparatively simple. In the critical case f3 ::::: I + 21N it is much harder to do, and therefore it is more convenient to construct an unbounded subsolution by an iterative method, as is done in the proof of the following assertion.
i'
E, > E,
Uo
(22)
a > 0 are constants.
1r{4cr
279
7 Blow-up regimes for sernilinear equatiolls
U"U,
.I), I>
n, x
E
R
2
Therefore if /U"l diverges at least at one point, the original problem has no global solution. Let us show that this is indeed the case. First of all we have
where 111ft) = Ao/(l + 41tOl) , E(t,x)::::: exp\-ao!.,f/(1 +411'(1)1. Let us now estimate other terms in the sequence.
§
IV Nonlinear equation with a source
280
This estimate is valid for k == I. Let it hold for all 1 ::: k ::: M, and let us show that (27) holds also for k == M + I. From (26) we have that
Let us prove by induction that U,,(t, x) :':
2:::
~,,(t)E'(t. x),
1/
(24)
== 2.3 .... ,
IIM+I (I) > - (M
k~"
where functions we easily obtain
+ ('
.In
dT
/I, ;:
0 will be defined below
(1'1
we already know). From (23)
x
1
7T(t·- T)1
exp
I
f; .f ~ II
:': I'IE
+
x ll47T(t-T)!
.lR'
,.1
.
+ 4aol)
I
x
AoE(I.
cxp{_IX-YI"}EIII(T'V)dV. 4ft - T) _.
.
6 M + 1 dT
tl(M + I-/).
(28)
/" I
=
x)
: . = ::.(1. x) = --·--In(1 + 4aol) 24rYo
(25)
The inner integrals in spatial variables are easily calculated for each k and are equal to
It is not hard to verify that L~~, I(M+ I -I) == M(M+ I)(M +2)/6> M(M + 1)2/6. so that (27) follows immediately from the inequality (28) for k = M + I. Thus. as 1/ -+ 00 we obtain the inequality
{_IX4ft-- } (2:::.'.'. /ldT)E' (T. y))" dy :': T)
I
+
(' InM-I(1 .10
r4
.lR'
281
7 Blow-up regimes for semilinear equations
1.2, ... , 1/
(29)
> O.
However, the series in (29) diverges for ::. = I, for example. for .1'=0 I), 1 = I., where Ao --In( 1+ 4aol.) = I, 24ao
(E(I, 0)
==
that is,
Therefore from
(25)
Therefore solution of the problem with an initial function of the form (22) exists 0 for time not exceeding I..
we obtain the estimate
Summing the series in (29), we can obtain an explicit form of the presumed "subsolution" u. (t, x), which has been constructed by the iterative procedure for the critical case f3 = I + 21 N, N = 2. It has quite an unusual spatio-temporal structure:
U'HI(t, x) :':
11(1 • .1'):::
that is, in (24) we can set (26)
u (1,.1')
=
1 An x p {anl.xl" } { 1- --exp Ao {aolx " } 1n(1 -e ,-----I + 4aol I + 4(Yol 24an I + 4aol
Let us show lhat hence we can obtain the following inequalities: k (I
Using inequality (13), derived in subsection I, it is possible to obtain explicit upper bounds on the time of existence of unbounded solutions. In the case of power type nonlinearity, this inequality has the form
. exp {IVI --'-
2
/.
}
lIo(.r+y)dy:::MI
11'/'-1/1/3-11
-
.
(30)
41
. It'
where M = (41T)N/2(f3 - I) 1/1f3-1 I. If this inequality is satisfied at a point (I" x,), then the unbounded solution exists for time not exceeding I,. Let lIo(.r) be an elementary perturbation: lIo(x) = 8 > 0 for all Ixi < a < 00, 2 lIo(x) == 0 for Ixl ~ a, Using the estimate exp{-IYI /(4t)} ::: (I - lyI2/(41))+, we obtain a lower bound for the integral in (30):
I (
1(1, x) >
-.111
min{a,
1(1) =
IVI2)
1--'41
lIo(.r
(here we must have I, > a 2 /4, that is, the energy IIl1oll!.1 IR") must not be too high) . This estimate is valid, for example, if IIl1olll'iK'l ::: 2M/a. Let us now consider the case I' ::: a 2 /4. Then 1(1) = 21 1/2 in (31), and the solution of the inequality (30) has the form
The estimate To :::
+ y)dy,
I.
holds if
I' :::
a 2 /4, that is, if
(31 )
1
21 /2}. 2
Clearly, in this case we can set x. = O. Let us assume initially that I. > a /4. Then 1(1.) = a, and the upper bound for the time of existence of the solution is determined from the equation
II l1ollt
l
(RN) -
M1 -1-
= MI
M
11'/2 1/1f3·-11
,I
11' + 2
3 Glohal solulions .lcJr f3 > I
+ 2/ N
We shall seek a bounded supersolution of equation (17) in the form (32)
11'/2
= 2rtN/2)(N a 1T 8. + 2)
For certain f3 this equation can be solved exactly. For example, for f3 = I + 2/ N
where 8+ (/;) = Aexp( -ae I and T, A, a are positive constants. Substitution of (32) in the condition A{tt+) ::: 0 results in an inequality, which can be brought to the form a(4(l' -
I)e + Af3
1
exp{a( I - f3lel ::: 2aN - 1/(f3 - I),
gE
R+.
(33)
From this we obtain the restrictions on the parameters A and a. First of all, the right-hand side of (33) must be positive, that is
This formula is correct if I, > a 2 /4, that is, M < IIl1oll/IIK'1 < 4M I/a!
I
+ M.
(34)
a> -------.
2N(f3-I)
If f3
= (4 + N)/(2 + N),
then Secondly,
I,
=
M+M
I
-·-,lIl1oll,.IIKN) lI/tolit'lltN)
4 < -,(M a-
+ M I )·
Let us note that this estimate shows that for f3 = (4 + N) / (2 + N) < I + 2/ N solutions corresponding to elementary initial perturbations with arbitrarily low energy,
A <
-
(
I)
2aN - - - -
f3--1
1/{f3-11
I - 4'
a<-
and from these inequalities we have the restriction f3 > I proved
+ 2/N.
(35)
Thus we have
~
IV Nonlinear equation with a source
284
7 Blow-up regimes for semilinear equations
Theorem 5. Let f3 > I + 2/ N alld let the illitial jimctioll Uo be such that uo(x)
:s
r!/(/3.!IAexp{-alx\2r- I }, x ERN,
~
illR j xR
(36)
N
I I I I I ! I I
I I I I I I
In this subsection we move on to a description of particular properties of unbounded solutions of problem (17): their spatio-temporal structure for timcs close to the blow-up time. A fundamental property of blow-up regimes, which does not depend on precise initial functions, is the property of localization. Equation (17) describes processes with infinite speed of propagation of perturbations, therefore, as in the case of the boundary value problem for the heat equation without a source term (see § 4 of eh. III), we shall introduce the concept of effective localization of combustion. Definition, An unbounded solution of the Cauchy problem (17) is called effectively localized if it goes to infinity as t -. Tii (To < 00 is the time of existence of the solution) on a bounded set
In conclusion, let us observe that the stable set 'W constructed here consists of functions Uo which decay exponentially as Ixl --> x. As in subsections 4, 5 of ~ 3, for f3 > I + 2/N we could construct a different set 'l(i with a weaker (power) decay rate of uo( x) at infinity. In this case the boundary of 'IV consists of global self-similar solutions of equation (17), which will be considered in subsection 2.6.
Oh:::
~
I I I I
where 1', A, a are constallts, the two last ones sati.living inequalities (34), (35). Theil problem (17) has a global solution, and furthermore
U(t'X):S(T+tr'I;I/IIIAexp{-;'I~:}
i i
285
I
Fig. 64. Effective localization
(L r
is the localization depth)
(that is, Lr is the extent of the domain in which the solution grows without bound as t --> To: see Figure 64). If u(t, x) becomes infinite at onc point, then L r ::: 0, which corresponds to the LS blOW-Up regime of combustion. Evolution of unbounded solutions of the problem (17) proceeds for (3 > I. as a rule, precisely in the LS-regime and u(t, x) --> 00 on a set OJ/. of measure zero. This is indicated, for example, by the estimates obtained in Theorem 2, in which we derived unbounded subsolutions that do evolve in the LS-regime. And, of course, this conclusion is corroborated by numcrical computations. In Figure 65 we present results of one such computation. It is clearly seen that in the blOW-Up process there arises a spatio-temporal structure with ever decreasing half-width and a conspicuous unique maximum in x of the spatial profile. Let us consider the spatio-temporal structure of unbounded solutions for times close to blow-up time. For that, by analogy with the quasi linear case (§ I), we can consider unbounded self-similar solutions
x) ::: oo},
11
where the function O.V< 1;) > 0 satisfies the ordinary differential equation which we shall call the localizatioll domain. N If, on the other hand, WI. is an unbounded domain (for example, W/. ::: R ) then we say that there is no effective localization. For our purposcs thc above definition is sufficient. In thc gencral case the N following blOW-Up set should be considered: lh::: (x E R 13 tIl --> To and x" --> x, such that u(t", x,,) -- 00 as II ,""""'> 001, which by definition of an unbounded solution is non-empty for bell-shaped data. In the following an effectively localizcd combustion process will be called simply localized. In the one-dimensional case it is convenient to introduce loca!i:atio/l
I f,N
----(~
Lr
= Il1cas (x E R! u(T,;, x)
::: 001
N
I
" , I 0,) - IO\~ - --01" 2· (3 - I
-,
(/,(O)
= 0,
0.\( (0)
+ 0\..13
::: O. f, > 0;
= O.
(39)
(40)
If we assume that the self-similar solution 1/.1' describes characteristic properties of LS blow-up regimes, then the amplitude of the solution 1/11/ (t) and the half-width of a symmetric domain of intensive combustion I'·f(r) can be estimated as t --> To according to I/II/(t)
vanishes at some point g = gl > gl), and is monotone on (go. gI)' Therefore (39), (42) has no non-monotone positive solutions having a point of minimum. We remind the reader that in the proof of Theorem 3, subsection 4, I, we made essential use of non-monotonicity of solutions. It is appropriate to recall that the analysis of that subsection concerning the self-similar equation (39). linearized around the homothermic solution H == «(3 I) I /I/!. II, shows that it has no non-monotone solutions. Let us observe that this argument correctly indicates non-existence of nontrivial solutions of the problem (39). (40) for any I < (3 ::: (N + 2)/(N - 2)+ 11971. Therefore for those values of (3 asymptotic evolution of unbounded solutions follows non-self-similar patterns. In distinction to (41), the half-width of the combustion domain changes as t ~ T(; according to I"f(t) ~ (1'1) - t)1/2lln(Tot) I' j2 (see Remarks). In the case (3 > (N + 2l/(N - 2)\ the problem (39), (40) may have non-tri~1al solutions. Then as t ~ Ti; , the solution evolves according to the self-similar laws of (41).
*
8
•
• • • • • • • • •
6
.5
•
•
Example. Let (3 = 2. Then for all N E (6,16) (note that here (3 > 1 + 2/N). there exists a solution of the problem (39), (40)
2 -
4
(43)
4'
• •
• ••
•• ••• •• • •C' .~
~
where 12
?
14
x
Fig. 65. Numerical solution of the problem (17) for {3 == 4. N == I: I: 12
== 0.269,3:
1.1
0.291,4:
14
== 0.299, 5:
I",
== 0.3006. 6:
1(,
II
== 0.236, 2:
= tUOI8, 7: 17 = 0.3022
There is, however. one significant difference between this and the quasi linear case. Proposition I. Lct N solwiol/s 11.1'(') > O.
=
I. ThclI
Fir
al/)' (3 > I the problcm (39), (40) has
I/O
aN, AN,
B N are positive constants: liN
= 2110( I + N/2) 1/2 -
AN
= 48110(1 + N/2)1!2 -
BN
= 24[(1 + N/2)1/2
(N
+ 14)\, + 14)J.
(N
- 21.
Non-existence of self-similar solutions in the one-dimensional case requires a substantial modification of the method of proof of effective localization for unbounded solutions. 5 Proof' of' ('fli'ctive lo('ali;'lItioll iI/ the rllle-dimellsiOlral
CliSC
Thus, for N = I the problem (39), (40) has no solutions. However, (39) admits solutions 11.1'«(;) with the asymptotics (44)
For the case (3 = 3 it has been proved in 1219]: for arbitrary (3 > I it has been established in 131 (sec also II, 2J, where I < (3 < 3). In the "course of the proof (sec 13]) it is shown that every solution of equation (39) in the domain
where 1<1«(;) ~ 0 as (; -..... 00 and C > 0 is a constant. Let us fix a C = C, in (44) and extend the solution into the domain of sufficiently small f Then. obviously.
*7 Blow-up regimes for semi linear equations
IV Nonlinear equation with a source
296
e
---
B
----
Fig.6!l. Solutions of equation
for I
(76)
<
/j
Fig. 69. Solutions of equation unique <
o
Let us also observe that for all f3 > I. the required solution than the decreasing solution of the equation without a source. ·,2
--
f3 - 2 I"~ 2(f3-1)"
----
-
I f3--1
-.,-.----.
I'
()~.
0:::.
-
>I). f( »)
,
(/s
= 0".
is not smaller
.
*
which was considered in detail in 4. Ch. Ill. There it was shown that the function 0 is determined from a certain algebraic equality. which allows us to derive lower bounds for 0,(0. Hence, for example in the case I < f3 < 2. we have
f(~) ~
meas supp 0, > meas supp I
J-f3'
0:::
2(
~-=-I)
(76)
for 11
>
2. Solutions of the prohlem
(70). (7\)
arc not
2
Therefore for f3 > 2 the solution is strictly positive in R+.
.1
297
12 {Jill 211!
which is not analytic at ~ = 0 and is shown in Figure 69. Numerical simulations of the problem (56). (57) indicate that the spatio-temporal structure of a.S.s. (68) does correctly describe the asymptotic properties of unbounded solutions and that the function (72) is realized in the LS-regime as r -+ Til' Validity of this "slicing" of equation (56) and passage to a first order equation has been checked by numerous numerical experiments. which demonstrated asymptotic convergence of the unbounded solution of problem (56). (57) to a.s.s. (68). In numerical computations. similarity representation of the solution UU. x) was determined from the formula
I Ii
(77)
Quite a good understanding of qualitative properties of the solution (/, can be gleaned from considering Figures 68. 69. which show the field of integral curves of the equation (76)
which is equivalent to (70), The thick linc denotes the solution Ii, ::: () with the asymptotics (72). which satisfies l:onditions (71). Here it has to be said that for f3 > 2, apart from a solution of the form (72). there exists another family of admissible positive solutions with different asymptotics:
where IIUlle, = sup,UU.x). O,((l) = (f3 - l)-I/II! II. As always, the spatiotemporal structure of a.s.s. (68) is implicit in the representation (77), and if uU. x) evolves according to the rules of (68). we must have the limiting equality lim
{}U, () :::,~
O,tCl.
(78)
,.
IllIllt,''>C
We emphasize that in this case it is hard to accomplish numerically the usual similarity normalization as. for example, in 4. Ch. III. since here the blOW-Up time is not known (/ priori. Convergenl:e in the sense of (78) to a.s.s. (68) for suflkiently general initial functions has been established numerically for different f3 E (1.21. For f3 > 2 (LS-regimc). as we know. the similarity funl:tion 0, = (I,(lgl) in (68) is not
uniquely defined. As an example (Figure 70) we show the results of similarity transformation (77) of the solution of problem (56), (57) for f3 = 2, when H" has a very simple form; see (74). Convergence (78) is clearly seen already when the solution grows by a factor of 10-20.
4 Three Iypes 1II'III/hllUI/ded ,1'01111;1111.\'
From (68) it follows that the spatio-temporal structure of a.s.s. depends on the sign of the difference f3 - 2. Thus, if f3 < 2, then U,U, x) --> 00 as 1 --> TIl simultaneously in the whole space: this is the HS-regime, and there is no localization. Thus we have found a semi linear equation with unbounded HS-solutions (recall that the equation u, = tlll + IIIl does not admit such solutions). Figure 71 shows the results of numerical computation of the one-dimensional problem (56).
L\
o
=
Fig. 70. Numerical verification of asymptotic staoility in the sense of (78) of a.s.s. (68) for {3 == 2 ({ 0= x), N =" I: I: II = 2.82. 2: I~ == ~.03. ~: '.1 == ~.12. 4: I~ == ~.16. 5: '5 = ~.18
j' .•
=
Fig. 71. Numerical solution of the problem (56), (57) for (3 1.~5 (HS-regime). N I: I: 11= 2.10. 2: I~ = 2.87.~: '1 = ~.OO, 4: I~ = 3.11. 5: '5 = ~,19. 6: 16 = 3.28. 7: 17 = 3.34. 8: IK == 3,40. 9: I') = 3,45, 10: 110 = 3.51
(57) for f3 < 2, which are in good agreement with (68), in particular, as regards the change in the amplitude and half-width of the solution. For f3 = 2. as follows from (68) and numerical results. the combustion process evolves in the S-regime; 1I(t, x) becomes infinite on a bounded set. In the case of a symmetric elementary initial perturbation, the localization domain of the Sregime is the support of the function (74): WI = lIxl < 1T) (Figure 72). Similarity transformation of this computation is shown in Figure 70. If localization domains corresponding to different perturbations are disjoint, combustion inside each one of them proceeds almost independently as 1 --> Til (Figure 73).
Remark. Proving localization of sufficiently arbitrary unbounded solutions of equation (56) for f3 ~ 2 is an interesting mathematical problem. In particular, in the one-dimensional case for f3 = 2. one could use to that end the method of intersection comparison of the solution under consideration and an interesting exact non-invariant solution (it is not invariant with respect to Lie-Backlund transforma-
rll:~;\S
~
IV Nonlincar cquation with a sourcc
300
asymptotic behaviour close to the blOW-Up time
U(t,.r)
(/>(1) = (1/2)(1'0 - 1)'111
80
111(1)
Therefore. as
I
->
301
7 Blow-up rcgimcs for scmilincar cquations I
+ O((T o -
= 1+ O((T o -
= To: I)lln(To - 1)1)1,
1)!ln(T o - 1)1).
To
50
U.(I. x)
= (To -
I) I cos 1 (.,j2)
+ 0(1 In(T o -
1)1),
that is. as the blOW-Up time is approached, the exact solution U, (I, x) converges uniformly (after the corresponding similarity transformation) to the approximate self-similar solution (6X). (74). However. in the general case the problems of localization for (3 = 2. N > I. remain for the most part open. If (3 > 2. then (6X) ensures unbounded growth of the solution at one point only. This is the localized LS-regime: a specific example is shown in Figure 74. It is clearly seen that everywhere, apart from one singular point. the solution U (t, x) of (56), (57), is bounded from above uniformly in I by some limiting profile
20
U(1'I~' x) < 00. x
o
x·
.9
Fig. 72. Numcrical solution of the problcm (56), (57) for f3 "" 2 (S-n:gimc). N = I: I: II
tion groups). This exact solution. which is 27T-periodic in space, has the following form: U,(I. x) =
4)(1), fll(l)
=I
3.
In those figures, dashed lines indicate the motion of the half-width XcI (I) of the thermal structure, combustion of which is initiated by the same perturbation. First the amplitude of the solution U(I. x) becomes smaller and the half-width increases. which corresponds to the process of spread of the non-resonance perturbation. Then. approaching linite time blOW-Up. x,. ( (I) starts to change in accordance with a.s.s. (6X): x,'I(I) ~ (To - I)W 11/121/1111, I -+ T(l' In particular. for (3 = 2 (S-regime) the half-width stabilizes, which can be clearly seen in Figure 72. Conclusions concerning stability of a.s.s" which we presented above. are in good agreement with the qualitative non-stationary averaging theory. As in 2, we shall seek an approximate solution 1I,(I, x) in the form U,(t,x) = fIJ(I)/J-({;). (; = x 14) (I ): I > 0, (; E R N , and let us demand that U. satisfy the conservation
*
laws
satisfy the system of nonlinear ordinary differential
·,DW,)dX=O. ./.R'
= -
I >
O.
as can he easily checked. In fact. this dynamical system is precisely the semilinear parabolic equation on the linear subspace'}' II. cos x J, which is invariant under the nonlinear operator U" + (U,)" + U1 . This system is equivalent to the lirst order equation It is easy to show that this equation has a family of trajectories corresponding to various unbounded solutions of equation (56) for N = I, (3 = 2, with the following
r
JR'\
U.DW,,)dx=O.
Then we arrive at a system of ordinary differential equations for the amplitude and the half-width (p(l):
111(1)
= IJlfI1 2 r/)N 2 + 1'2 11,ll cp N, it (1IJ2r//v), = _1'\fI12
(79)
where IJ , (i = 1.2. ,., .5) arc some functionals of /J- (their exact form can be easily wrillen down). From the system we pass to the single equation
problem. Finally. the third critical value is f3 = 2 + 2/ N. For large amplitudes r/J we can neglect constant terms in the numerator and the denominator of (80). As a result we have the approximate equation
(80)
where (/1, (/2, hi, h2 arc some constants, which we take to be positive based on natural requirements on the behaviour of the trajectories. Equation (80) is more complicated than the one considered in ~ 2. The main difference is essentially the following: (80) contains three independent critical values of the parameter f3, which "control" the general structure of the phase plane. 2; the criterion f3 ~ 2 determines the presence, or lack, of localization First is f3 of unbounded solutions (in the cases f3 ::: 2 and f3 < 2 the behaviour of the integral curves is completely different). Secondly, f3 = I +2IN; for f3 < I +2/N equation (80) has no globally defined trajectories, while on the other hand, for f3 > I +2/N there arc trajectories to which there correspond global solutions of the original
=
which is the same as the one considered in ~ 2 for (T = I. Therefore the phase plane behaviour depends on the condition f3 ~ (r + I + 2/N = 2 + 21N. For f3 < 2 + 2/N all unhounded trajectories converge to the "separatrix" generated hy a.s.s. (68):
'11 (1)
~ (To -
II' :::: 4)
2/11i 21.
III
tr
II. t{J(t)
~~ (T o - 1)111-21/121 IJIII
l / IIJ
--> 00:
(81)
as t -> To. On the other hand, if f3 :::: 2 + 2/N. then the asymptotics of unbounded solutions as I -> I'll is a non-self-similar one (see ~ 2).
304
~
IV Nonlinear equation with a source
Fig. 75. Intcgral curves of equation (l'O) for {3
E
(1.2). {3 > I
+ 2/N
= exp{(T o -- n I/I/! "fI,(~)I- I. § = 1.11/(1'11 ..- nil! :2111:2(/3 III.
11.,(1.1)
2 (the case f3 > I + 2/N for N
Fig. 76. Integral curves of equation (l'O) for {3 '/1, '" (h 2 /
Most of these results are in good qualitative agreement with the conclusions we arrived at earlier. The estimates (H I) give us, in addition. quantitative agreement for I < 13 < 2 + 2/ N, which supplements the evidence that the construction of unbounded a.s.s. (6H) is valid. Figures 75-77 show schematically the fields of integral curves of equation (XO) in, respectively. the cases I < 13 < 2 (HS blow-up regime). 13 = 2 (S-regime). 13 > 2 (I~S-regime). In all three figures the parameters 13, N have been so chosen, 'tbat, first, 13 > I + 2/ N, so that there is a class of global trajectories. and. second. 13 < 2 + 2/N, that is, unbounded trajcctories behave according to (X I). We denote by I/Js the separatrix, which separates fami lies of global and unbounded trajectories. and by I/!O and III"". the isoclines of zero and infinity, respectively. Thus. localization of unbounded solutions of the problem (56), (57) occurs for 13 = 2 (S-regime) and 13 > 2 (LS-regimel. while for 13 <: 2 there is no localization. Clearly, this classification remains the same if we go back to the original problem (54), (55). Setting 11., = exp( V,) - I, we obtain the following expression for a.s.s. of equation (54):
305
7 Blow-up regimes for semilinear equations
lH~~~~~~
o
Fig. 77. Integral curves of equation (l'Oj for {3
>
2).
rp
>
2 (the case I -I- 2/N
< {3
<:
2 + 2/N)
(X2)
It is not hard to see that it satisfies the following nonlinear first order equation of Hamilton-Jacobi type:
(X3)
Let us note that in the original variables t . .I. II the evolution of the blowup process looks different. In particular, from (X2) it is not hard to obtain an expression for the dependence on time of the half-width of the structure, 1,,/(1).
§ 1. Numerical examples of evolution of S-, HS.., and LS-regimcs are taken from
for all f3 > I. (84) In this sense there is no difference between the three finite time blOW-Up regimes. At the same time, for f3 < 2 the solutions are not localized and 1I,(t, x) ......,. 00 in R N , I --l- To. On the other hand, for f3 :::: 2 solutions are localized. The estimate (84) holds in all the cases. Therefore in numerical simulations of the original problem (54). (55) the difference between the HS- and S- . LS-regimes becomes apparent only once the amplitude of the unbounded solution has grown significantly (by order of tens of hundreds). The logarithmic change of variable U = In( I + II) removes this inconvenience (see Figures 71. 72. where in order to identify the blOW-Up regime it is sufflcient for the amplitude to grow by a factor of 5-10). 5 Global a.s.s. For f3 > I + 21 N there exist global a.s.s. of the problem (54), (55). From the method of construction of bounded superso)utions in Theorem 9 it follows that at the asymptotic stage we have to neglect the term IVlJI 2 in equation (56). Therefore the global a.s.s. lJ, satisfy the parabolic equation
asymptotic properties of the solutions of which are well known: see subsection 2.6. Going hack to the original notation, we see that the global a.s.s. II, = exp(U,1 ~ satisfies the following parabolic equation: iJujiJl
= fiu,
-- IVu,1 2I( I + II,) + (1 +
II,)
In#( I +
II,).
(X5)
Therefore equation (54) has the following interesting property: asymptotic behaviour of its unbounded and global solutions is described by vastly dissimilar nonlinear equations of different orders, (X3) and (85), respectively.
Remarks and comments on the literature The first qualitative and numerical results for unbounded solutions of the problem (D. I ). (D.2) were obtained in [349, 353, 391, 89, 92, 268, 2761. These papers also contain a preliminary analysis of unbounded self-similar solutions and flrst formulate the concepts of localization in finite time blOW-Up regimes in heat conducting media with volumetric energy production.
[391, 92, 353 J. The idea of describing non-monotonicity of the functions (Js(~) for f3 > IT + I close to the homothermic solution by linearizing the equation, appears in [349, 891 (see also [90, 267, 268j, which present numerous examples of numerical construction of the families of solutions Wsl for various IT > 0 and f3 > IT + I. N = I). This idea was then exploited in [I. 21. where existence of a flnite set of self-similar functions fJs(~) of the LS-regime is proved for N = I (the methods of [ I. 2J arc different from those of § I). The asymptotic expansion (33) can be proved by the methods of [210] and 139, 40. 41. 370\ (for N = I it was done in II, 21). § 2. The idea of methods of non-stationary averaging is due to the authors of [91]: for more on this method sec also 189.90.2681. Let us note that simplicity, constructiveness and sufflcient trustworthiness of the method make it applicable in a number of other problems, for example. in the study of nonlinear problems of thermochemistry 1561. There are reasons to consider it as a version of the method of radially spherical decomposition of the function space (as opposed to the method of spherical decomposition [335, 336\). § 3. [n the presentation of subsections 1-3 we mainly follow [152]. We notelhat
for IT = 0 Theorem 2 gives the familiar result 1296, 112J (the same is true for Theorem 3). though, of course, the proof in the quasilinear case is substantially different from the semi linear case. Theorem 2 is also tflle for the critical exponent f3 = IT+ I +21N 1138]. Let us brielly mention a moditled simpler argument based on inequality (19) (see a slightly different approach in [2441J. As in the proof of Theorcm 2, there exists a solution liS of (10) such that u ? liS everywhere. Integrating equation (1) with f3 = IT + 1 + 21N over RN we conclude that
~
dl.
.r
/. u(r. xli/x
!
=.
ufJ(r. xli/x
?:
!
.
vf(r. x)dx
=
('
(7'1+1)
Hence. u(r. xli/x ?: (' In( 7' I + I) ......,. 00 as I ....... 00. Therefore. there exists I such that M 1 u(r, x)i/x» I. By comparison we have u(r. x)?: v(r, x) for I > II. where v(r.x) solves (to) with initial data U(rl' x). It follows from asymptotic stability of the self-similar solution (II) (! I071J that for h » II there holds V(r2. x) ~ V,(r2. x) where v, has the same mass MI » I. Therefore. the constant 7)0 in (12) satislies 7)0 = 17o(M I) » I. Finally, the profile V,(r., , x) satisfies the property (19), and hence u(r. x) blows up in a llnite time. The survey [2901 (sec also [2921J cOlllains a large number of nonlinear equations and systems thereof, for which there exists a critical value of the parameter of the source term (in the sense of Fujita [112\). The assertions of subsection 4 were obtained for (T = 0 in 12101. The basis of the proof of Lemma I is the well known Pohozaev inequality [332, 333J: the scheme of the proof is taken from [21 OJ. The family of solutions (50) was discovered in [298j, where the equation fiu /lINt21/1N 21 = 0 was considered.
I»
=.r
Remarks and comments
IV Nonlinear equation with a source
308
Existence of such a family has to do with invariance of the elliptic equation with respect to a conformal transformation; see the bibliography in [221], as well as [220]. Results of subsections 5-6 are obtained using the methods of [125, 127. 1621· In the passage to the limit T - , 00 in (75) we follow an idea from 1178. 1791 where a more detailed analysis is given. § 4. Intersection comparison theorems of subsection I were first applied to study unbounded solutions of equation (I) in 11291 (see also 1130\. where Theorem I' <subsection 2) is proved). The subtler Theorem I is proved in [132. 139]. Let us note that intersection comparison with the explicit solution 11.(1. x) for {3 = (T + I establishes the following upper bound for an arbitrary unbounded solution: 11(1, x) :s sup, 11.(1, x) =" 14J(I)(if/(l) + I )]1/" in (0. TI)) (see 1135]), which is optimal, since it is allained on the solution 11.,(1, x). Furthermore. the same method is used to prove results concerning the structure of the blOW-Up set H,. (see [140]): if .1'1,.1'" E V, =" R\lh and 1.1'1 -~ .1'"1 < L.\, then [XI, .1'"1 C V,. It is found that meas lh :::: L s for any initial function with compact support. An essentially different example of combustion in the S-regime is presented in 11091 (see also 114]), where the boundary problem with zero Dirichlet conditions on the boundary is considered for the equation II, = 11"(11" + II). The authors prove localization of the unbounded solution in the localization domain HI = {lxl ::: 17' /2). Unlike the example considered above, the problem of asymptotic behaviour of the solution in H,. close to the blow-up time, remains largely open. It is proved that 11(1, x) """ A(I) cos x in lh. but the behaviour of the amplitude A(I) as I -+ T1i is at this stage unknown. Note that the conjecture [ 1091 A(I) ~ (T -I) 1 g(l) with g(t) = [In Iln(T -.1)11 10 , 1--> 1', seems to be true. since thc factor g(l) is a natural one for the equation without a source term. v, = v" v", which governs the process 1 in a small neighbourhood of the end point of HI, x = TT /2. where Ill" II" I » 1/ . We invoke now a formal matching argument. There exists an explicit solution of the log-travelling wave type, v.(f, x) = (T - I ) 10/(r/l. TJ;;:;: .1'- TT/2 + Aln(T -~ I) (ef. Ch. II, ~ 12), where / solves the ODE .f'" I" + AI' - II2 = O. Since .I'(r/)'~ (-TJ)lIn(~TJ)IIO as TJ -, -ex;, by slightly perturbing v, (for instance, by assuming that A = A(I) is a slowly decaying function) we have the following OilIer
°
expansion for x""" TT/2: 1'(1, x) """
A(I)(T
--I)
1011n(T -l)lg(llll -
(.1 -
17'/2)/A(I)
Iln(T -1)11·
Then a matching procedure with the inner expansion 11(1. x) """ A(I) cos x ror x""" TT/2 (for instance, by taking II, """ v, at x;;:;: TT/2) yields A(I) = (T _1)-IOg(l). Proof of localization of unbounded solutions for {3 > rT + I (subsection 3) is given in 11291. The presentation of subsection 4 uses results of [1301· As we mentioned earlier. Proposition I, on the number of spatial intersections (or number of changes of sign of the difference of two solutions) is a natural consequence of the Strong Maximum Principle for linear parabolic equations and
309
has been known for a long time: see the first quite general results by 13681 and 1316, 3551. as well as various examples of analysis of the zero set of solutions of parabolic equations in [303. 315. 13. 17 L 175, 180, 2631 and others. Among the general results contained in these papers, let us note those [131 and [2631. where the zero set of solutions of linear parabolic equations is studied under quite weak restrictions on coefficients. The idea of intersection comparison turned out to be very fruitful in the analysis of unbounded solutions of a wide class of equations. The above results were obtained by studying the variation with time of the number of intersections. More subtle results. obtained for different types of equations with a source in [170. 17 L 175. 1801. deal with analysis not only of the number. but also of the character of points of intersection. One of the main results of these papers is the following conclusion. which has a simple geometric interpretation: in certain conditions a point of "inflection" with a stationary solution can arise, as the solution evolves, from at least three intersection points. Such a comparison with a family of stationary solutions allows us, for example. to show that for an equation of the general form II, = (4J(II))" +Q(II). a solution that has become sufficienLly large at a point .I' = XI) at time I = II). can only increase in time: II, (I . .1'1)) :::: 0 for I > '0: see [175. 180]. Other applications of intersection comparison with radial stationary solutions in the multi-dimensional case can be found in 11371. [16 4 1 contains a general description of applications of the method of stationary states (m.s.s.) in the study of unbounded solutions of nonlinear parabolic equations and systems: see also other applications in 1172. 1741 (for other details see ~ I, eh. VII). M.s.s. provides sufficient conditions for absence of localization of unbounded solutions with arbitrary coefficients. ~ 5. The main results were obtained in 1130 I: sec also [1291. Everywhere in ~ 5 we are concerned with the determination of an attracting set 'W of an unstable stationary solution. In the case of equation (23) it has the form 'W = {HI) = Ti/"I1I)(.r). where III):::: 0 satisfies (9), (10) and () < To < ex; is the blOW-Up time of the solution of the problem (I), (2) with the given initial function 110(.1') I· It is of interest that 'W is unbounded and contains functions (1), the difference of which with Hoi in R can be arbitrarily large. It is important to note that 'TV is an infinite-dimensional set. which, of course, is not dense in L". As of now there arc relatively few complete results concerning the structure of attracting sets of unstable stationary solutions of nonlinear parabolic equations. In this direction, let us mention 1226, 15X, 169, 170. 197,3141: results of 147[ (fast diffusion equation in a bounded domain) are discussed in comments to Ch. II. A large number of papers deals with asymptotic stability of regular (without points of "singularity" in time) solutions of parabolic equations: sec, for example 14 2. 378. 158, 162. IX4. 3X.l 241. 242, 10, I I. 20, 21, 22. 50. 107, 20X. 213. 234. 235, 30X. 344, 356, 359 I and references contained therein. We should also like
to mention the interesting ideas of 181 and 1284] concerning stabilization without constructing Liapunov functions. The problem that arises most frequently in this context is that of finding an attracting set that contains a neighbourhood of the stationary solution (that is, is dense). In [1581 we obtain conditions for asymptotic stability of unboundcd self-similar solutions of quasi linear cquations with a source, for which boundary value problcms in bounded domains with moving boundaries were formulated. This ensured asymptotic stability of thc corresponding stationary solutions. Lct us note that the cstimate (20) (or (26)), which plays an important part in the proof of Theorem 2 dealing with stabilization, holds also without the restrictions of the form (9), (10) on the compactly supportcd function lIo(.\} (see 1130, 171 j). Asymptotic stability of self-similar profiles for 13 :> IT + I is proved in 11411. § 6. Results presented here are contained in [131, 1231. Statement I of Theorem 3. dealing with the semi linear equation for IT O. P :> {f3 - I)N /2, has been
=
proved in [3791 (see also 126, 2131 and comments on eh. VI!). Of utmost importance in demonstrating localization in blow-up regimes is the derivation of upper hounds for the solution //(1, x). In § 6 we present an approach bascd on intersection comparison with a localized self-similar solution with the samc hlow-up time. It is, however, not without disadvantages. In particular, its usc throws up a restriction on the maximal valuc of the source parameter 13· In this contcxt, very effective is an idea that first appears in [lORI, where it is used to prove blow-up at a single point for semilincar equations //, == /:1// + Q(//). A certain modification of this approach (sec [172, 173 j), applied to radial solutions of quasilinear equations of general form. //, = /:1cp(lI) + Q{//), which consists of deriving conditions under which ]/I{/, x) == r N Icf/(//)/// + r N F(II) :::: 0 in (0. To) x R+ for a special "optimal" choicc of the non-negative function nil) (it satisfies a certain ordinary differcntial cquation), allows 11731 to prove an estimate of the form (9) for a spccial class of //0 for arbitrary 13 :> CT + I. The same approach to e4uations of general form [172, 1771 provides conditions of localization of unboundcd solutions in terms of the coefficients cf)(//), Q(//). It is interesting that practically in all cases this approach gives upper bounds that coincide precisely with the real asymptotic behaviour of a widc class of unbounded solutions. Questions of ('(llIIp/e/c blow-up, i.e .. of possible extension of a blOW-Up solution for I > T via construction of a certain minimal solution (as the limit of solutions to truncated equations) have becn considered in 1271 for semi linear equations Il, = /:1u + Q(Il). A criterion of il/('(IIII/i/ele blOW-Up for a general quasilinear cquation 1l(J) = (cf)(lI))" + Q(//) has been derived in 11931 via intersection comparison with the set of travelling wave solutions. For equation (2). N = I. it is P + IT ~ I. IT E (-I,O).
§ 7.
In the prescntation of most of thc results of subsections I and 2.1-2.3 we follow 11501. Inequality (12) was obtained earlier by a different mcthod in
311
Remarks and comments
IV Nonlinear equation with a source
[112, 1141, where Theorem I is proved. Theorem 3 is proved in [I 12] (on this see also 1296, 254j). For the cases N = I and N == 2 Theorem 4 is proved in [212]; a generalization of the method of [212], which is used in subsection 2.1, to cover the case of arbitrary N :::: 1. is contained in [22, 254]. For 13 = 3 Proposition I was proved in [2191; the case of arbitrary 13 E (l. (0) was considered in [3, 21· Solution (43) was constructed in [158], using the ideas of [341. where a solution of a similar structure was found for an equation (39) with a sink _0 2 (13 = 2). instead of a source. In the conditions of Proposition 1 the similarity representation fl(t, ~) == (To 1)1/(!3-ll lI (t, ~(To - 1)112) stabilizes. for a large class of lIo(.l'), as I -> Ttl on any set lIxl < ('(To - 1)1/2) to the uni4ue non-trivial solution 0/1 == (13 - Ir l /(!3-11 of equation (39) in R 1169.170.1711. In [170,1711 the estimate lI(t,X) :::: j.L. (To - I) -I;W- II in (O, To) x R was obtained under the following rcstrictions on lIo: sup I/o < 00, lIo is uniformly Lipschitz continuous in R; these conditions are weaker than the ones in [380. 3811. The results of 1170 \' as well as of 11691, were obtained by applying comparison theorems for different solutions 1/ and v. based not only on the time dependence of the number of their spatial intersections, but also on the nature of those intersections (for example, in [170J under ceralin restrictions. a theorem of the following form is proved: if w(Jo, xo) == I/(Jo, xo) v(to. xo) = O. then lIJ,{to, xo) i: 0). These theorems are fairly general: they hold true for a large class of quasi linear (degenerate) parabolic e4uations, including the multi-dimensional casc, II == lI(J, Ixl). Sce also general results for linear parabolic e4uations in [13] and [2631. Let us note that if there are lower and upper bounds for lI{J, x), the stabilization (J{J.~) -> 0/1, which is uniform on all compact sets in R, follows from the results of [197\; they also consider the multi-dimensional case. In 11971 it is shown that under these conditions, stabilization occurs for any I < 13 :::: 13. == (N + 2)/(N - 2)+, which has to do with non-existence of non-trivial solutions fl(~) '¥= 0/1 of the elliptic e4uation
/:1~-O - ~V'~fJ'~' -- H/(f3-
I)
+ f}!3 == O. ~
ERN
(for H = fl(I~J) it bccomes (39)), if 1 < 13 :::: 13•. This is proved in 11971 by deriving Pohozacv type ine4ualities 1332, 333]. Estimates of sup, II{/, x), as well as the structure of the blOW-Up set, were later studied in [19R, 1991· Results concerning existcncc of non-trivial self-similar functions Os '¥= const (see (39)) in the supcrcritical casc 13 >- 13. werc obtained in 1286, 287], where it is shown that for •
13.
<
13
<
13 ""
IN
N ~ 2(N --- 1)1/2 4 2(N _ I)
I.,'
N :::: 3,
there exists an inlinite number of solutions [2861 (see also the exact solutions 11581 for 6 < N < 16. 13 = 2 and the existence thcorem of [3701 for N = 3.
312
IV NOlllinear equatioll with a source
6 S (3 S 12), while for {3* S (3 < 1+ 6/(N -- 10)+ 1287\ ranges of {3 with any finite number of solutions are determined. In this connection, let us mention the result of 139, 40\, who show that for {3 > {3' the asymptotic behaviour of a solution as ( -+ Til is not self-similar if it satisfies everywhere the condition u, .=.:: 0, that is. in this case we can say that the non-trivial self-similar solution in unstable in this class. The proof uses results of intersection comparison with a singular stationary solution. The resulting non-self-similar asymptotics will be presented below. A more accurate qualitative analysis shows that the spatio-temporal structure of u(t. x) as I -+ Tel is described by a.s.s. of the form (,(1.7)) == (To t) 1/!/3.II(,(T/). T/ == x/(1'o - tjl/2lln(T o - ()lIJ2. where the function f,(7)) > 0 satisfies in R the first order equation-f'7)/2 - f/{{3 - I) + ff.! = 0, f(±oo) == O. It has a whole family of non-trivial solutions f(7)) = ({3.- I + C7)2)-I/1{.! II. e == const > O. Such an a.s.s. was first introduced in [219J. A similar phenomenon of convergence to a self-similar solution of a first order Hamilton-Jacobi equation occurs for solutions of the porous medium equation with strong absorption 11881· From the requirement of analyticity of the corresponding similarity representation f(l, 7)) == (To -- tjl/(iJ 11 11 (1, T/{T o - tjl/2 1 In(T o - tjll/2) at the point ( == To, 7) = 0, it follows that as ( -+ Tel only one solution in the family tIl is realized: to it corresponds C = C. = ({3 - 1)2 /(4{3), and we have the stabilization f(l, 7)) -+ f,(rJl as I -+ Til' which was verified numerically. These conclusions for {3 == 3 were derived in 1219] (which contains some results for N = 2); the analysis of arbitrary {3 > I was performed in 1169. 170 I· A rigorous justification of the non-self-similar asymptotics mentioned above .,has been carried out by di fferent methods in 136, 215, 2161 (see also the results of \97,143]). Upper bounds, which are exactly the same as this asymptotic behaviour, have been derived earlier by lin, 177]. A similar situation occurs in the semilinear equation with an exponential source, II, = 6./1 + e". The lirst qualitative result concerning non-self-similar asymptotics was obtained in 17X1; see also [791· Nonexistence of non-trivial self-similar solutions of the form liS (I . x) = -In(To -I) + 05(x/(T o - tj l 12) was established in 137. 871 for N = I, 2. Such solutions can exist for N ~ 3 1881. The whole spectrum of results obtained here is presented in [40]. Justificatioll of the non-self-similar asymptotics as I -+ I'll is can'ied out in 152, 36, 215, 216, 217 \; see also 11431 (whose approach is also used in the problem for a semi linear equation with strong absorption in the study of the total extinetion phenomenon in finite time 1218, 1441). Theorem 6 is proved in 11691. In [1701 it is shown that the criticality of the solution condition (II, .=.:: 0 in (0, To) x R) can be dispensed with. In 1170, 1711 it is removed by the following quite general result: there exists a constant M I ;> 0, such that if 11([0, .10) > M" then 11,([ . .1'0) .=.:: 0 for aliI E 110. To)· In 13KOj, under fairly severe restrictions on UO(.I) and (3 ;> 2, it is shown that the unhounded solution 11(1, !xJ) of equation ( 17) satisfies the condition meas oJr = O. so that II{T(I ' x) = 00
Remarks am! commenls
313
only at one point. As we already mentioned, a very effective approach to proofs of localization is that of 11081, which was used in a wide variety of equations; see the references in 140. 104, 133. 164, In. 173, 1771; for its applications in a problem of total extinction see! 106, 1441. A spccial role in the study of semilinear heat equations with a source is played by methods of analysis of the set of zeros in spaces. or, which is the same. by intersection comparison methods. This approach allows us to obtain reasonably complete results concerning the structure of the blOW-Up set for semi linear equations with a source (see 162, 104. 115, 65j) and of the total extinction set 1661· Theorem 7 is taken from 12101. The main results of suhsection:l were obtained in 1150, 127, 347]. The exact solution presented here for {3 = 2. N = I was constructed in [134, 176]. convergence as I -+ 1'(1 to the self-similar solution of a Hamilton-Jacobi equation and localization for a large class of equations were established in 1189] (a similar asymptotic technique hased on a general stability theorem for perturbed dynamical systems 11901 can be used for {3 < 2 and (3 > 2). There it is shown. for example, that meas B r = 2tr. Ahsence of localization in the boundary value problem in a hounded domain for I < {3 < 2 was first demonstrated by 12811; localization at one point for {3 > 2 and upper bounds corresponding to spatio-temporal. structure of a.s.s. were estahlished in 1177\; see also the references there. The asymptotic behaviour of hlow-up in the three parameter ranges, {3 < 2, {3 = 2 and {3 > 2 for a more general quasi linear heat equation has heen established in 11921. Proof of convergence of some classes of global solutions of heat equations with a source to a.s.s. which satisfy nonlinear Hamilton-Jacohi type equations can be found in 11601· Let us note that for equations of other types the problem of determining the blOW-Up set is formulated in a different way. The structure of the so-called degeneracy surface in ([, x) space was studied in 160, 61. 1101 for hyperbolic equations UII = /iu + 1"(11) and in 1III1 for Hamilton-Jacobi equations II, + H(f),u) = F(u). We do not consider here in detail questions of fine structure of quasilinear parabolic equations. In this context, we mention 189. 90. 267. 268, 270, 271, 2n, 274, 276, 349\ (see also ~~ 3, 4. eh. VII). Multi-dimensional non-symmetric eigenfunctions of nonlinear elliptic problems, which arise in the construction of unbounded self-similar solutions, have up till now heen studied only numerically 1274, 2761. They can have a varied spatial structure: for example, a "star-shaped" localization domain 1274]. Group-theoretic analysis of multi-dimensional nonlinear heat equations with a source was carried out in 183, 84, 851. General ideas concerning the role of eigenfunctions of nonlinear continua in mathematical physics are developed in 1267, 268, 2751. For applications, consult 1267, 392, 3501, as well as the survey of 1269j, which contains, in particular, a bihliography of applications of hlow-up processes in the theory of self-organization of nonlinear systems.
Interesting properties are also exhibited by unbounded solutions of a different parabolic equation with power form nonlinearities: II,
= V· (IVIII"VII) + II fJ ;
IT> 0.13>
I: II =
1I(t. x):::
O.
Conditions for unboundedness of solutions of the boundary value problem were obtained, for example. in [371, 2931 (see the survey of [1571)· The Cauchy problem was considered in 1128], where it is shown that for 13 E (I. IT+ J +(IT+ 2)1N) all the non-trivial solutions II 0# 0 are unbounded (it is also true for 13 = cr+ I +(IT+2)/N [138 D, while for 13 > IT + I + (IT + 2) IN there is a class of small global solutions. There it is also shown that for 13 :::: IT + I unbounded solutions are localized. while for I < 13 < IT + I there is no localization. The localization property for 13 = IT + I (S-regime) is illustrated by the separable self-similar solution constructed in [128]: tis = (T o - t) I/,,(}(x) > 0 for Ixl < Ls12, 11=0 for 1.1'1 ::: Ls12. where Ls is the fundamental length of the S-regime: L s = 7T(lT + ) )1/(,,+21[ITsin(7TI(IT + 2))\ 1. Fine structure of localized self-similar sulutions fur 13 > IT + I (LS-regime). Us = (To - t) I/(/J I )(J(~), ~ = xl(To - I)"', III = [13 - (IT + I) ]/1 (IT + 2)(13 - I) I. was studied in [1551. where the elliptic problem for the function {}(~) ::: 0 is considered:
It is shown that even in the symmetric case, () = (J(I~I), it has quite a complicated spectrum of solutions, which consists. roughly speaking, of four families of solutions: three discrete (two countable) families and one discrete continuum of solutions. Existence theorems for self-similar solutions are proved in 11561· Localization of unbounded solutions for 13 = IT + I and an estimate for the thermal front of the compactly supported solution, h f (I) ::: h) (0) + Ls, are proved in [1341 by intersection comparison with the above exact self-similar solution. Asymptotic stability of the self-similar solution is proved by the methods of 5. Blow-up at a single point for 13 > IT + I has been established in [1331 (for N = I) and in [1811 (arbitrary N ::: I).
*
Open problems
(*
1. I) Show that the Ilumber of positive solutions of the problem (5 )-(7) is finite for I < 13 < ((T + I)NI(N - 2)1' Are the predictions of the linearization procedure in subsection 4.1 correct') 2. I) Prove existence of solutions for the self-similar problem (5)-(7) for
(*
13 :::
(IT
+ I)(N + 2)/(N
- 2)
I'
315
Open problems
IV Nonlinear equation with a source
314
(*
3. I) Prove existence of radially non-symmetric solutions of the elliptic equation (2) in R N for 13 > cr+ I, IT> 0, N > I (for IT = 0, I < 13::: (N+2)/(N-2l.+ they do not exist [1971l. Determine the number and spatial structure of such solutions' (qualitative and numerical analyses are carried out in [2741). 4. I) Prove uniqueness of the strictly monotone solution constructed in Theorem 4 (it is important for the stability results discussed in 5) and of any symmetric
(*
*
solution having a given number of maxima. 5. 4) Demonstrate localization of unbounded solutions of the Cauchy problem for II, = V· (II"VII) + II/!. I > 0, x E R N for 13 :::: IT + I, IT > 0 in the case of arbitrary initial functions 110. Is it possible to derive. as in the one-dimensional ease 4). an estimate of supp II(Tii. x) in terms of supp 110 and the time of existence of the solution? 6. 4) Prove effective localization in the case 13 :::: IT + I for arbitrary (noncompactly supported) 1I0(x) -> o. Ixl ->X. 7. 5) Prove asymptotic stability of unbounded self-similar solutions of the LS-regime. 13 > IT + I for N > I (for N :::::: I see 1141]l. 8. 7) Prove that the asymptotic behaviour of blOW-Up solutions of equation (17) is stable with respect to "small" nonlinear perturbations of the equation, whefi it becomes II, = V· (k(II)VII) + Q(u). For which k(lI) is there non symmetric single point blow-up (see [1671 for such examples)') 9. (§ 7) Justify in the general case the asymptotics of unbounded solutions of the problem (17). 1I(t. x) ::::: (To - 1)',I/t/H1f.(x/(To - t) 112 lln(T o - tll l / 2 ) as I -> Tii, which was suggested. at a qualitative level. in [219, 1691 (this problem is partially solved in 136.971 and 1215. 216]). What is the structure of the attracting set of non-trivial self-similar solutions, which exist in the supercritical case 13 >
(*
(*
(* (* (*
(N
+ 2)/(N
(*
- 2),')
10. 7) Prove effective localization of arbitrary unbounded solutions of the N Cauchy problem for the equation II, = Li.1I + (I + II) In#( I + II). I > 0, X E R • for 13 :::: 2 (see the partial results of 11891 for 13 = 2 and 11771 for 13 > 2 and also [192]).
(*
11. 7) Determine conditions, under which asymptotic behaviour of nonsymmetric unbounded solutions of the problem (54), (55) is described by the degenerate a.s.s. (68) (see the result of 11891 for the case 13 = 2 and general analysis of symmetric solutions in [1921l.
~
I Criticality conditions and a dircct solutions comparison thcorcm 11(1. x)
Chapter V
= 111(1. x)?: O.
0 < 1< T, x E
317
an.
(3)
Let us assume that 1I0Cr) -+ O. l/\(I. x) .-+ 0 as Ixl -+ 00 and l/(I. x) -+ 0 as Ixi -+ oc for any 0 < 1< T. The problem (1)-(3) includes the Cauchy problem as a particular case (then simply n = R N and (3) should be omitted). The function Up.lj. 1') in (I) is defined and once differentiable in all its arguments; furthermore iJLlilr > 0 in R j x R+ x R which means that the equation is parabolic. We shall also assume that there exists a real valued funetion I' = I(p.lj, Y) which satisfies the identity
Methods of generalized comparison of solutions of different nonlinear parabolic equations and their applications
Up.lj./(p.lj. Y))
==
(4)
R, x R+ x R
Y. (p.lj. Y) E
The function I is differentiable in all its arguments in view of the smoothness of L and the parabolicity condition. From (4) we obtain the following identities: In this chapter we prove comparison theorems for solutions of different parabolic equations, based on special poinwtise estimates of the highest order spatial derivative of the majorizing solution in terms of the lower order derivatives. Derivation of such estimates is done under conditions of criticality of the problem (§ I, 2). In § 3 we consider the more general III-criticality conditions. In § 4. 5, using an operator version of the comparison theorem, we study the heat localization phenomenon in media with an arbitrary thermal conductivity. In § 6 the results of § 1-3 are used to study unbounded solutions of quasi linear parabolic equations. In § 7 we obtain conditions for criticality of finite difference solutions. Using these conditions, we prove a direct comparison theorem for implicit finite difference methods for different nonlinear heat equations.
LI(p.lj. /(p.lj. /(p.lj. Y))
+ L , (p.lj./(p.lj.
Ldp.
(here and below we are using the notation L 1 L , = aLI iJr and so forth). Let us sel lo( p. lj)
By
(4)
A(II)
where K (p.
II) > 0,
N (II.
1/)
Formulation of the boundary value problem and the Cauchy problem I(p. II. Y) =
= IIO(X)
?: O. x
E
n;
= ilLlap.
II
== 0
= alli/p,
L"
= aLI(lq,
= I(p. lj. 0).
(6)
== o.
(4')
(p.lj) E R+ x R+.
= K(li.
1'Vl/I)~l/
+ N(II.IVIII).
(7)
are given sufficiently smooth functions. In this case I N(p.lj) N(p.q)I-----.---. lo(p.lj) = - - - - - . K(p. q) K(p, q)
(8)
For the operator of nonlinear heat conduction with a source.
= A(II) = L(u. l'Vul. ~II)
11(0. x)
IY -
with a smooth
(here !'VIII = Igrad, IIi), let us consider the boundary value problem (0. T) x n with the conditions
L,(p.lj. 1(·))/"(·)
(5)
the function 10 satisfies the identity
theorem
II,
q./(·))/)(·)
== O. == I.
The above requirements are satisficd, for example, by the quasilinear operator
§ 1 Criticality conditions and a direct solutions comparison
Let n be an arbitrary domain (not necessarily bounded) in R boundary an. For a nonlinear parabolic equation
+
L"(p. 1/./(·))
L(p.lj./o(p.lj))
N
y))/dp.lj. Y)
(I)
111
lur
=
AlII) = 'V. (k(II)'Vl/)
+ Q(l/) = k(u)~u + k'(II)I'Vu!" + Q(l/). k
> O.
(9)
the analogous expressions have the form I(p.lj. Y)
* I Criticality conditions and a direct solutions comparison theorem
V Methods of generalized comparison
318
We shall assume that in (Vr there exists a positive classical solution of problem ( I )-( 3), and that it is unique. Everywhere below we shall take the following restrictions on the function L to hold: a) the operator A is parabolic: ilL/ilr > 0; b) there exist functions I. 10 satisfying. respectively. the identities (4). (4').
2 Conditions for criticality of the problem
Theorem 1. For criticality
1I(t.
x) ~: 10 (11(1. x),
In particular, for the operator
(7)
IVu(l . .1)1) in Wr·
(15)
iJII1(t. x)/iJt:::: O. (I. x) E (0,7') x iJn.
( 16)
Proot: Necessity of the conditions of the theorem is obvious. Let us prove sufficiency. Let us set 11,(1. x) :=: Z in wr. Then L(II, IVul. ~II) :=: Z and therefore ~II :=: 1(11. IVul. Z). From (I) it follows that the function z satisfies in WT the
Here VII . Vz is the scalar product of two vectors obtained as a result of differentiation: IVul, :=: (Vu . Vz)/IVIII. Formally this equation is a linear homogeneous parabolic equation with bounded coefficients. which is ensured by sufficient smoothness of the solution 11(1, x) and of the function L. Therefore by the ~ax imum Principle (sec I. Ch. 1) z :::: 0 everywhere in the domain wr as soo~ as Z :::: 0 on its parabolic boundary Yr = II E (0.7'). x E ilnl U II = O. x E HI· This completes the proof. 0
*
In the following, inequalities (15), (16) will he called the crilim/ilY cOllditiolls for the boundary data of the prohlem (I )-(3).
Remark 1. It follows from the theorem that criticality of the problem does not impose any restrictions on the elliptic operator of equation (I) (if it does not depend explicitly on the variable I); it is fully deflned hy properties of the boundary data. For an operator of a more general form, A(II) "" L(II.IVIII. ~II;
~II
Vz/IVIII)
(17)
However. L\ (p. if, r) > 0, and therefore the inequality (II ') can be solved for ilu. This leads to the estimate tlll(l.
L(lIo.IVllol, ~1I0) :::: O.
x) crilical if
Condition (II) will he used to derive a pointwise estimate of the highest order spatial derivative (Laplacian) of the solution, tlll(l. x). in terms of the lower order derivatives. IVu(l . .1)1 and 1I(t. x). Indeed. by (I). condition (11) is equivalent to the inequality 0 in wr·
==
.lEn.
A(1I0)
equation
(II)
/.(11(1, x), IVu(t, .1)1, ~II(1. x)) ::::
the prohlem (I )-(3) it is Ilecessary alld sl~lficiellt
thaI
z,
Definition. We shall call a problem (1 )-(3) and its solution everywhere in wr it satisfies the condition
(It"
319
( 14)
Pointwise estimates of the form (12)--(14) are the hasis of the approach to comparison of solutions of different paraholic equations we propose helow. For simplicity. let us assume initially that 110 E CC(O)nc(!'1), III E c~;Oqo, n x an). II E C~;'I(wr)nC);C(wr). This allows us to differentiate eyuation (I) once in t in (Ur.
I. x)
the same statement is no longer true. Following the proof of Theorem I. it is not hard to see that in this case for criticality of the problem we need in addition the following inequality: iI ijiL(p.q.r;l,x)lrd"I/,.,/:fII::::O. (p.q)ER+xR.r . (1,.1) EWr.
For example. if A{tt)
:=:
V· Ik(t/; I) VII 1+ Q(II; I),
then condition (18) can be written in the following form:
fk ( iJliJp
iJk
!~.I)
ilp ill k
if2
+~g .~ ill
ilk Q > ill k -
(J.
(18)
* I Criticality conditions and a direct solutions comparison theorem
V Methods of generalized comparison
320
= U;{'I(X) ::': 0, x E n; [/"1(1, x) = U\"I(I, x). 0 < I <
u'l') (0, x)
In view of the fact that the quantities p and q vary independently, this relation decomposes into the two inequalities
lIi"1 E C([O, T) x
32\
U~{'I E C(!!);
(20)
T, x E iin;
(21 )
an),
The functions Iy'l(p, £1,1') are assumed to be sufflciently smooth. As in subsection L we denote by fi21(p, £I, n (/i121 = I'21(p, £I, 0)) the solution of the equation Remark 2. Let us show that the smoothness restrictions on the solution II used in the proof, may be weakened substantially, Under the natural assumption that U E C~;2(wrlnCCiirn, the proof follows exactly the same lines, with the difference that instead of the function: = 1//(1, x) we consider the finite difference :(1, x)
I
= -Iu(l
+ 7. x)
- 1/(1 .
.I'll.
(I. x) E lur"T'
T
where 7 E (0, T) is a fixed constant. Then a parabolic equation of the form (17) satisfied by : can be derived in a similar manner. As far as boundary data are concerned, in this case under the same conditions (15), (16), for (I. x) E (0. T - T) x iin we have :,(1, x)
Furthermore, for
I
=
I -IUIlI
+ T, x)
Z(O, x)
Theorem 2. LeI
11 121 ::': II' II
) X E I/o121 ( x ) -~ I/o(II (x,
(l
1,
(22)
T
= III(T, x) -
1/0(.1')1/7. x E
III addilioll. leI Ihe SOllllioll I!I" Ihe problem (19)-(21) for v = 2 he critical (Ihis mealls Ihat u;21 ::': 0 ill wr), alld Ihal for all (p, £I, 1') E R, x R I X RIve hm'e the
n.
However the function v(l. x) == I/o(.r) is, in view of (15), a subsolution of the equation (I); in addition, v = 1/1(0. x) ::: 111(1, x) on an. Therefore 1/::': von Yr, and thus u(l, x) ::': v(l, x) == I/o(.r) in wr. The last condition is equivalent to the condition z(O, x) ::': in for any 7 E (0, T). Thus the function z satisfies a parabolic equation in wr r and z. ::': 0 on the parabolic boundary Yrr' Then: ::': 0 in Wlr, from which it follows that for all
illequalilies '_ji) r I'
n
(1,.1') E wr .
1//(1, x)
Oil Yr, Ihal is.
ulll, x)J ::': O.
-
=0
°
Let there be positive classical solutions of these problems in lur, and assume that lion Yr. In the following assertion we staW two sufficient conditions for a 121 direct comparison of the solutions of the problems (19)-(21), under which 11 ::': 11"1 everywhere in lur. Let us emphasize that we are talking about comparing solutions of two substantially different equations. u'21 ::': 1/'
= r--O' !tm
11(1
+ T,
- 11(1, x) T
(23)
nI21 (p. £I)) ::: o.
(24)
L111(p. If,
Proof Let us set
> O.
IL(21(p, £I, 1') - L'II(p. £I, 1')1::': 0,
1/'21 -
1/' II
= z.
Then the function: satisfies in wr the equation
-
Obviously, using this method of proof. we can replace condition (16) by a weaker one: 1/1 (I, x) docs not decrease in I on an. We can also weaken the smoothness requirements on the initial function 1/0(.1')·
and, by (22), z ::': 0 on Yr. Linearizing the right-hand side of (25) with respect to the function z and its derivatives, we obtain for z a linear parabolic equation with bounded coefficients:
3 A theorem on direct comparison of solutions I
,ll (1/ 1'1 -.
:, - ' 1
Let us consider in wr boundary value problems for two different parabolic equations (Il = 1,2): (19 )
InvI/ (OIl A - . v.du:.
111 - I -2 (II (°1 -.
_ L\II(VI, 1'11/ 121
= LI21(1I1'1.IVI/'211, /l1l(21)
-
1.
InvV2 I,ull A 1'1 1 - )( n vV2 . n v:.J /In vV2/l1/(21):
where the bounded smooth functions Vj, j = 1,2.3 (some average values). depend on the solutions III II, IF). 121 Let us consider the right-hand side of (26). By criticality of the solution /1 . we have the pointwise estimates (see subsection 2) (27) Condition (23) means that the function lyl(p, if. r)-L(lI(p. q. in its third argument. Therefore. using (27), we obtain
1')
is non-decreasing
Criticality conditions and a direct solutions comparison theorem
323
due to independence of p and if in the second of the inequalities (28), these inequalities can be written as
kI21 (p)
- klll(p) :::
O.
(30)
I ' , I(p) - k l , (p)kl-l(p) :::
D.
(31 )
QI21(p)k lll (p) - QIII(p)k 12J (p) :::
n.
(32)
.., ' 1 l
kl-l
(p)k
The inequality (31 ) can be put into a more compact form: (31 ')
== O. Hence by (24) The comparison conditions (23). (24) (or (28)) of solutions of equations (33) have the form (34)
Therefore from (26) we have that (II"
•. / - L 1
u.:': -
iJ I21 (p)1/1'(p) _ iJll'(p)((111(p) ::: V t,111(V' t72'V'Z) -lllll~.>_() . • I v v11
everywhere in wr. Since z > 0 on Yr. invoking the Maximum Principle, we conclude that:,: ::: 0 in wr, that is /1(2) ::: /1111 everywhere in that domain. 0 Let us see what form the comparison conditions (23), (24) take in the case of particular parabolic operators.
Example L The inequality (23) depends, in general, on the three variables p. q. r. However. for quasilinear operators of the form
n.
and look much simpler than (30)-(32) (at least they do not contain derivatives of the functions entering them). At the same time equations (29) can be reduced to the form (33) by simple transformations. Indeed. let us set H(l"(p)
= /" ./0
kl,'I(T/)
dT/. p > O.
/l
= I, 2.
and denote by hll" the functions inverse to HII'I. so that HII'I(hl"'(p» == p (hil'I exist at least for all small enough p by monotonicity of H(l'I; the latter is ensured by the conditions k ll " > 0), Let us make a change of variable in the equations (29). (35)
it depends. as docs (24). only on p. q. The inequalities (23). (24) in this case have the form
KI21 (p.
q) - K11i(p. if) :::
O.
K I21 (p. if)NI11(p. if) - K111(p. q)N111(p.l!l :::
D.
Then. taking into account the fact that functions VI"1 the equations
kl"'(hll')(p))hll'I'(p)
==
I, we obtain for the
(28) which are the same as (33), if we set in those equations
Example 2. For the nonlinear heat equation with a source. (29)
d"I(p)
=,
iJiI"(p)
= QiI'J(hil'l(p»)kI1'I(hl"I(p)).
kl"'(h1t'I(p»),
(36)
~ 2
Y Methods of generalized comparison
324
From (35) it follows that the inequality vl 2 )
?: Vii)
is equivalent to the inequality (37)
As a result we obtain that under the conditions
using degenerate equations, which, as is well known (see § 3, Ch. I), do not necessarily have a classical solution. We shall consider in most detail the onedimensional case, though all the results hold for equations in many independent variables (corresponding examples are given below).
1 Criticality conditions for solutions of degenerate parabolic equations Let us consider in
and also under the other conditions of Theorem 2, in particular the inequality
(JJr
= (0, T)
x R+ the boundary value problem
(40)
holds. The comparison conditions (38), (39) can be written as:
(37)
k I21 (p) _ kIIIWIIU/121(p))!?:
QI21(p)._
QIII[h ll )(lJI21(p))1
11(0. x)
?: 0, p:>
(39')
0.
Therefore, by comparing not the solutions 1//'1 themselves, but rather some nonlinear functions of these solutions (see (37) or (37')), we managed to simplify the comparison conditions considerably: instead of the three inequalities (30)-(32), only two remain: either (38), (39), or (38'). (39'). We shall call this generalized comparison method the operator or the .Iilllclio//al comparison method. It will be consKlered in more detail in the following section. Remark. It is not hard to see that the comparison conditions (30)-(32) will be satisfied if and only if the functions k( I) and Q(li can be represented as klll(p)
== kl21 (pll 1+ ,u(p)]
QIII{J));;;;: Q121(pll
I,
1+ ,u(pWII I + dp)!
x:>
Prllll:::: (37')
in Wr.
?: O.
0: sup 110 <
{(t,
x)
E
wr
where ,u. 1/ arc arbitrary smooth non-negative functions: in addition ,u is a nondecreasing function.
§ 2 The operator (functional) comparison method for solutions
of parabolic equations In this section we prove a more general comparison theorem for solutions of two different nonlinear parabolic equations than that of in § I. We present the material
sup 1IC/)(lIo I, I < x; T.
\U(I,
x)
:>
° and
(2)
(3)
k E
01
and it can happen that at points of
SrlllJ :::: Prlll!\Prl/l!\dwT (degeneracy points) not all the derivatives in (I) are defined. There the function is continuous in x in R 1 for all fixed I E «), n. As in § I, we shall call the problem ( I )-(3) and its solution criliC(/I if II, (I, x) ?: o everywhere in Prill!. 2 1.1. For convenience. we shall assume below that /10 E C everywhere where /10:> 0, and thatul E C 1 (\0, T)) (these restrictions can be weakened substantially). Under the assumptions made. we have the following theorem.
k(II(t, x))u,(I, x)
Theorem 1. For crilicality O/Ihe prohlem I
OO.
Let the equation (I) be degenerate for II :::: 0, that is k(O) C 2 (Rt ) ncqo. (0)). The solution of the problem is classical in
Then the inequality (37) can be written in the following form: 11 121 ?: hI2)IHIII(IIII))1
lIo(X)
11(1,0):::: 1I1(t):> 0,0 < 1<
(38')
0,
==
(I )-(3)
il is necessary and sl(flicienl
Ihal
(k(II0)II;/ ?: 0, 11'1 (I)
x
E Ix :> 0 Iuo(x) :> 0),
?: 0,0
< I <
T.
(4)
(5)
Proof Let us prove sufficiency of conditions (4), (5). Let us make the preliminary observation that a critical initial function 1I0( x), bounded in R +. is non-increasing. Indeed, by (4). it cannot reach a positive maximum in R I · Let us assume that at a point x' E R I ' where 110(.1"') :> 0, it is increasing, that is 11;1(.1"') > O. Then IIO(X) :> 0, 11;1(.1') :> 0 for all x :> x', anti we have from (4) that
that is. uo(x) grows without bound as x -, 00, which contradicts the assumption thatllo E e(it t ) (supuo <(0). Thus the initial function Uo is non-increasing. and we are entitled to conclude that the set of degeneracy points Srlullies on only one curve (0. T) x !x = ((I)), such that furthermore (6)
k(u(l. ((I))u,(I. ((I)) = 0.0 < t < T.
The function x = ((I) is nondecreasing and continuous in [0. T) (see. for example. 118,252.3281)· Theorem I can be established in a number of ways. Below we briefly present one of the proofs. which makes substantial use of the property (6) and the assumption that II E C'=>,·I(Prlll\}. Let us set; = u, = (k(u)u,) ,. The function; satisfies in Prllli the formally linear parabolic equation ;/ = Ik (u); I" . Then ;:(0. x) ~ 0 in (0 < x < ((Ol} by (4) and, as follows from (5), we can assume that ;(1. 0) ~ 0 for r E (0. T). It remains to verify that, roughly speaking, ; ~ 0 near the curve (0, T) x (x = ((1)1, the right lateral boundary of Prlul· This follows directly from the equality; = (k( U)II,),. Integrating this equality over a small interval (((I) - E, ((I)), E > 0 (it is not hard to check that this makes sense), by (6) we have '(([1
;(1, ./ (III"
x) dx
2 Thc operator (functional) comparison mcthod
= -(4)(u)),I([.11I1
fl'
==
["
k(TJ) dTJ·
(6')
. 0
Since 4)(u(l, x)) > 0 on (((I) .- E. ((I)) and ((/1(u(l, x))), --'> 0 as x --'> ( ' (I). we can always lind an arbitrarily small E > 0, such that (4)(u)) , < 0 at the point (I, ((I) E) and therefore
Therefore at any arbitrarily small left half-neighbourhood of the point x = ((I) we can lind x,,(1) E (((I) -- E. ((I)), such that zU, x.(I)) > O. The function; is a classical solution of equation (7) in the domain (0, T) x to < .\ < .\,(1)). such that, furthermore, ; ::: 0 on its parabolic boundary. Then, by the Strong Maximum Principle. z ~ 0 at all interior points of the domain
E >
327
0 can be arbitrarily small, we obtain that
0 in Prlu].
0
Remark 1. It is not hard to show that we can take the set (0. T) x (x = x.(lJ} to be a continuous curve. Moreover, under the conditions of the theorem u, > 0 in a neighbourhood ((). T) x (((I) - [) < x < (U)} (see 1252 1). Remark 2. Using the same method. it possible to prove that under the conditions of Theorem 1 (8) u, (I. x) :::.0 in Prlul·
We note that the functions; = u, and; = u, satisfy in P r [lI] the same linear parabolic equation (7). 1.2. An assertion. similar to Theorem I. is true for a degenerate parabolic equation more general than (I), II,
= V'. (k(u)V'u)
+ QUo ==
/:;'
+ Q(u).
(9)
where Q E e l ([0. (0)), Q(O) = 0, is a given function. Let n be an arbitrary domain in R N with a smooth boundary an. For the equation (9) let us consider. N for example, the boundary value problem (or the Cauchy problem if n = R ) with the conditions u(o. x) u(l. x)
= 1I0(r) ~ O. = 0, 0 < t <
(10)
.lEn;
T. x
E
(I I)
an.
For our aims. the following assertion concerning the criticality of solutions of the problem (9)·_·( II). which is far from being optimal in terms of requirements on uo(r), will be sufficicnt. Theorem 2. Let Q E e l ([0. (0)), Q(O) = 0 olld Q'(u) ~ O.liJI' U > O. Let the domaill no = SIIIJ{J Uo C n have a sl/looth /](}l1/u/ary ano alld Uo E e~(no) neerh Theil .fill" critica/it\' of u(l, x) ir is /Joth 11 ece,1 so 1'." al/(I sulficiellr rhcH
( 12) Proof As our point of departure we take the fact that the generalized solution u(l, x) can be obtained as the limit as E -'> 0'1 of a sequence of strictly positive classical solutions u, of equation (9) in lur = (0, T) x n with the conditions
u,(O.
x)
= u.oCr) == 4)-1 (c/)(uo) + E)
-....
Uo
uniformly in n as E--' 0". II, = c/) I (E) on (0, T) x an. Let us fix a small enough 0 and let us consider the function ;:.(1, x) = 111.(1 + 7. x) ~ 11.(1. x)\/7, which
7>
V Mcthods of gcncralizcd comparison
328
satisfies in wr . T the linear parabolic equation
where we have denoted by a, iJ the smooth functions
. 1 1
a==
.0
iJ ==
t
Jo
Furthenl1ore, ::., ::;;; 0 on [0, l' - T] X an. Let us consider the function ;-., (0, x) == Ill, (T, x) _. II,O( x) liT in n. Since by the Maximum Principle LI, ::: qJ I (E) in lvr (recall that Q(II) ::: 0 for all II ::: 0), then for all x E n\n o we have ::.,(0, x) := ILI,(T, x) -lVI(E)]/T::: o. Furthermore, let us consider the function v(l, x), a classical solution of equation (9) in (0, T) x no with the conditions v(O, x) = Il,o(x) in no, l) ::;;; lpl(E) on (0, Tl x an o. It is critical. since by (12)
Let the functions in the statement of the problems (13 )-( 15) satisfy all the requirements of subsection I concerning the functions k, Ilo, III in the statement of problem (I )-(3), and assume that there are in wr non-negative generalized solutions of the problem we are considering. Let us introduce a function E(p), which is twice continuously differentiable for all p > 0, such that, moreover, E(O) = 0, £(00) == 00 and E'(p) > 0 for all p > D. The last condition means that E is a bijection R I. r-> R+. Therefore the inverse mapping E' I (p) is defined on R f; it satisfies all the requirements made on the function £(p). Let 11 1]1 ::: E )(Il l )') on Yr. The problem of the opel'll/or (functional) comparison of solutions III]) and III I' is to determine conditions under which 11121 ::: £'.1(11 111 ) everywhere in wr. In the following theorem we use to that end pointwise estimates of the highest order derivative of the majorizing solution, which follow from its criticality.
Theorem 3.
LeI Ill]' :::
£
I (u l I') 011
Yr, Ihal is.
(Llii '(x)),
II;,]I(X) :::
E
I
1I11])(l) :::
E
1(11\"(1)),
Moreover. leI Ihe SOllilioll of' problem (13 )-(
and therefore v(l, x) ::: lI(O, x) in (0, T) x no. From the comparison theorem we then obtain Il, ::: 11 ::: Il,o in (0, Tl x no. Therefore ::.,(0, x) > 0 in no for any T E (0, Tl. Thus, '., ::: 0 on the parabolic boundary of the domain Wr-T' and by the Maximum Principlc z, ::: 0 in «Jr'T' Hence, by passing to the limit E -> 0' and T -> 0+, we obtain that 11(1, x) does not decrease in I in wr and therefore Il, ::: 0 in Prllli. Necessity of condition (12) is obvious. 0
Iha/ JiJr al/ p >
hold. 1'111'1111 121
*
(13)
1/"1(1,0)
= 1l\"'(I)
::: 0, x>
0;
> 0,0 < I < T.
:::
Viii r
We shall first demonstrate the possibilities of the operator method of comparison using relatively simple equations. The comparison theorem we obtain in the process will be used in 4 in the study of the localization in boundary blOW-Up regimes. Let us consider in (vr boundary value problems for two different (degenerate) parabolic equations (11 = I, 2):
( 14)
D<
and, by (16),
E
(\6)
< T.
15) be crilicalJiJl'
/l
= 2 alld aSSllIlle
D,
( 17)
Ikl]l(p)lklll(!:'(p))E'(P)( ::: 0,
( 18)
:::
1(11 11 ') evellwl1cre illlvr· I (III II)
= Vii)
The function
= LIII(VIII IVIIII VIII) .' \ • .\.\
1l (2 )::: VIII
is critical. that is, pointwise bound
I
•
Ihe colldiliolls
Let us set E'
2 An operator comparison of solutions theorem
= Il;;)(x)
0
x E Rf
kl]l(p) - klll(E(p)
Proof
//"1(0, x)
329
~ 2 Thc opcrator (functional) comparison mcthou
VIII
satisfies in wr the equation
= -
on Yr. The solution
11<21(1. x)
of the equation
It I ::: 0 in Pr IIPII, which ensures that in Pr[I/( 21 1 we have the (21 )
It is not hard to see that the inequalities (17). (18) are the conditions for direct comparison of solutions of parabolic equations (19). (20). if we have the estimate (21) (see Theorem 2 of I). Let us use the fact that generalized solutions of the problems ( 13 )-( 15) can be obtained as limits as k -+ 00 of sequences of classical strictly positive and bounded solutions lui"11 of the corresponding equations. It is not hard to see that monotone 21 decreasing with k sequences of inlinitely differentiable functions {ui (0. x») and lui~'(t. 0)), converging to the functions II;)~I(X) and U\21(t), can be chosen to be critical. Existence of the sequence {[(~~'(t. 0ll with the required properties is obvious. As far as the initial function is concerned. this problem reduces to approximation , . . . 1'1 1'1 121 I ~I" of a pieceWIse smooth convex function U o- (x) ;::: eP - (u o (x») (U o ::: 0 at all points where U;)21 >- 0) by a sequence of smooth convex positive functions (pI21 (ufl (0, x») I uniformly bounded away frolll zero, Clearly. this can always be done. In the construction of the approximating slllooth solutions {[(i"I) it is not hard . . II,I~I to have u (2) ::: Viii , ;::: [-.\ : (u,III ) on Yr I'or any k· ';::: I . ')_. ' , .. 1'11e I'unctions l are critical and satisfy the inequality (21). Therefore by Theorem 2 ui~1 ::: viii in (ur for each k ;::: 1,2." .. Hence by passing to the limit k -+ 00 we have that u l21 ::: VIII in Wr. 0
*
from which, if (23) holds, it follows that VI I I is a subsolution of the problem (13)-(15) for II;::: 2, and since UI~1 ::: Vii' on Yr, ul~'::: Vii) everywhere in wr· In the proof of Theorem 3 we used our ability to approximate a critical initial function U~)~I (x) by a sequence of positive smooth critical functions. Under the assumption that this can be done (see Theorem 2), the comparison theorem is valid in the case of boundary value problems for parabolic equations with a source:
Ui"l;:::
(22) and
[(12) :::
E
1([(111)
Oil
Yr, LeI
[(.I'
n.
p >
o.
-+- QII"U/"').
1';:::
1.2.
(24)
Theorem 4. LeI [(I~I is, U;~I ::: 0
ill
::: CI(UIII)
on Yr, and lellhe sollllioll u(2) be crilical, Ihal
Prllll~1 I. AS,I'ullle, lIIoreover, Ihal
11'1'
have Ihe inequalilies
kl~l(p) - klll(E(p»
::: O.
(25)
IkI21(p)/lkll'(E(p)E'(p)Il' :::
o.
(26)
QI~'(p)klll(E(p» - QII'(E(p»)k(2'(p)/E'(p) :::
o.
(27)
Pro()j: To prove the theorem, it is convenient to write the equation satisfied by the function Vi II ;::: E'I (u l II) in the form Viii;::: V' 'lklll(E)V'VII)j
-+-
k(\)(E)E"
1
--'-:-IV'V(\)1-
E'
QI)I(E)
-+- - - - .
E'
(28)
Then the inequalities (25)-(27) are the conditions for a direct comparison of solutions of equations (24) for I' ;::: 2 and (28) (see Theorem 2 in I). 0
*
Using the Maximum Principle it is not hard to check the validity of the following simplest possible version of the operator comparison theorem:
have, jilrlherlllore. Ihal
f:'''(p) :::
V', (kl"I(I/"I)'h/"I)
where Q"'I E CI(\O. x») are given functions.
{
Corollary. LeI Ihe jilllclion E he such Ihal
331
Ifl-criticality conditions
(23)
Corollary. LeI Ihe jill/clion E he such Ihol Ihe incquolily (22) holds, and leI I/I~I::: £-1(1/111) on Yr. ASSI/I/lc, I/lOreOI'er, 11/(/1 for al/ p>- 0 E"(p) < O. QI~I(p) ::: QIII(E(p»)/E'(p).
The inequality (23) is equivalent to the following one: (23') Let us note that in the conditions of the corollary there is no assumption of criticality of u121 . Validity of the inequality UI~1 ::: E- I (1/111) == Viii in (ur follows from a direct comparison of equations (Il) and (20). The first of these can be written in the form
§ 3 f/J-criticality conditions In this section we present one possible generalization of the concept of criticality of a solution. and derive a new class of pointwise estimates of the highest order derivative of a solution of a quasi linear parabolic equation in terms of lower order ones. These results can be used to prove more general comparison theorems for solutions of different equations; they can also be applied in other areas.
~
V Mcthods of gencralized comparison
332
The inequality (8) can be written in the more compact form
Definition of a "I-critical problem Let n be a bounded domain in R N with a smooth boundary an. In this section we consider a boundary value problem for a quasilinear parabolic equation: II, == A(II) == V·
+ Q(II).I
(k(u)VII)
> O. x E
n:
== lIoLt) ::: O. x E n; "'~ 0.0 < I < T. x E an.
( I)
(3)
For simplicity we shall assume that the positive in (ur solution of the problem ( I )._(3) is classical. Suppose we are given a function ~/(p), which is twice continuously differentiable for f! > 0, I/!«)) == 0, III E C
Definition. The problem (\ )--(3) and its solution will be called III-crilica/ respecI 10 a gil'en lilllelion III J. if everywhere in wr we have the inequality II, (I. x)-
'11(11(1. X)) :::
(8')
Let us note that nn makes sense also for values (8) is defined at points where Q == 0).
(wilh
Let us set: == lI, - 111(1I). From that: ::: () on Yr. Using the equalities
Proot:
lI,
!ill ;>
--
-----IVlIlk(lI)
111(11) - Q(II) .
+ - -k(lI) ---
III
wr·
(5)
2 Sufficient conditions for "'·criticality of a problem Let lIO E C 2 (n) n C (ri). II E C;~4 «(u,) n C ~ ~ 2 (Wr l. Let us note that the smoothness requirements on the solution in (ur (and, in part, on the initial function 110) can in principle be weakened. Under these assumptions we have For
~/-erilicaIiIY of' Ihe Ilmblem (I )-(3) il
k'(II) k'(II) _: __.: + --(2,11(1I)
==
{
k
alit! Ihal
Fir al/
p >
n \1'1'
0, x
,)
(II)~'-(II) -
) Q'(II)
it follows
"(lI)IVIII-I.
(9)
+ Q.' (1I) + k(lI)
[k(lI)~/(lIJ]'} ---Q(lI)
1
k(lI)
+ k(lI)
[k'(II)]' --k(lI)
lVIII'' } ==
[(k(lI)III(II))'] , k(lI)
(\ 0) )
IVIII'·
Conditions (7). (8) guarantee that the right-hand side of equation (10) is nonnegative, which by the Maximulll Principle ensures non-negativity of the function : = II, .- ~/(II) in wr if::: ::: 0 on yr. 0
Remark I. In the case III == M == const < O. criticality conditions for an operator A have the following form: (k'/k)' :::. 0, IIQ/M -- I)/kl' ::: O. They hold, for example. for k(lI) == (I + II)". IT > 0, and Q == O. Therefore the solution of the equation II, = V· « I + II)"VII). which satisfies conditions (2), (3), has the following interesting property 11,(1, x) :::
n.
- Q(lI) - k
0,
inf
(6)
that is, for
I
11/(0. x) :=
inf(V· ( 1+ lIo)"Vlloll(x).
,,:0
> 0 the solution cannot decrease in I faster than it did at the initial
time.
have Ihe ineqllalilies
I (kill)' /k I' (p)
==
To derive this equation. it suffices to notice that ::., == 11 11 - ~/(1I)1I/ and then to determine from equation ( I ) the derivative 1I". simplifying by using the equalities
is slIfficie/ll Ih(lllhe/ill/Clio/l
E
- Q(II))
k(lI)
,ell
III(UO(X)) :::
(II(O)
2k'(II)VII . V: -
:,-k(lI)!i: -
110 salisfies Ihe cO/lt!ilio/l
A(lIo(x)) -
and the condition
(9).
wUsing estimates of the form (5) with a sufficiently general function ~I widens the scope of the direct and operator methods of solution comparison.
Theorem I.
(6)
,1 == : + (II(II). L111 = - I : + W(II)
{ k(lI)
*
,
== 0 (similarly,
it is not hard to obtain the parabolic equation satisfied by::
*
k'(II)
(11(p)
k(lI)
(4)
O.
In accordance with the definition of 1. we shall call a zero-critical problem (that is, III-critical with respect to III == 0) simply critical. From inequality (4) follow more general estimates of the highest order spatial derivative of the solution in terms of the lower order ones, than those obtained in I: .
0 where
p >
(2)
11(0. x) 11(1. x)
333
3 III-criticality conditions
:::
n.
Ik'II1 2 - Q2(klll/Q)'](fI) :::
o.
Remark 2. Let ~/(fI) > 0 for p > O. Then from (4) it follows that everywhere in (ur the solution of a ~/-critical problem is related to the initial function in the
*4 Heal localization in problems for arbitrary parabolic nonlinear heat equations
V Methods of generalized comparison
:1:14
t E (0, T). Under the restrictions of Remark 4, the same statement also holds if
following way: "11,.\1 /,
.11,,(.1)
dry
.
- ' - > I 0 < 1< 7 I/I(ry) -
,
(II)
*
§ 4 Heat localization in problems for arbitrary parabolic
nonlinear heat equations
Remark 3. As we already mentioned above, the smoothness requirement II E C;;4 (Wr) n C~ ;"Cwr) can be weakened, if we prove the claim tirst for the function I = -111(1 + T, x)
~(I. x)
- 11(1. x)] - 1/,(11(1. x)). T E
(0. T).
(Vr
T
to the limit as
T
->
0+.
Remark 4. Theorem I holds also in the case of generalized solutions of the problem ( I )-0). If. for example, we use the regularization of the proof of Theorem 2, 2, then under the assumption 4)(110) E c 2 (?lJ. /:"(/)(110) + Q(lIO) ::: 1/1(110) in n. for initial functions 11.0 = (V I (C/)(1I0) + 1') ::: (p-I (1') in n we obtain
*
/:"(p(II,O)
+ Q(II,O)
- I/I(U.O)
==
/:"c!)(lIO)
+ Q(u,o)
(0, T) x
an.
1 Formulation of the problem
= 0(1)
Here (u.)t -1{1(1I.) == -1{J(cP· I (I')) = 0(1) as I' -+ 0 on Therefore if inequalities (7), (8) hold, by the Maximum Principle we
The main result of this section is the proof of existence of the heat localization phenomenon in media with arbitrary dependence of the thermal conductivity coefficient on temperature. In 5 we obtain a slightly less general result on non-existence of localization. Proofs of these assertions are based on the operator comparison theorem formulated in 2. A different approach to the study of the heat localization phenomenon in general media is suggested in Ch. VI.
*
T
and then pass in the inequality ~ ::: 0 in
II
is a generalized solution of the Cauchy problem with compact support.
.
This inequality provides us with a pointwise lower bound from for the solution (for an application of this kind of bound, sec 6).
as
3:15
1
n.
We are going to consider in wr = (0, T) x R f • T < problem for a degenerate parabolic equation:
00,
the first boundary value
have that (II,), - 1/1(11.) :::
inf{ (II.), y,
- 1{1(1I.)} = o(
I)
11(0. x) = 110(.1)
I' -+ oj we derive the inequality in Prllll· It is possible to give a different proof. Let us consider, for example, the case of radially symmetric solutions, II = 11(1, 1'), I' = Ix!. Let Prl u] = (0. n x {O ::: Ixl < ;(1)1. As in the proof of Theorem I. 2, we use continuity of the derivative: (c!)(II)),' -+ 0 as I' -+ { (I). Then we can lind in Prill] a subset (0, x (0 ::: I' < I'.(lll, such that. lirst, 1'.(1) E (;(1) - 1', C(I)) for aliI E (0, and, second, lit - 1/,(11) ::: -8 at the point (1,1'.(1)), where I' > 0, 8 > 0 can be arbitrarily small. In the derivation of the last inequality, the value I' = 1'.(1) is chosen from the condition IIdl, 1', (I)) > 0, while the estimate II, - I/I( II) ::: -8 for I' = 1'.(1) follows from the conditions 1',(1) E (;(t) -I', ;(1)).1/1(0) = O. Therefore the argument of the proof of Theorem I shows that by the Maximum Principle ::. == II, - '/1(11) ::: -0 in all interior points of the set (0 . n x (O ::: I' < 1'.(1)1. Hence, letting I' and 0 go to zero, we obtain that ~ ::: 0 in Prllll·
as
I' -+
lit -
Of in wr. Now, passing to the limit
1/1(11) ::: 0
11(1.0)
n.
Remark 5. It is not hard to extend the proof of Theorem I to the Cauchy problem. In this case 1I0(X) must be such that ~ = II, - 1/1(11) -+ 0 as Ixl -+ 00 for all
UI
where the boundary function
*
n
'c"
:0:
O. x E R , : 110 E C(R t
(I) >
III (I)
O. 0 <
I <
T: III
E
).
sup 110 <
C(lO. T»).
00:
(2)
(3)
blows up in finite time: ex).
111(1) ->
I
->
(4)
T .
The function k(lI) (thermal conductivity coefficient) is sufficiently smooth: k E n C([O. (0)) and is positive for II > D, k(O) = O. Moreover, we shall assumc that the inequality
(,2(R t )
/' ~~!!-l
./(1
till < 00
(5)
II
holds: this is a necessary and sufllcient condition for tinite speed of propagation of disturbances in processes described by equation (I) (see 3, Ch. I). We shall also assume that k'(II) > D for all sufficiently small II > O. The initial function 110(.1) will be taken to have compact support, which by (5) ensurcs that the solution of the problem (I )-( 3) has compact support in x for each 0 < t < T. Further;more. let sup Ilcp(tIO)!,1 < 00.
*
~ 4 Heat localization in problems for arbitrary parabolic nonlinear heal equations
V Methods of generalized comparison
336
Under these assumptions there exists in W r = (0. T) x R" T < T. a unique nonnegative generalized solution of the problem (I )-(3). We remind the reader (see § I. Ch. III) that by definition problem (I )--(3) with the given boundary regime with blOW-Up will exhibit heat localization if there exists a constant I' <Xi. such that (6) meas supp 11(1, x) ::: 1',0::: I < T. The smallest possible value of I' in (6) is called the lom/izl/lion deplh. If on the other hand meas supp 11(1, x) --> 00 as I .~ T-. then Ihere IS /111 heat localization in the problem (in this case the thermal wave heats to infinite temperature the whole half-space x > 0). In this subsection we solve the following problem: given a (.I"llf/icienlly arhitrary) Ihermal conductivilY coefficienl k(lI) in equalion (I) find the classes bOllndary regimes wilh blOW-LIp (UI (I) I lI'hich lel/d to heat localization. To that end we use the method of generalized comparison of solutions of two different parabolic equations (see § 2).
or
2 Sufficient conditions for heat localization
The main result of this subsection is Theorem 1. Lei Ihe Ihermal condllClivily coellicienl const > 0 Ihe condilion
Ik" l' (0) <
k(lI)
sillis/.\·
jill'
some a
=
00.
¥Then Ihere exisl boundary blow-up regimes, which lead problem (I )._( 3).
III
heal localizalio/l in Ihe
The operator method will be used to compare the solution of the problem (1)(3) and the separable solution of equation (9) (S-regime): (I' - I ) 1/"( I - . \./ .1'0 . )21". + . .to
_ IIlfn ( 1,.\.) _
.-,>
00.
inequalities
= (II"U,),.
pfT [
k(E(p))E'(p)
IT
>
0 is a constant.
]' >
(12)
0, P > O.
-
We remind the reader that the mapping I:" : R t I--> R, must be bijective and monotone. that is E'(p) > 0 for p > 0, 1:"(0) = 0,1:'(00) = 00. Inequalities (II). (12) follow directly from the comparison conditions (17), I21 (IX) of Theorem 3. § 2. if we set there k lll (lI) = k(II). k (1I) = II". The following assertion gives necessary conditions for solvability of the system of inequalities (11), (12). Lemma 1. Lei the Iherml/I clllldllClivit." cOl~lIicienl .II/lis!"\' (7) .fill' sOllie (l' > O. Then .fil)' 1111." 0 < IT ::: ITO = 1/ It there exists 1/ SollliiO/l E or the system or inequalilies ( II), (12). Proof For convenience let us set I:"
I
= H.
Then the inequalities ( II J, (12) take (II')
kip) ::: 1-1", Ii> 0,
-J'
__k_'(_P_) [ HfT(p)H'(p)
Let us set
k (p)
< -
n
( 12')
p > O.
'
p > O.
_ w(p)
0, w'(p) :::
IIJ(P) >
The function
H(p)
n,
p:>
( 13)
n.
has the following form:
(9\ H(/I)
where
(11 )
p" -- k(l:"(p)) ::: O. p > 0,
Then, clearly, inequality (12) will be satisfied if
In this case the localization effect will be analyzed using the operator comparison method for solutions of equation (I) and an equation with power nonlinearity, II,
(10)
This solution was studied in detail in Ch. III. §§ 2. 3. It graphically illustrates the heat localization property. Here I' = .1'0. Note that the function lI lf ri is critical, since (iJ/fJt)lIl.TI(t. x) ::: 0 almost everywhere in WI· Given a thennal conductivity coefficient k in ( I). let us find out which functions (operators) E ensure that if the inequality lI(rn(l, x) ::: r;1(U(I. x)) holds on Yr. then it holds in WT. It follows from Theorem 3. § 2. that for that it suffices to find at least for one IT > 0 a solution I:"(p) of the system of ordinary differential
HfTH'(p) k(lI) ..", 00, II
= 17( - IT + -7)/ IT ]112 .
the form
Therefore the existence of the heat localization effect is independent of the behaviour of k(IIJ as II --> 00. Naturally, the form of the boundary regime with blOW-Up. which leads to localization is primarily determined by the behaviour of the thermal conductivity at high temperalUres. Sharp estimates for classes of localized boundary regimes will be obtained below. 2.1. Let us consider lirst the Cl/se or 11I/1101I/1£ll'd cOI'!ficiellIs k, when
~ 4 Heat localization in problems for arbitrary parabolic nonlinear heat equations
V Methods of generalized comparison
338
By condition
(7)
and the assumption if :::
(To,
Therefore we can always find a smooth function (13), such that
we have
lIJ(
p),
satisfying the inequalities
ltJ(p) 2: [kl/"(p)]', P > 0,
(15)
since the expression in the right-hand side is bounded for all p 2: O. Let us substitute into (14) an arbitrary function lIJ, which satisfies conditions (13), (\5), and let us show that the operator (14) constructed in this way is a solution of the system of inequalities (II'), (12'). Indeed, in the new notation (II') takes the form
339
Let us note that there exist coefficients k for which condition (7) does not hold for any a > 0 and system (II), (12) has no solutions. This is true, for example, for a coefficient k which has for II E (0,1"1, I" < I, the form k(II) = (-lnu)ll, where J.L < -I is a constant. At the same time condition (5) for finite speed of propagation of perturbations is satisfied. Let us observe also that solvability of the system of inequalities (II), (12), which defines conditions for heat localization in this problem, depends only on the 1 behaviour of the thermal conductivity coefficient k(u) for small u ....... 0 • From the method of proof of Lemma I we immediately have Corollary. LeT condiTion (8) hold, and aSS/III11' ThaT Ihere exisls sllch Ihal
\k"(p) Then I:'
=
I.:
I(pli"), lI'here
I"
(I
constanl a > 0, (16)
2: 0, p> 0,
I.: I is Ihe ./illlcliun im'cl'Se
10 k
(1.:,1 exislS due to
l/Io/lOtoniciT\' or 1.:: This FIIIOII'.\· .fi'OI/l ( I (j) i, is a solulion or The S,I'sTcm or inequaliTies
(II), (12)f(}r and by (15) is satisfied for all p > O. Next, from (14) it immediately follows that H(O) = 0, H'(p) > 0 for p > 0 (the latter inequality is ensured by the first condition in (13»). Moreover, using (15), we have from (14) that H(p) 2:
[(I
+
(T),["
d7711/11 ,,,' == kl/IT(p).
= Ila.
*
Now using Lemma I and the operator comparison theorem from 2, we <:jln formulate sufficient conditions for heat localization in the problem (I )-(3). Theorem 2. LeT The Iherl/lal ('{)ndIlclil'il,l' coefficienT k(lI) salis!'y (7) Fir some 0: > 0: leI E be some sollilion or The s,l'slel/l or ineqllalilies (II), (12), corresponding 10 a .fixed
k(7]) [k l/ "(77)]'
(f
(T
E (0,
II l1' I.
Moreover, leT Ihe bOlilldary condiliol/s
,:/
Hence by (8) H(oo) = 00. Thus the function H : R+ -> R j , defined by (14) satisfies the system (I I'), (12'). Therefore E = HI: R, -> R+ is a solution of the original system (II), (12). 0
1I0(X) :::
E
Remark 2. LeI us show that from (7) it follows that condition (5) holds. Indeed, by (7) there exists a constant M > 0, such that k(lI) < MIII/" for all 0 < II < I. and therefore -I k(lI) .[ 0
Remark l. It is not hard to see that the claim of the lemma is still valid without the restriction (8) on the coefficient k, since no conditions on the rate of increase of the function ltJ(p) (apart from (13), (15») arise in the course of the proof. Therefore we can always choose the function ltJ so that the integral in the right-hand side of (14) grows without bound as p ._+ 00, which ensures thaI H(oo) = 00, or, which is the same, that 1:'(00) = 00.
01' prohlem
salisl'1' Ihe il/e(IUalilies
E! II Uri (T,
x) I, II'here
11 1,,1
is defined ill (10).
There./iJre There is h('aT locali::.aTioll lI'ith defiTh I' ::: Xo ill flmblem (I )-(3).
Remark. In § 4, Ch. III it was shown that solution of the problem for equation (9) with the boundary regime II(T, 0) = (T- I)-I/IT, I E (0, T) and an initial function 11(0, x) E e(R+) is bounded uniformlY in I E (0, TJ fOr all x > Xo = 12(if + 2) I (T lin Using this result, it is not har~1 to show that the restriction (17) on 1I0(X) in the theorem is not essential: if all the other conditions hold, in order to have localization, it is sufficient for 110 to be a function with compact support. A method 10 prove this type of assertion will be presented below. The result of Theorem 2 prOves Theorem I in the case of unbounded thermal conductivity coefficients 1.:, which satisfy (8) .
§
Y Methods of gencralized comparison
340
Let us consider now some specific examples of the use of Theorem 2 (all the coefficients of the examples below satisfy the condition of finite speed of propagation of perturbations).
4 Hcat localization in problems for arbitrary parabolic nonlinear hcat cquations
to
(T :::
*
lIo(.\') ::: IIl1rl(O. x), X E
Rj
:
111(1) ::: (1' - I)
0
1/"
< 1<
T.
Example 2. Let k(lI) = !e" - 11,1. where A > 0 is a fixed constant. In this case inequality (16) holds for (t ::: 1/,1.. and thercfore a solution of the system of inequalities (II), (12) with IT = A is the transformation
Here the required operator E corresponding
I is
xli in R, and
III1(x) ::: E[IIII)((l.
then heat localization with the depth I' ::: ./6 occurs in the problem (I )-( 3). Using the methods of the theory of a.s.s. (see eh. VI). it can be shown that the estimates of localized blOW-Up regimes obtained in Examples 2-4. are optimal and cannot be improved. Let us consider now an example for which this is not the case. Example 5. Let k(lI) ::: In A ( I + II). A > 0 is a fixed constant. In this case condition (7) is satisfied, for example. for a ::: 1/,1., and therefore we conclude from Lemma 1 that for any IT E (0. AI there exists a solution of the system of inequalities (II), (12). Following the proof of Lemma I. let us construct the desired E. Let us fix arbitrary IT E (0, AI. Inequality (15) is equivalent to the inequality
E(p):::k l(flA)==ln(l+fI). fI>O.
IV(p)
A In,\(" 1(1 + p) ? ---~~--, fI> O. I
(T
Thcn from Theorem 2 we conclude that boundary conditions that satisfy the inequalities lIo(x) ::::.
III (I) :::
A(A-
CM
lVr ).
Example 3. Let us consider the coefficient k( 1I) ::: II expll/ I. which satisfies condition (16) for fl' = I. Therefore the transformation E(p) = k l(fI) is the required one. and if inequalities (17), (18) hold. there is heat localization in the problem ( I )-<(3) with the depth I' ::: ./6. Let us estimate the asymptotic behaviour as I --* l' of the boundary regime leading to localization. Since
(19)
+ II
The function in the right-hand side is bounded from above by the quantity
lnll+ 111,11(0, x)! . .IE R,: Inll + (1' - I) 1/,1\, 0 < I < T.
ensure occurrence of heat localization in the problem (I )--(3) with the depth /' ::: 12(,1. + 2)/,1.1 1/2 (let us observe that here 11(1, x) ::: In[1 -+ 111,11(1, x)! everywhere in
+ Inti + fI)!·
E(fI)::: k l(fI)::: Inll
and therefore if Example 1. Let k(lI) ::: II" /[1 + jJ.(II) I. (T > O. where jJ.(II) is an arbitrary smooth function. satisfying jJ.(II) ?. 0, jJ.' (II) ? O. II > O. In this case a solution of the system of inequalities (II). (12) is the identity transformation E(fI) == fl. This is equivalent to an application of the direct comparison theorem to solutions of equations ( I ). (9) (see Theorem 2 in I and Example I considered there). Therefore by Theorem 2. there is heat localization with depth l' ::: XII ::: [2((T + 2)/(T!I;:' in problem (I )-(3) with boundary conditions that satisfy the inequalities
= exp{e"-Ij- I.
Example 4. Let k(lI)
::: -
IT
-
)
I ,1/"
I
exp
{
A} .
I - -
IT
IT
< A. C,\,\ ::: I.
IT
Hence, taking into account conditions (13), we conclude that to achieve the maximal growth rate of E(fI) as fI --->x (and therefore the maximal admissible growth rate of boundary blOW-Up regimes 111(1) :::~ EI(T - I) 1/"1 as 1---> l' ). it is necessary to set w(fI) == C M in (14). Recall that H ::: E I, and therefore the slower H(p) grows as p ---> x, the faster will E(p) grow. Thus, from (14) for IT E (0, AI we obtain
*4 Heat localization in problems for arbitrary parabolic nonlinear heat equations
V Methods of generalized comparison
342
From Theorem 2 we conclude that in this problem localization is produced by any boundary blOW-Up regimes that satisfy as t -, T the condition lll(l) :::
EI(7' - tl
1/ "1::::::
(I,\(~lltrIIIT/(IT
+ I)IA(T -
tr (IHrI/"1 In(T -
This result will be used in the operator comparison of solutions of equations (I) and (23); comparison conditions are inequality (22) and the inequality
tli A. (20)
Recall that here the value of the parameter (T E (0. A I is arbitrary. In particular. decreasing (T we have that any power law boundary blOW-Up regimes 1lI(l)=(T-tl". 0<1<1': ll=const
= exp((T -
n 1/(IIAI\.
0 <
1
IH'(p)/k(p)]' ::: O. p > O.
Hence E
I (p)
:= H(p)
= /.,' k(T/)w(T/ldT/.
(22)
*
In 4. Ch. III we slUdied the action of houndary blOW-Up regimes on a medium with constant thermo-physical properties. diffusion of heat in which is described by the linear equation Vt
V(O. x)
= [' \.\'
0
< I <
7'. x
E
(23)
R,:
== () (which is not essential by the superposition principle).
("" k(r]lu)(T/)dT/
./0
0)
= expl( I' - nil.
0 <
I <
T.
leads to effective heat localization with depth L' = 2. This means that v(l, x) -.... as t -, 7' for all 0 ::: .I' ::: 2, while for .I' > 2 the solution is bounded from ahove uniformly in I E (0, T): 00
v(r, x) < v(T, .1)
= 7T
1/2
k(T/)dT/
= 00.
(28')
in order to have the maximal growth rate of E(p) (or minimal for E-I(p)) as p -.... 00. we have to require that the non-decreasing function III be bounded in R.e • for example. by setting W == I. Thus. if conditions (13). (28) hold. operator E in (27) guarantees comparison of solutions of equations (I) and (23). Without loss of generality we can consider only the case lIo == 0 in R I . Then. since the boundary condition (24) is critical. we conclude from the operator comparison theorem (see 2) that the boundary blOW-Up regime (29) lll(l)::: E(exp((T - nill. 0 < 1 < 1',
*
leads to effective heat localization in the problem (I )·(3) with depth L* ::: 2. such thal. furthermore
Elv(T.x)l, 0 < 1< T. x> 2.
(30)
In the next theorem we "pass" from effective heat localization to localization in the strict sense. Theorem 3. A.I·SIII/II' that ill the probll'lII (I )-(3) 1I0(X) == 0, (1/1(1 that the !JolIl/dary hlo\l'-llp regillle sllti.l:fies cOllditioll (29), where E: R I. --> R+ is a sollIliol/ (d' the systelll or illequ(llities (22). (26). Theil Ihere is heal loca/i';atioll il/ the prohlelll,
=
[I _Uf]
(28)
/ .0
ll(l.X) <
(24)
= cx:;.
The restriction (28) ensures that E(N) = N. From (27) it follows that in the case
It was shown. in particular, that the houndary hlow-up regime v(r,
(27)
where lll( p) is an arbitrary bounded function. which satisfies ( 13) and the condition
.">-
k(I)) ::: I. II> O.
P > O.
.0
< T.
which agrees on the whole with the fact that the exponent II < 0 in the family (21) of localized regimes is arbitrary. 2.2. Let us now consider the Cllse oj' hOlillded c(le/./iciellls k. Without loss of generality we shall assume that
(26)
p > 0
(this is equivalent to (12) if IT = 0). Setting H = E I. we rewrite (26) as
*
lIl(I)
o.
Ik(E(p))E'(p)]' :::
(21 )
will be localized. However. the right-hand side of (20) does not allow rigorous passage to the 1 limit as IT -> 0 1 (because. among others. the estimate I' ::: Xo = 12(IT + 2)/(TI /C does not make any sense then) and therefore we cannot obtain in this way the precise boundary of localized regimes. Such a boundary will be determined in 2. Ch. VI by constructing approximate self-similar solutions of this equation. To this boundary there corresponds a function of exponential form.
343
(25) 1/2 ["" .
(,' w41/14TI
I'
"T/ 1/2 dT/ < 00.
IIlld there nists
(I
('OIlStalll
1* > 0, sllch that
11(1. x)
= 0 ill
(0. Tl X {.I > /'1·
(31 )
~ 4 Heat localization in prohlems for arhitrary parabolic nonlinear heat equations
V Methods of generalized comparison
344 Pro(~j:
Let us lix arbitrary x' :> 2. Then it follows from (30) that u(t, x·) < E[ v( T ,x') I < 00 for any 0 < I < T, and therefore by the comparison theorem for solutions of parabolic equations (see I, 3, Ch. I) in (0, T) x Ix :> x*l the function 1I(t, x) does not exceed the solution of the problem
*
U,
U(O, x)
= 0,
= (k(U)U ,)"
,
0 < I < T, x:>x,
x:> x"; U(t,,t')
= Elv
Let us note that there is no restriction (7) on the thermal conductivity coefficient in Theorem 3: it is sufficient that the condition for finite speed propagation of perturbations holds. Let us cOllsider some examples. Example 6. Let k(lI) = 27T -, arctan u, Then condition (22) is satisfied (inequality (5) also holds). Selling LV = I in (27), we obtain
,x'lI, 0 < ( < 1'.
E-1(p)
*
The solution of this problem is a self-similar one (see 3, Ch. II) and has the form U(t, x) = I(?), where I; = (x- X')/III1. The function I is determined from the following boundary value problem for an ordinary differential equation: (k(j')!")' .. I{O)
1]_(_/1] ::::- ~ _'_I . P ---> I + 21] In( I + 1]) 2 In p
= f"
< 00.
Therefore the problem (I )···(3) exhibits heat localization in the sense of (6), while for localization depth we have the estimate
'11::::- exp(T -
Example 7. Let us consider the coefficient case condition (22) holds, since 2uln( I + therefore
.
Hence we immediately infer that SUPPII(t, x)::S
2 [ arctanp-7T] = p+ -p
T
--->
k(u)::S
meas
k(1])d1]
7
UI(t)::S
*
::s
=
for sufliciently large p. Hence E(p) ::::- p + 2/7T In p, p ---> 00, and by Theorem 3 we conclude that localization with depth (34) is produced by boundary regimes, which satisfy the estimate
= O.
Existence of a solution of the problem (33) for any finite I(O) :> 0 has been established in 123,681 (see I, Ch. I). There it is also shown that under condition (5), the function I(?) has compact support, that is, I(?) = 0 for all ? ::: 1;0 «(0 = I;o(.r') < 00 depends on the choice of x* :> 2). Thus, everywhere in (0, T) x (x > x*) we have the inequality 1I(t, x)
345
./0
,
<
I,
II
In this
> 0 and
I
== I we have 00.
Hence E(p) ::::- 2p In p, and therefore the localized regimes satisfy UIU)::S
E(exp(T -
t)
Ill::::-
2(1' - t) I exp(T - t)
II. (-,
T
(34) which completes the proof.
o
Example 8. Let k(u) = 211/( I + II"), In this case conditions (22), (28') hold. Selling in (27) LV == I, we have E I (p) = In(l + ,J"), E(p) = (e" - 1)'12. Therefore any boundary regime of the form
*
Remark. In 4, Ch. III it was shown that the opposite "passage," from strict (for a function 1I()(x) of compact support) to effective localization (when 1I0(.r) E C(R t ) is an arbitrary function) is also possible. It is made by deriving a special energy estimate for the difference of two solutions of equation (I) which correspond to the same boundary condition.
IIdt)::S
lexplexp{(7' -
t)
111-
11 1/1::::- exp
{~eXP(T -- t)
II}' (-, T
leads to strict localization with depth (34) (and to effective localization with L'
Estimates of localized boundary blow-up regimes obtained in Examples 7, 8 arc not optimaL Sharp estimates for these cases will be derived in Ch. VI. Example 9. Let k(lI) = 11/( I +11'). Then condition (28') is not satisfied. Choosing w(p) = p (conditions (13) arc satisfied), we obtain from (27) E I (p)
I 1 £(p) = (1 = -In(1 + V),
e'/> -
3
I ) In :::::
(,I',
p
-'>-
00,
and therefore the localized regimes arc "1(1):,:,:
exp{expl(T --
Setting now w(p) ::::: p/ In pas p obtain from (27) E'I (p)
:::::
In In
-'>-
00
I) Ill.
T .
(-'>-
(conditions
(13)
exp(el').
-'>-
00.
(-'>-
r.
p. E(p) :::::
p
are still satisfied), we
and localization is produced by blOW-Up regimes
IIdl)::: exp{exp{exp{(T As on.
cv
we could also take the function
I) 1111,
cv(p) :::::
p
-'>-
00
exp{exp ... lexpl(T
-I)-Ill ... }.
(-'>-
r.
!""
k(7]) ---d7] < 00
= meas Ix E
R r I lim
",. r
11(1. x)
= 001
<
00
(36)
(1-" is the localization depth). In this definition there is no requirement on the initial function 1I0(X) to have compact support; it is only necessary for it to be bounded in R+. Moreover, condition (5), of finite speed of propagation of perturbations, is not necessary. To study effective localization, it is not hard to modify Theorems 2. 3, as well as the results obtained in Examples 1-9. Then analysis of unbounded coefficients k(lI) uses the operator comparison methods and the derivations of § 4. Ch. III. As a result, for the localization depth we obtain the estimate 1-"::: 12(cT+2)/er]IO. where CT E (0. 1/ a 1 is the parameter in the system of inequalities (II), (12). The case of bounded coefficients is analyzed as in subsection 2. Note that the boundary blOW-Up regimes mentioned in Examples 6-9 lead w effective localization with depth L' ::: 2. cv
== I in (27). we obtain £(p) = e'~-I.
lead to effective localization with depth L' ::: 2 (this upper bound for localized boundary regimes is not optimal; see § 2, Ch. VI).
4 Heat localization in the Cauchy problem
with any finite number of exponents in the right-hand side, will be localized. In § 2. Ch. VI we shall obtain results showing that under the condition
. I
L'
and so
Proceeding in this fashion. we conclude that in this case all the blOW-Up regimes of the form 111(1) <
A solution of problem (I )-(3), which blows up in finite time, is called effectively localized if it becomes infinite as ( -'> r on a set of finite measure:
Example 10. Let k(lI) = 1/( I +U). Setting Hence the boundary regimes
p/lln p(lnln p)1 as
347
The solution of the Cauchy problem for equation (I) with an initial function of compact support (35)
7]
all boundary blow-up regimes arc localized. (Actually. this can be proved by comparison with a solution of travelling wave type, which, if (35) holds, blow up in finite time; see § 3. Ch. I). It is not hard to see that the coefficient k of Example 9 satisfies this condition.
3 Effective heat localization All the results of the previous subsection can be used to analyse effective localization.
11(0, x)
= uo(x)
:::
O.
x E
R; liE erR). 110 ¢ O.
(37)
is called locali::.ed if its support is stationary for some finite time, that is, there exists T' E (0, (0), such that suppu(t.
x)
= suppuo(.r).
0 < ( < 1'"
(38)
(herc, naturally. we assumc that condition (5) is satisfied). It is well known (see Remarks) that stationarity of a front point of a generalized solution of equation (I) is determined by the asymptotics of the initial function in ,a neighbourhood of that point. However. the magnitude of the localization time T'. which is of physical importance, depends on the "global" spatial structure of.. the function lIo(X). This dependence is reflected in the theorem stated below. \
~
Y Methods of generaliz,ed comparison
348
Theorem 4. LeI the cOI~tticiellt k sati,I:f.v condilion (7) for some a > 0, and leI E be a solution lJllhe system III inequalilies (II). (12), which corresponds to a fixed IT E (0, I I a]. LeI un(x) sali5:/.)· Ihe inequalilies
349
5 Conditions for absencc of hcal localization
that is, as I -> 1'- heat penetrates arbitrarily far from the boundary x = O. Let us remark that (4) is equivalent to the condition in R-j.
u(l. x) -> 00
I ->
(5)
l'
(the truth of this statement is proved by the method used in the proof of Theorem where
x,,, are fixed posilive constlwls. Then Ihe sollllilJ/l or Ihe Cauchy problem Ilxl < x,,,) and .f(Jr localizatioll lime \VI' !lave Ihe eSlimale P ::?: (Tx~I/[2u:~,(IT + 2)1· 111/1,
3 of
*4).
(I), (37) is localized in the domain
The validity of this statement is deduced from Theorem 2 using a technique applied in 3. Ch. 1\1 to a similar analysis of equation (9). To illustrate Theorem 4, let us consider
*
2 Sufficient conditions for the absence of heat localization Let us denote by earity
Ullrl U. x)
UI
Example 11. Let k(u) = 1''' - I. In this case a solution of the system of inequalities (1 I), (12) for IT = I is the function E(p) = In(1 + pl. Then from Theorem 4 we have that the solution generatcd by the initial function
the solution of the equation with a power law nonlin=
(11"11.,),.
which satisfies thc boundary condition u(,,)(t.Ol
= (1' -
1)".0 < 1< UllrI(O. x) E
"llfIU, x)
1 Formulation of the problem k'(u) >
*
As in 4, we shall consider in ltJr the first boundary value problem for a degenerate parabolic equation: (I) U, = (k(U)lI,)\: x E Rt
: IIU,O)
= Il, (I)
> O. IE (0.
n.
(2)
whcre the boundary function III E C I ([0.7'». II', :::: 0, blows lip in finite time. Let all thc restrictions imposed on the function k(lI) in subsection I of 4 hold. In particular, we assume that the condition for finite speed of propagation of perturbations is satisfied:
*
,1 k(7])
(3)
- - d(7]) < 00. / .n
IIlIrI
1)1 1+lIIf1/2 -> 00. 1-> T·.
o.
U
> 0; k(oo)
= 00.
k(/J) [
-jo
00. I ->
l' .
(8)
(4)
(9)
*
7]
(((I. x)
is not localized, and that
are satisfied. Conditions for thc abscnce of localization in the case of bounded coefficients k(lI) will be obtained by a different method in Ch. VI. Suppose we are given an arbitrary coefficient k E C"(R , ) n C([O. (0», which satisfies conditions (3), (9). Let us find what functions E enable us to apply operator comparison methods to the solution Ullfl of equation (6) and the solution of the original problem (I), (2), that is, when docs the inequality uU. x) ::?: E-llullriU. xl] hold everywhere in lUr· Since thc solution u(I. x) is critical. from Thcorem 3, 2 we have that to that cnd wc must find the solution E(p): R I ~ R I of the system of ordinary differential inequalitics
There will be no localization in the problem (I), (2) if meas supp
::?: an(T -
< -lilT.
This result will be used in the comparison of solutions of equations (6) and (I). Below we shall assume that the conditions
§ 5 Conditions for absence of heat localization
== O.
= const
C(R.).
*
meas supp
U(O. x)
1': n
In 2. 3 of Ch. III it is shown that the function there exists a constant an > 0, such that
is localized in the domain Ilxl < XIII} for time not less than x~,/(61l1/l).
These inequalities coincide with the comparison conditions (17), (18) of Theorem 3 of 2, if we set there kll )(II) == II", k<21(U) == k(II). Sufficient conditions for solvability of the system (10), (II) are given by the following
*
Lemma I. ASSII//1C Ihal cOlldilio//s Q' > 0, such Ihal
It is not hard to sec that if (9), (15) hold, then the function k defines a bijective mapping [0,00) ---> [0,00). Therefore the mapping E = k" has the same properties. Using Lemma I and the operator comparison theorem, we state
(9) are sali,ljicd alld Ihallhere cxiSIS a COlls1lI1I1
III addilioll, leI Ihcjill/clio// IklTj"(u) IWl'ejiJrll > 0 ali//ile //u//1her oI:eros. Then .f{1/" IT == I/a Ihcre exisls a SolUlio// E of Ihc SYSle//1 o{ i//cqualilics (10). (II),
= II a.
E
(12) is
be ({ Solulio// of Ihe sysle/ll o{ i//equalilies (10),
IOil/" sufficielllly large I <
UI(l)::: E-,II(T -
1)"1. //
l'
ha\'e
lI'e
= const (I),
(16)
Lei us set k(p)/IE"(p)E'(p)1 == I/lv(p). The inequality (II) will be satisfied if ( 13) w(p) > O. w'(p) :::: n, p > o.
Ihe// there is
Then
Proof The proof is based on comparing in (T. n x R+ the solution 1I(t, x) of the problem (I), (2) with v,. == VUI,n(t, xv"";!), a self-similar solution of equation (6), The constants T E (0.1'), v > O. are chosen from the condition II(T, x) ::: v,.(it. x) in R+. Since V,.(T. x) -+ 0 and supp V,,(T, x) ---> 10j as v ---> 0+, this can always be achieved. Then thc claim of the thcorem follows from the inequality II ::: E~·I (v,.) in (T. T) x R t , Furthcrmore, in (17) 00 == lIOll'O. where ao = (10(11, IT) > 0 is the 0 constant in (8).
Pr(}(~r
E(p)
==
[
(I
+ IT)
] 11< I
"
r .10
,"I
k(T])lV(T]) dT]
Inequality ( 10) then lakes the form (14)
Assumption ( 12) for a == lilT enables us 10 construct the function IV( p) satisfying conditions (13) and the inequality IV(p):::: lkl/"]'(p) for all sufficiently small p > O. The second condition of the lemma means that the function I k I I"]' (p) is monotonc for all sufficiently large p > p, > 0 (this condition, obviously, is not optimal). Therefore there exists lim 1' .• ""lk l /"I'(p) = K. If K > 0, we can set for p > p, £1)(p) = inf"llo.I',illk'/"]'(P)}. If on the other hand K = 0, we set w(p) = Ikl/,,!'(p) for p > p,. In both cases such an extension of the function w(p) for large p > 0, while preserving conditions (13), (14), allows us to achieve the equality F(oo) = 00. 0
Remark. There exist coefficients k, for which condition (12) is not satislied for any O. This is true, for example, for the function k (u) = expl-II"}, II = const < n. From the method of proof of the lemma we deduce
ill
R+
(IS
I
//0
--->
heal lo('(/Ii::'(llio// ill Ihe proble/ll
< -lilT.
T',
(2); 1I(t. x) --->CXJ evel)'where
a//d Ihere exists a co//slmll 00 >
0, sllch Ihal
Example 1. Lct k(lI) = uIT11 + J.l(u)l, IT > O. where J.l E C 2(Rt ) satisfies the conditions J.l ::: 0, J.l' ::: O. In this case a solution of the system (10), (11) is E(p) == p. which is cquivalent to applying the dircct solution comparison theorem to equations (6), (I) (see Theorem 2, I). From Theorcm 1 we then obtain that boundary regimes 111(1)::: (1'-1)". I --> l' ,where II < -"-lilT, do not lead to heat localization.
*
Example 2. Let us consider the coefficient k(u) :=: InA(1 + II), where A > 0 is a fixed constant. Thcn for a= I I A, condition (15) holds. and therefore the function
a >
is a solution of the systcm of incqualitics (10), (II), which corresponds to I/a = A. Thus thcre is no heat localization in the problem (I), (2) if
(T
=
Corollary. LeI co//dilio//s (9) hold, a//d ICI Ihere exiSI a co//sla//I a > O. such Ihal III (I) Ik lT ( p) 1" Thc// Ihc jill/clio// I:' for IT
==
I/a.
==
: :. n,
p > O.
k"(p) is a sollllio// of Ihe sysle/ll of i//clillalilies (10).
( 15)
where (II)
II <
:::
exp \(1'
.~
1)"1- I.
-I I A. There exists a constant Go
0>
1--->
T.
(18)
0, such that (19)
V
352
Example 3. Let k(u) = Inri operator is the function
+ In(l + 11)1.
which satislies (10), (II) for
E 1(p)
IT ::::
U
(II) :::
= exp (el'
- I) - I.
In this section the methods of ~~ 1-3 are used to derive conditions of global insolvability of quasi linear parabolic equations of the form
-In ---
I} - I.
(->
(2D)
I' .
-I lead to absence of heat localization:
meas suPPU(l ..I)
:::
{ioU'
1)11'"1/2 -" x. (-" I' .
(21)
The lower bounds of non-localized boundary blow-up regimes we have computed in Examples 2, 3 are not optimal. In ~ 2, Ch. VI. by constructing a.s.s. for equations under consideration, we shall establish sharp bounds for such regimes. In particular, it will be shown that in Example 2 there is no localization for any 11 < -1/(1 + t\) in (18), and unlike (19) meas supp 11(1,
x) :::: bo(T -
nll!1I1 ItAil12
-"
·x.
I --'"
I' .
II,
{(T - 1)"llIln(T ..... 1)1
I}. ( -" I' .
= V'.
(k(u)V'u)
+ Q(u) ==
/:l1jJ(1I)
+ Q(u).
(I)
where k, Q are sufficiently smooth non-negative functions, such thaI Q(u) > 0 for II > D and Q(II) C': O. We consider two problems for equation (I): a boundary value problem for I > O. x E n (n is a bounded domain in R N with a smooth boundary an) with the conditions u(O. x)
= 110(.1')
1I(t.
= O.
x)
I >
:::: D, .r D. x
E
E
n;
110
E
C(n),
an.
(2) (3)
and the Cauchy problem with the initial condition
where bo :> 0 is a constant, which depends only on 11, t\. In Example 3 absence of localization is caused by boundary blow-up regimes that are weaker than (20): III (t) C': exp
353
§ 6 Some approaches to the determination of conditions for
0 in R , . the required
I. Therefore boundary regimes
1I1(t):::: explexpl(T 11 <'
Since k
6 Some approaches to the determination of unboundedness
unboundedness of solutions of quasilinear parabolic equations
E( p) :::: k(p).
for
~
Methods o!' generalized comparison
(2')
It is assumed that the function Q satisfies the inequality
(22)
where 11 <' -I. Furthermore, estimate (21) holds for some (io :::: llo(lI) > O. ~et us stress that the limiting exponents in these non-localized boundary regimes, 11= -1/(1 + t\) in (18) and 11 = -I in (22), are sharp and cannot be replaced by larger ones. Example 4. Let k(lI) = 111.'/1. In this case condition (12) holds for every a E (0, II (note that (15) does not hold for any a > 0). Therefore it follows from Lemma 1 that for any IT :::: 1/ a :::: I there exists a suitable solution of the system of inequalities (10), (II). For example, let us set IT = I. Then conditions (13), (14) are satisfied for IV =' I and the transformation E has the form
..... ./ I
dry - - <' CXJ
Q(ry)'
(4)
,
*
2, Ch. I). is a necessary condition for existence of which, as we know (sec unbounded solutions of the problems we are considering. In the following we shall use extensively results obtained for the equation with power type nonlinearities, II, which appear in
= V'. (II"V'II) + u li .
*3, eh.
IT
> 0,
f3
> I,
(5)
IV.
1 A method based on l/J-criticality of the problem E(p)
= [2.f'
Tiel/dry] 1/2 :::
12p('I'11 .
p-' 00.
Hence E" 1(p) ::: 21n p .,- In(4ln p) for large p, and therefore under the inlluence of boundary regimes of the form
Let us consider the boundary value problem (I )-(3). A solution of the problem is called Ill-crilical, if everywhere in the domain we have 11,(1. x) - 111(11(1, x»
1II(1)::::2In[(T-1) 11--ln{4Inl
I.
::::
n.
(6)
Sufficient conditions of III-criticality of the problem were established in Theorem 3. Assuming that the solution is sufficiently regular (it is also assumed that
Example l. Let kIp) == I. and let the function Q be convex: Q"(p) ::: 0 for all p > O. Then as I{I we can take 111(p) = IIQ(p), where II E (0, I) is a constant. Indeed, inequalities (8), (9) are satisfied. while (10) holds in view of (4). As an initial function satisfying (7). we can take a solution of the boundary value problem (17) ~IIO + (I .- /I)Q(lIo) = O. .I' E n: lIoLlo = O.
We shall use the inequality (6) to determine conditions for unboundedness of solutions to the problem. Let the function III be positive in R+ and ."
355
6 Some approaches to the determination of unboundedness
N
This problem does not always have a non-trivial solution. For example, if f1 E R , N ::: 3. is star-shaped with respect to some point (in particular, if it is convex) and Q(lI) = 11 13 . then for (3 ::: (N + 2)/(N - 2) there are no solutions (see 3. Ch. IV). At the same time, in the case of annular domains n = {O < (/ < 1.1'1 < iJ < oo}. a solution exists for all f3 > I (see the Comments section). Thus, let the initial function 1I0 satisfy (17). Then it follows from (13) that the solution of the problem grows without bound in finite time not longer than
Then, if the solution of the problem is VI-critical. it follows from (6) that the function 111/1 (I) = max.,(cllll(l, .1') will be for all , > 0 not smaller than the solution Y(t) of the following Cauchy problem for the ordinary differential equation,
dY
- ( I ) - VI(Y(I))
dt
= 111/1(0)
Y(O)
= O.
*
(\ I)
, > 0,
( 12)
= max 1I0(x) > O. lEO
"
By (10) the function Y(t) is defined on the bounded interval (0. ,*), where
r'
= ("- ~.-./",,,(0)
111(1)
Hence it follows that the original problem there exists To :::: such that
,*,
(I )-(3)
max 11(1,
.
has no global solutions and that
f.
\/. (k(lIo)\/lIo)
+ Q(1I0)
= 1{I(1I0), x EO: lIolall
x
, "tl 'I
,ell
As 1I0 we can take any non-trivial non-negative solution of the boundary value problem for the quasilinear elliptic equation
= 0.
d1) --
<
00:
1I1/1(0)
= max 1I0·
Q(1)
Let us observe that 111(p) = IIQ(p) satisfies the inequality (9) also for arbitrary non-increasing coefficients k (when k'(p) :::: 0 for p > 0). Let us see now what can be deduced from the estimate (16). Let the solution be VI-critical with respect to the function 111(p) = /lQ(p}, II E (0. I). Then
( 14)
x) = 00.
1.
"-
. ","(ill
( 13)
00
-
:::: 1"\
In particular. I) if Q(II)
= /I/i, f3
d1).. . -.- - ::: P( To _. t) 111 (0, To) x
n.
Q(Tl)
> J. thell
( 15)
In the one-dimensional case this equation can be integrated in quadratures, which allows us to give a reasonably detailed description of the spatial structure of its solutions (see subsection 2). For N ::: 2 the question of solvability of the problem (15) is an interesting and sufficiently complicated problem in its own right (see 3, Ch. IV). Let us indicate another application of the inequality (6). If (10) holds then from (6) we can deduce the following upper bound for the unbounded solution (it
*
2) if Q(II) = (I
3) if
Q(lI)
= e".
+ /I) In/'( 1 + II), f3
then
> I (Q" ::: 0
in R t
),
then
~
V Methods of generaliz.ed comparison
356
6 Some approaches to the determination of unboundedness
357
All these estimates are sharp in their dependence on (To - I). This is demonstrated by comparing l/(t, x) with the spatially homogeneous solution v: v'(t) :::: Q(v(t)). v(T(}) :::::; 00. By the comparison theorem of 4, Ch. IV, l/(t, x) must intersect v(t) for any t E (0, To). and therefore sup, l/(t, x) > v(t) in (0, To), which produces the following lower bounds for the amplitude: I)sup,l/:> (f!--l) 1/1(3-II(To _t) 1/1(3-11: 2) SUP,II:> exp{(,8-lr l / W · 1I (To--1) I/I/HI} --I; 3) sup, II :> In\(T o - 1).1]. t E (0, Tid. Let us consider two examples of degenerate parabolic equations (the possibilities of derivation of III-criticality conditions are discussed in 3).
2 Unbounded solutions of the Cauchy problem with a critical initial function
Example 2. Let us consider equation (5) for ,8 :> iT + I. Let us set ~/( p) :::::; I' pfl, where /l :> O. a E (1.,8 - ITI are constants. Inequalities (8). (9). which reduce. respectively. to (ex _. I)(a + IT) ::: 0 and (TIl -- (a + iT - f!) p(3-" :': O. are satisfied. Since ex > I, condition (10) holds as well. Choosing the initial function in the form I/o :::: I(iT + I )vo]I/IIf+ II, where Vo ¢ 0 is a solution of the problem
In particular, a critical function is any non-negative solution of the boundary problem (21 ) 'V. (k(l/O)'VIIO) + QUlo) :::::; O. x E n; l/olall :::::; D
*
*
tl.vo
+ I(IT + l)volfJ/llflll
we have that for some To:S
-'I'I(IT
+
I)Vol"/llftll
= 0 in H, Vo:::::; 0 on an,
t':::::; Imaxl/oll-fl/!JJ«(Y -
V'. (k(l/o)'Vl/ o ) + Q(uo) :':
O. x
E
(x E R
N
luoCr)
:>
Of.
(20)
(f! C R N is an arbitrary bounded domain with a smooth boundary iln) extended by z.ero into RN\n. When n is a ball. all radially symmetric solutions of the problem (21) can be determined from the equation
In this subsection we show that criticality of solutions of a wide range of equations (I) frequently leads to global insolvability of the Cauchy problem. An initial function and the solution of the problem (I). (2') are called critical. if l/, ::: 0 in Prluj. Thus l/ does not decrease in t in (0, TJ x R N As can be seen from the results of § 2. for criticality of a solution it is. in general. sufficient that the initial function satisfies the inequality
1':::::;
Ixl > 0,
(22)
I'
'V·(ln(1 +u)'Vu)+(1 +1I)ln/J (1 +11),
(18)
where f! :> 2 is a constant. Let us take ~/(p) "" 1'( I + p) In" (I + pl. ~I :> 0, E (I, f! - I) (for (l' :> I the integral in (10) converges). Conditions (8) and (9) assume. respectively. the form (Y" - I ::: 0 and II - (ll' + I - ,8) In/!-" (I + p) ::: O. and since by assumption (Y E (I, ,8 - I), hold for p > O. Therefore if 110 satislies inequality (7), the solution of the problem becomes unbounded after time To which is not larger than
and the boundary conditions (23)
f
t';;:: (lnll +maxlioil l "/111(a- I)J <
where II", > 0 is an arbitrary constant. In thc one-dimensional case equation (22) can be integrated and the solution of the problem (22), (23) has the form (24)
00.
There is a close connection between the results of Examples 2, 3. Let us make the change of variables II :::::; ell -- I in (18). Then the function U satisfies the equation
where X kJ is the function inverse to
(25)
(19)
which differs from (5) for IT :::::; I by the presence of the additional non-negative term UI'VUI" in the right-hand side. Thercl'ore solutions of equations (19) and (5) for IT:::::; I are related by UU, x) ::: 1I(t. x) if it holds for t = 0 (here we are in fact using the simplest version of the operator comparison theorem). This argument allows us to derive unboundedness conditions for solutions of one equation from similar conditions for another one (see subsection 3).
Hence it follows that the function (24) IS delined and strictly positive for all Ixl < 1'0(11",), where I
Equalities (24)-(26) give us an idea about the nature of the dependence of the spatial structure of a critical initial function I/o on the magnitude of its maximum 11," and the coefficients k, Q. For arbitrary N ::': I, equation (22) can be integrated, for example, if Q(I/) = v 0, wbere (P(I/)
=
r
}o
k(ry) dry.
1/
> 0,
(27)
Then by a change of variable Uo --> (P(I/o) it reduces to a linear equation, the solution of which is the function ,.111'11/".1 11'1 111!(I'I/1r). where .1 11\' "Ii" is the Bessel function. Let us now state the main results, Let us fix an arbitrary R > O. We denote by nil the ball (ixi <: RI and introduce the function (28)
where il/illl is the derivative in the outer normal direction to aH/i.
However,
r/JR = O. iNIl/OIl ::: 0 on on ll (the last statement follows from positivity of r/J/i in
H II ). Therefore, taking into account (30). we obtain from (34)
If. on the other hand. 1/ is a generalized solution. then the estimate (33) is proved by a standard regularization argument. 0 Using (33) to estimate the right-hand side of (32). we deduce the inequality (35) which forms the basis of the following analysis. Lemmu
Q salis/iI'S cOlldirioll (4), as lI'ell as rhl' ill-
2. Assll/Ill' rhar rhe fil/lClioll
eql/aliry Q"(I/) ::':
which is positive in nil: here
alld Ihal rhe,.l' exisls (f COllslmll f.l.. III
359
§ 6 SOl1le approaches to the determination of unboundedness
V Methods or generalized comparison
158
:>
O.
1/
>
(36)
O.
0, sl/ch Ihal
(29)
"
A=!:N/RI,
and :~I is the first (smallest) positive root of the Bessel function .1 11'1-"1/1' The constant Co is determined by the condition 1I'(I/iIII,IIII N I = I. It is not hard to verify that f(I/i(X) solves the problem
(37) Theil Ihl' soll/liOIl or Ihl' Cal/ch.\' p,.ohll'11I ( I), (2') is UllhOUlldl'd (fm/exisrs or 11I0sr fil,. ri11ll' (38)
c:'I(I/i
+ MI/i = 0, ~//i = 0 on
iln/i.
(30)
We set '(I/i = 0 in RN\n/i. Let us consider the function H/i(l)
Proof: Let us choose the constant R > :~1f.l..111 such that H/i(O) > 0 (this can always be done if l/o¢O), By (37). from inequality (35) we have
= (1/(1. X).I('[() ==
r
1/(1, X)fP/i(X) dx.
(31 )
Taking scalar products in L"(n[() of both sides of equation (I) with 1('11 and assuming for simplicity that the function H /i(l) is differentiable in I, we obtain the equality (32) Lemmu I. Fo,. all admissihle I
:>
dr
~>
H1 _ (_I ~ I) 2f.l.. ~t W(I/).I(I/i), I > O.
(39)
--
Hence, using Jensen's inequality for convex functions 1211 J. (Q(II). ~'R) > QI(tI.I('/i)! = Q(H/i) (recall that 111(1/i1l/11l!,,.1 = I). we derive the estimate
0
(33) Pro(lj: If the solution
d H [(
}II"
From this estimate. in its turn. we obtain
1
'11"111
0 is a classical one, (33) is immediately verifled. Indeed, using Green's formula. we have II :>
(34)
,IINIOI
drl
. ~?~:.:._(:~1)1J.L, r>O.
Q(ry) :::.
R2'
Hence H[«(I) is unbounded on (0,1'1. which, in view of ensures that (14) holds.
11(1, x).
(40) H/i(l)
:::
maX,tEII 0
•
§ 6 Somc approachcs to thc dctcrmination of unhoundcdncss
V Methods of generalized comparison
360
Theorem 1. Let cOlldition,l' (4), (36) hold, lind lIssume that there exist COllstllnts J.L > 0, h > 0, sllch that.!iJr all II> h ineqllaliry (37) holds. Let the illitilll jllnction 110 he critical, lind that, moreover, 1I0(X) ) h Fir 1I11 Ixl ~ a, where the constant a > 0 is sllch that (41 )
where f' :;:;: tit'! £I?;. Existence and uniqueness of the solution of problem (47) for sufficiently arbitrary coefficients k have been established in 123. 24. 68J. There it is also shown that if (43) holds. the function f(?) has compact support: {o = meas supp f < 00 and whercver it is positive. it is strictly decreasing. This guarantecs the existence of a point { = ?/II E R I • such that /((/II) = 1./ 11 ,12.
Theorem 2. Assllme rhclI the cOlldiriolls jilllCtiol1 Q N) is 1l011-tlecrellsiI1R. that is,
Theil the soilltioll (If Ihe Clillchy problem (I). (2/) exists at 1Il0sr Fir rillle (42)
00.
Proof By criticality of the initial function. we have the inequality 11(1. x) ~ h for all 1.11 S a and admissible I > O. Let us take R :;:;: a. Then tP(lI) S J.LQUt) everywhere in B a . Therefore from (39) follows validity of the inequality
(4). (36).
Q'(II)qJ(II) - Q(II)k(lI) ~
O.
(43) lire slitis/ied 1I1ld that the
II ;>
(48)
O.
Lei the il1itial jilllcrioll 110 he critim/ l , al1d max 1I0(X) = 11/11 ;> O. 71/('n jilr N the soluriol1 of Ihe Callchy problem (I J. (2/) exists lit most Fw time < 00.
dNa tit
which by 2).
(41)
ensures global insolvabiJity of the problem (see the proof of Lemma 0
A stronger result than that of Theorem I will be obtained for N shall assume that the condition 1 [,
.0
k(T/) - - dT/ < T/
=
00
Proof: For definiteness. let sup 110 be attained at x = O. that is. 110(0) = 11/11' Then by criticality of the initial function we have the inequality 1I(t. 0) ~ 11/11 for all admissible I;> O. Comparing equation (I) (for N = I) and (45). we see that
(43)
everywhere in the domain of definition of the solution of the Cauchy problem we arc considering. The validity of this conclusion follows from the Maximum Principle: also taken into account arc condition (46) and thc assumption (43); in addition. wc make usc of the fact that V.I(I, Ixi) -> 0 as t -> 0+ everywherc in R\IO}.
Let us set h/II = II m/2. J.Lm = t/J(h/ll)/Q(h,I/)' Thcn by (4X) for all II ~ h m incquality (37) holds for J.L = J.Lm' From (50) we have that for all t ~ tT. where
, _ ')fJ..1I1
II -
the self-similar solution of the equation x E
(0)
= O.
.
(45)
t> 0
(46)
RI
which satisfies vs(l. 0) :;:;:
11 111 ,
vs(l,
(49)
(50)
*
O.
=
I. Here we
holds. which ensures that the solution of the problem has compact support in x if mcas Suppllo(.r) < 00 (scc 3. Ch. I). Let us denote by (44)
v, :::: (k(v)v.),.1 ;>
361
~
(,III)" _ ,'·N -
{~I
') qJ(III1Jll_(,III)" - . ' ) , '·N •
Q(II/11! ~){~I
(51 )
on the interval 1.1'1 S lI/11 = (/11(11 )1/2 we have the inequality lI(r. x) ~ h/ll = 11/11/2. This choice of the constants hm. fJ../I/. lI/11 ensures thal (41) holds (here fJ..1II(:.~I/alll)" = 1/2 < I). By Theorem I this guarantees unbounded growth of the function H"",(lj + t) in time which does not exceed
(here 11/11 ;> 0 is a fixed constant). The function f(?;) is determined from the following boundary value problem for an ordinary differential equation:
(k(nt)' + ~ t?:;:;:
o. ?'
> 0: f(O)
=
I Without II/II'
f(oo) :;:;: O.
(47)
that
(24)
luss or gcncrality we can takc 110 to be dctincd hy (24). It is not hard to show is the minimal nitical initial function having a maximum 11 111 > n.
~ (, SOll1e approaches to the detenllination of unhoundedness
Y Methods of generalized comparison
(in the derivation of the last inequality we used the obvious estimate H"", (I;) > tI ,,,/2). Adding (51) and (52) together, we arrive at the required result. 0 The conditions of Theorem 2 are satisfied, for example. by coefficients of equation (5) for {3 ::: IT + I (this restriction is connected with (48»). It is known, however (see ~ 3. Ch. IV), that for I < {3 < IT + 3 all non-negative non-trivial solutions of the Cauchy problem (5), (2') are unbounded in the case N = I. Hence we have
°
Theorem 3. LeI Ihe illililll./ill/cliol/ 1/0'/= oj' Ihe prohlelll (5), N = I, be criliclIl, Theil Ihe prohlelll hlls I/O glolwl soll/Iiol/s.
(2')
3 Application of generalized l'omparison theorems In this subsection unbounded solutions of equation (I) are studied by comparing them with various solutions of equation (5), properties of which were studied in detail in ~ 3. Ch. IV. For convenience. let us introduce the function (53)
o :': I
Fir {3 > I.
This theorem can be extended to the multi-dimensional problem (5). (2') under the condition I < {3 < (IT+ I)(N +2)/(N - 2) I ' Let us brielly explain the method of proof. If I/o(.r) is a critical initial function with compact support. then assuming that N 1/(1. x) is a global solution leads to the following conclusion: 1/(1, x) --> ex) in R as I -> 00. If that is not the case. then two possibilities arise: either /{(I. x) is bounded uniformly in I in R N , or 1/(1, x) stabilizes from below as I --> ex; to a singular stationary solution ii,(x). defined, for example. in RN\{O}. and ii,(O) = 00 (such solutions exist for {3 > «(1'+ I )N/(N --2)+; see 1201. 227]). By monotonicity of 1/(1. x) in I (which implies existence of a Liapunov function - .I~·I/(I, x) dx with an arbitrary compact subset KeRN. and hence the estimate .I~ 11/,(1)1 E L 1« I, (0», which would be suflicient to pass to the limit as I = Ik --> (0). the first assumption leads by standard arguments to the conclusion that 1/(1, x) stabilizes from below to a stationary solution /(.(x) > 0 in R N , which does not exit for {3 < (IT + I)(N + 2)/(N - 2) I ' (For the case 1/. = I/,(Ixl) this has been established in ~ 3, Ch. IV; non-existence of asymmetric in Ixl solutions of the equation t,,1/~ I I + I/~ = 0 in R N has been proved in 120 II·) The second assumption reduces to the lirst one. It means that 1/ --> (X) as I -> (X) only at the point x = O. However. 1/(1, x) < ii,(.r) in R+ x R/', since /{, > 0 in P(>.;I/{I by the Maximum Principle 11011 (z = /(, > 0 in a neigh6ourhood of any point where the equation for z is uniformly parabolic). Therefore there exists Xo ERN. such that 1/(1, x) S: ii,(x+ xo}, and therefore I/(t. 0) S: ii.(xo) < (X) N for all I > O. which means that the solution 1/ is uniformly bounded in R+ x R Thus. if 1/ is a global critical solution with compact support. then 1/ ->00 in R N as I -+ 00. Therefore sooner or later 1/(1, x) wi II satisfy the conditions of Theorem I (the case {3 ? IT + I) and eventually will become unbounded. For I < {3 < IT + I every solution 1/ '/= 0 of the Cauchy problem for (5) is unbounded (sec ~ 3. Ch. IV).
363
where T, relations
1I,
<
T, x
E R
4 A" IT (/C
All
;
IT
> O. {3 > I,
A are some positive constants, the latter two of which satisfy the
In ~ 3. Ch. IV it is shown (Theorem I) that if the inequalities (54) hold, the function (53) is an unbounded subsolution of equation (5). We shall use the fact that for I < {3 < IT + I + 2/ N all the non-trivial solutions of the Cauchy problem (5), (2') are unbounded (Theorem 2. § 3, eh. IV). 3.1. First we use the direct solutions comparison theorem (Theorem 2, ~ I).
Theorem 3. Lei Ihe coefficienls k, Q in (I) sati,v!'y for al/
where IT > 0, {3 E (I. IT + I + 2/ N) (Ire fixed conslwl/s. Assllme Ihal Ihe inilial fimclioll IIn(x) '/= 0 ill (2') is crilical. Theil Ihe Cal/chy problem (I). (2') has no global .wlll/ions.
Example 4. Let us consider the equation (56)
It is not hard to see that conditions (55) are satislled with
II' =: I. Then it follows from Theorem 4 that in the case of I < {3 < 2( N + 1)/ N any critical initial function 110'/= () generates an unbounded solution of the problem.
* 7 Criticality conditions for linite difference solutions
V Methods of generalized comparison
364
Let us consider in more detail the problem (56), (2') for N c!)(II} =: [
"
k(T/}
drl
I· = "?k'r .-
. 0
=
In § 7, Ch. IV it is shown that for (T = O. {3 > I + 21N for sufficiently small initial functions 110 equation (6I) has global solutions. More detailed information concerning unbounded solutions of equations of the type of (61) is obtained in § 7, Ch. IV by a different method.
I. Here
I}, II> O.
-
It is easy to check that
sgn([Q/4Jj'(II}):::: sgn [.B1I13
1(1'''' -
I) - 211 /HI ] ?: sgnl({3-2)II N ,II:::: sgn({3-2}.
Hence it follows that for .B ::: 2 inequality (48) holds, and then, using Theorem 2, we eonelude that for N = I any solution of the problem (56), corresponding to a critieal initial function (defined, for example, by (24)). is unbounded. 3.2. Below we apply to the Cauchy problem (I), (2 ' ) the operator comparison method in the form presented in the corollary to Theorem 4, 2. The following claim is proved using the results of § 3, Ch. IV.
en
*
Theorem 4. Let there exist a m(}//(}tone illcreasing .filllCTioll E : R+ That E"(II) k(lI)
Theil: I) if I < {3 < has no global solurio/l.\,2; 2)
SO,
->
(T
+ I + 21 N
alld
110
R+,
Example 6. Now let E( II) = II / Ink 2 + Ill. It is not hard to check that the function E is monotone increasing and concave in R!. Then conditions (58) are satisfied by the functions k(lI)
= II" /
In"
(e
2
+ II). Q(II)
?: 211 13 1n
l
N(e2
+ II}.
Therefore the Cauchy problem for equation (I) with such coel'licients does not have global solutions in the case I < {3 < (T + I + 21N. 110 jE O. As other examples of functions E : Ii f -> R" which satisfy (57). we could take, say, £(11) = 1I 1/ 2 In(e· l + II), E(II) = exp{ln 1f2 ( 1 + IIll - I and so on.
sllch
(57) (58)
§ 7 Criticality conditions and a comparison theorem for finite
difference solutions of nonlinear heat equations
jE 0, The Callchy prohlelll (I), (2')
iF (59)
.fin' some T < 00 (E I is The.filllcTion illl'erse TO E), Theil the sollltion (~(the problem iiJexists fi)/' time To S T, alld for all 0 < t < To we hal'e the estimate (60)
Let us consider some examples. Example 5. Let E(II} :::: In( I + II). Condition (57) is satisfied, and examples of functions k, Q which fulfill the requirements (58) are the coefficients of the equation N (61 ) II, = \7. (In"( I + 11)\711) + (I + II) In ( I + II).
Therefore for I .B < (T + I + 21N all non-trivial solutions of this equation are unbounded. In the case {3 ?: (T + I + 21N a solution will be unbounded if the inequality (59) holds: here it has the form
In this section we show that the assertions concerning criticality and the comparison theorem can be extended without requiring major modifications to cover the case of finite difference solutions of the same paraholic equations. that is, the solutions of implicit difference schemes constructed using an approximation of the paraholic operator in divergence (conservative) form. This indicates that quite suhtle properties of the heat transfer process arc shared by its cOITectly constructed finite difference approximation. This fact is of supreme importance for us, since at all stages of the investigation of nonlinear proeesses of heat conduction and comhustion we use numerical methods extensively. The theory of comparison of solutions of different paraholic equations is one of the main tools of our study. Therefore it is important that this theory can be used almost as freely at the level of finite difference solutions. Below we present our analysis using as an example the nonlinear heat equation (I)
for which we shall consider a boundary value problem in (0,1') x (0, /), where T. I are fixed positive constants, with the conditions 11(0.
21n this case the interval of time of existence of the problem satisfies the upper bound obtained in Theorem 2 of 3, Ch. IV,
*7 Criticality conditions for finite difference solutions
V Methods of gcncralized comparison
366
The non-negative function (p E C 2 (R,)nC(\0. 00» is strictly increasing: (1/(11) > o for u > 0, 4)(0) = O. The functions III), III, 112 are taken to be sufficiently smooth, 110(0)
= 111(0), 110(/) = 112(0).
Let us introduce a spatial grid W"
= lx, = kh. h
> 0: k
and a (non-uniform) grid in time
Let us denote by
= lIo(kh).
110"
=
II,
(t
- I: hM
= Il
generated by a system of time intervals
.i=O.I. .... N:
\TJ •
II'T
(V T •
= I. 2, ., .. M
7 1)
•
L
7
1=Tl.
367
1 A Maximum Principle
Let us denote by i)w T " the parabolic boundary of W r ", that is, iJw r " = It = o. x E WI,} U (t E W r • x = 01 U (t E W r • X = Il. lih = Ix = kh, k = 0.... , MI· In the following lemma (it is repeatedly used below) we obtain restrictions on the tlnite difference approximation of the parabolic operator. under which the solution of the problem cannot take negative values, that is. it satisfies a weak Maximum Principle. Lemma 1. Lef the grid .Iilllctioll :.{ he the solllfiOlI
or the prohlem (8)
0 < k < M.
0 :::
II :::
N. i = 1.2.
lI'here fhe .IilllctiOlI fJ,,(a. h. c). which is Comillilo/ls
Oil
R x R x
R.
is silch that
roO
the corresponding grid functions. which coincide with floCr) and 11,(1) at the nodes of the grids (v" and W T ' respectively. Corresponding to the problem (\ )-(3), let us set up the implicit (nonlinear) difference scheme: (4)
fJ,,(a. h. c) ~ fJ,,(h. iJ. h) ~ D. a ~ h. c ~ h. Let :.{ ~ 0 Oil i1w r h' Theil ::[ ~ 0 ill
(9)
WTh·
= max(j l:.~ ~ O. .r E (V,,: 0::: i ::~ .il < N. Then there exists .r (a point of negJative minimum in x in WI,). such that ~ :.{il. Therefore, using (8), we obtain from (9)
Proo( Let./
:.n.: v~ =
110"
iJ o
~
n. x E
= lilT
(u,,;
~ O. iJ M
vg = 110(0). V~I = 110(/): = 112T ~ O. f E W T ·
==
I
(4)(['d)"
= ,1(p(iJ, h-
I) -
2r/)(vd
+ q..(iJHill·
(6)
(7)
The problem (4)-(6) is a system of nonlinear algebraic equations. Questions of existence and uniqueness of the solution of the resulting tinite difference problem are considered in 5. Ch. VII. Iterative methods of solution of similar nonlinear finite difference problems arc considered in detail in 13461· We shall assume that the solution of the differem:e problem (4)-(6) is defined everywhere in W T ". and that at each .i-th step in time the system of equations is solvable for all 0 < T ::: T}. We shall also assume that in some class of grid functions Vii I close to vi the mapping T-> v[+ I is a bijection and continuous in C(lu,,) and that vi'l -> v[ as T -). 0 j in w". The last requirement is natural: the two preceding conditions are not restrictive and arc satistled in a more general setting (see 5. eh. VII).
..::
-T }
Therefore :.{'
I
2.
~
= (I"
(:.{
f : •
:.{I I. ::{.: :) ~ (Jt, (::{' I. :.{II . :.{II) ~
::f, which contradicts the choice of the number ./.
Here we have introduced notation which is standard in the theory of difference schemes [346}: iJ = v[' I, V, = vi is the unknown function, (V,) \ = (v, - Vk I) / h, (vd, = (v" I - vd/h arc first order difference operators. so that
p,,(u, I. U,' iJ"I)
E Wh _1+1 ~k
(II) with the boundary conditions (5), (6). where rll(v) = (p(lvl)sgn v. It is not hard to check that for this problem all the conditions of Lemma I are satisfied, so that ::[ ~ 0 in Wrl,. However, for~, ~ 0 equation (II) is the same as (4), so that by uniqueness of solutions (see 5. Ch. VII) v[ == :.[ ~ 0 in (I)r'" The second assertion follows immediately from the analysis of equation (10) with right-hand 0 side (II).
*
--.~
Y Methods of generalized comparison
368
7 Criticality conditions for finite difference solutions
3 A comparison theorem
2 Sufficient conditions for criticality of the finite difference solution
Definition.
A
solution of the problem
(4)-(6)
will be called critical if for all
Let us consider in W r " the two finite difference problems corresponding to boundary value problems for two different (/I = 1,2) nonlinear parabolic equations with operators of the form (1):
( 12)
(18)
IE w r
In the case of the problem (4)·,·(6). the theorem concerning criticality of the solution is to all intents and purposes the same as its differential analogue of ~ I. Theorem 1. Crilicalitv ot" sollllion or Ihe pmblem
(4)-(6) is el/sllred by Ihe il/'
equalities (\ 3) 11'1 (I) :::
o. II~(I)
:::
O.
I E
( 14)
10. TI·
Of course. (14) can be replaced by a condition of non-decreasing of the functions u,(t) (i "" 1,2). Pro(lt: Let us set ~, ::;: z"
(h - vd/T). Then
z'U
=
i'lir T,
Theorem 2. LeI
lind let rhe bOllI/dory cOl/dilio!lS lit" the prohlem Fir
/l
()
II>'
x E Iu,,;
T, i
=
(p) ::::
o.
(22)
[etPI'(p)/(I)III'(p)]' :::
O.
(23)
O. O:~
1:-::
I,V,+TJ~,.V'jl+TI~'II)j
u~, V~tl)'
I)
I.
(\ 5)
.,
1/)1,)
!
(p)
(I) (I I
I
1.2.
(I, x) E Iu r ,,;
X E lu,,;
,g = z'~~r = 0:
III,. i'l c, - 112, '"AI ::;: - - ' - - . IE
(u"
( 16)
(17)
. ". I )ro(lI,. •Selllllg
·1 ') = Uk'
-
lwe b ' f'or tllS I' gn.d I'unction , tle I pro)l Iem 0 tam
·1 I Uk •
TJ
To deline the initial function in (16), we have introduced an additional fictitious time level with index j = --I: T I > 0,
in W". Let J = maxLilz' k > O.X E (0,,: 0:-:: i:-:: j) <: N. Since z{ > 0 in w". we deduce from (4) that (11)(v{)h, "" z,{ > 0 in (tI". Let us consider (15) for j = J. By assumption. ~,TJ '=' Vk -- v~ --+ 0 as T 1 --+ Of in C(ItJI,). Therefore there exists Tj > O. such that Tj ~k = 0 for some .r E WiJ and T 1 ~H 1 ::: () (T 1 > 0 hy the condition ((1)( vd h, <> 0 in (OiJ). Then from equations (15), (7) we arrive at a contradiction, since its right-hand side (i),Uk I. O. Zk II) is non-negative by the conditions 1//(11) > 0 for II > 0, ~u I ::: o. 0
and by (21) ~k ::: 0 on i!tur": "\: > 0 in lu". Using the method of proof of Theorem l. we have that~, > 0 in (0" if (i),(~k 1,0. ~kfl) 2:: 0 where ~Hl 2:: 0 (here we have introduced the notation ~, ::;: ,,{ II, J = max(j I z'k > 0 in (u,,: 0 :-:: i ::: j}. Since
From the criticality of the solution i)~2) it follows that in inequality
lV r "
Remarks and comments on the literature
we have the Results of
* I, as well as of subsection
*
I of 2, are presented in detail in Comparison theorems proved here, which are based on special pointwise estimates of the highest order spatial derivative of the majorizing solution, can be considered as generalizations of well-known assertions concerning the relations among subsolutions and supersolutions of equations or systems of equations of parabolic type (see, for example, [10 I. 338, 361, 3781)· Earlier criticality theorems for solutions of semilinear parabolic equations were used in [356.357. 37H1. Criticality conditions for a generalized solution of a scalar quasilinear heat equation were obtained in 12951. Particular comparison theorems for solutions of specific quasilinear parabolic equations, proof of which uses the sign of the second spatial derivative of one of the solutions, can be found in [252J. Slightly after [147, 148, 1511, the same method was used in [451 to establish the comparison theorem for operators (33) with bl/- I == 0 (see I): the comparison condition then has the form of the (irst inequality (34). The operator comparison theorem, particular cases of which are Theorems 3. 4 of 2, stated for sufficiently arbitrary nonlinear parabolic equations, which lases estimates following from III-criticality of the majorizing solution. was proved in II 17. I 181 (in [118] the results are presented using an example of quasi linear equations with a source). Sufflcient conditions of III-criticality of solutions (Theorem I, 3) of problems in one space variable were obtained in [117,1181. In [117] in the case of the Cauchy problem the dependence Ilf = III( U. /I \) was analyzed (the setting of the Cauchy problem allows us to determine the sign of the function ;(0, x) = /1,(0, x)!/J( III1(.t), U;I(X))). Later concepts equivalent to Ilf-criticality werc introduced in 1364, 365]: these papers contain applications to unbounded solutions See other applications to explicit solutions in 11391. Most of the results of 4, 5 are contained in [146, 1541. A different approach to the analysis of the localization phenomenon is used in Ch. VI. Theorem 4 of 4, concerning heal localization in the Cauchy problem, was proved in 11461· Conditions for immobility for a finite length of time of the front point of generalized solutions of equations with k(u) not of power type. were studied in [2521· For kUtl = u", rT > 0 such studies are to be found in [58, 232, 2481· For the multidimensional equation some results in the same direction are contained in [59]. There (see also [3281l the authors also study the properties of the degeneracy surface, which corresponds to an arbitrary generalized solution of the Cauchy problem for the equation lit = Doll"! I. Most of these results are analyzed from a general standpoint in the monograph 11031· The method of analyzing unbounded solutions of parabolic equations with a source (subsection I, 61. based on III-criticality conditions of the problem, was presented in [120, 1251. To establish upper bounds a similar approach was used [147,148,15 11.
Therefore (25)
where (r/PI) I is the function inverse to rl)12!. Since e/pi - r/illl docs not decrease in R, (sec (22)), using (25) we obtain
r/P'(n~21) -
r//II(f{i) ::::
~ [r/PI(n~211)
- rIl'l I (e/p) I
+
e/PI(n~2: I)]
*
-
(~(e/P(V~211) + r/P(fJ~2;1))))'
*
Substituting this estimate into (24). we obtain
*
(26)
Let us introduce the notation rIP I(f{; I) be written in the form .'1,;;1 f!~21 _ .'1:;,11 i)~21 :::
-i
1~2
= 1/1H I.
Then the last inequality can
{r/)III (r/PI) ~ 1 (1/1, I ;
*
11',+ 1) ) (27)
[e// II «¢I2I)"I(W' 1))+r//II«r//:'I) I(W HI ) ) ] } '
However, inequality (23) ensures that the function e// II ((e/)121)' I(p)) is concave, since
where T/ = (r/PI) I(p). Since f«T/1 + T/2)/2) ::: If(T/I) + flT/:.)!/2 for any concave function f(T/) and any T/I, T/2 > 0, we have that the right-hand side of (27) is non-negative, which completes the proof. 0
*
*
•
V Methods of generalized comparison
372
independently by [1081 and in a large number of consequent papers. In proving assertions of ~ 2, we use a method (called in the terminology of [289J the method of eigenfunctions), which was also applied for a similar analysis of boundary value problems in bounded domains for semilinear (k{u) == I) [243. 289], quasi linear [120, 121, 125. 225] parabolic equations and systems thereof [1611· Proof of Theorems I--3 in ~ 6. which uses a new idea. viz.. criticality of the initial function, is given in 1120. 1271. Part of the results of subsection 3 of ~ 6 is to be found 1150J. For other methods applicable to the study of unbounded solutions see eh. IV. VII. For short surveys concerning unbounded solutions of evolution problems see [157.289.3341. The results of ~ 7 are a particular case of the statements proved in 1126 J. where criticality of finite difference solutions has been established for parabolic equations of general form U t :::: L{lI. U,. u,,). [1261 also contains comparison theorems for solutions of implicit difference schemes for two different equations of the form
//"1:::: r
Chapter VI
Approximate self-similar solutions of nonlinear heat equations and their applications in the study of the localization effect
§ 1 Introduction. Main directions of inquiry
L{uli') .1.\.\, /,("1) .
In this chapter we propose a general approach to the study of asymptotic behaviour of solutions of quasilinear parabolic equations lit ::::
(k{lI)u,),: k{u) >
O.
II
> 0.
(I)
where k E C"((O. oo))nC{IO, (0)). For this equation we shall consider a boundary value problem in WJ :::: W. T) X R+. T < 00, with the initial condition u{O. x) = lIoer)::: 0 in R I , and with the fixed boundary behaviour 11(1.0) = lIJ(I) > 0, f E to. T), which shows blOW-Up behaviour:
(2) We shall he interested in the asymptotic properties of solutions of the problem under consideration, which are expressed at times sufficiently close to the blOW-Up time I = T'·. and in particular the restrictions on u(l. 0) = 1I1 (I) under which the problem admits or does not admit localization of heat (understood in either the strict or the ellective sense; see ~ I. Ch. III). We want to undertake such a study for sufficiently arbitrary boundary regimes 1/1 (I). and for a wide class of coefficients k{lI) as well. As we have mentioned already (see eh. I-III), an efficient method of studying such problems consists of constructing and analyzing self-similar or some other invariant solutions of equation (I). which satisfy some ordinary differential equations. These particular solutions have a simple spatio-temporal structure, which defines the form of the boundary condition u(l.O) = UI(t) and supply us with the necessary information concerning the asymptotic behaviour of the process. Reasonably detailed information concerning invariant solutions of equation (I) is contained in Ch. I--III.
However, the group classification of equation (I) performed in [321, 3221 shows that the number of solutions of this equation invariant with respect to a Lie group of point transformations is not large. The more general Lie-Backlund transformations do not signilicantly enlarge the class of invariant solutions; new possibilities here arise only for k(u) = (I +" u) 2 (see 1221, 51, 2621l. For k(lI) not of power (k(lI) t- lit', IT = const) or not of exponential (k(lI) t- e") type, equation (I) admits only two types of nontrivial invariant solutions: liS (I . x) = fs(x /( 1+1) liC)) and IIs(l, x) = fs(x - I). Of these only the second one, in the case when the integral J;""" (k(T/)/T/) dT/ converges, is generated by the boundary blOW-Up regime. Equations (I) with a power law or exponential nonlinearity admit other invariant solutions of interest for us (sec ~ 3. Ch. II). The fact that certain parabolic equations of the form (I) admit a wider class of invariant solutions provided one of the main stimuli to the development of the special comparison theory for solutions of various nonlinear parabolic equations (see Ch. V). However, the upper or lower bounds of solutions of equation (I) of general type, obtained in the framework of comparison are frequently not sharp enough, and thus do not allow us to describe the asymptotic stage of evolution of the process. Using the theorems proved in eh. V. it is not always possible to single out. using upper and lower bounds, a sufficiently narrow "corridor" of the solution's evolution in time (narrow enough to enable us to speak about a correctly determined asymptotic behaviour of the solution). This is mainly connected to the paucity of invariant solutions of equation (I). To determine asymptotic behaviour of solutions of equation (I), we use in the present chapter approximate self-similar solutions (a.s.s.). which, though they do not satisfy equation (I), do describe the asymptotic properties of its solutions correctly. In each of thc following sections we shall describe a different method of construction of a.s.s. The main problem is that of determining the principal (or we can say, defining) operator in the right-hand side of the equation, which dominates the fully developed stage of evolution of a boundary regime with blOW-Up, Of particular interest arc the results of ~ 2, where we determine a class of coefflcients Ik(II11. for which the defining operator is a first order operator, and finally the asymptotics of a heat transfer process is described by invariant solutions of an equation of Hamilton-Jacobi type. It has to be noted in particular that every non-trivial self-similar solution of the nonlinear heat equation (I) is, as a rule, asymptotically stable with respect to small perturbations not only of boundary conditions, which is quite natural. but also of the equation itself (that is, to perturbations in the coefficient k(lI) with . respect to the corresponding invariant dependencies). It has to be said that in the laller case the term "small" docs not have to be taken literally. since frequently an a.s.s. obtained from an invariant solution as a result of a small perturbation of k(lIl. docs not look anything like it.
*2 Approximate self-similar solutions in the degenerate case
375
As discovered in [184, 185, 186, 187], the set of sufficiently "regular" asymptotic behaviours of solutions of equation (I), which grow unboundedly, can be subdivided into three classes. depending on the character of growth of k(lI) for large II; each of these classes consists of three subclasses, ordered by the form of the boundary functions (lid!)}. The first class, k(lI) of "weakly linear" form, is considered in ~ 2; the second class, of !dll) "close" to power law dependence, is studied in ~ 3. In ~ 4 we propose another method of constructing a.s.S" applicability of which is perhaps more restrictcd: this study leads, nonetheless, to intriguing general conclusions. In this chapter we do not consider the third class. of nonlinearities k(lI) close to exponential. sincc in the analysis of asymptotic stability of the corresponding a.s.s .. a boundary value problem in a bounded domain with moving boundaries has to be considered [186 \. so that this case cannot be applied in the study of heat localization in half-space. Questions related to construction of a.s.s. connected with usual boundary regimes without blow-up: III (I) -> Xl as 1 -+ Xl, are not considered here. In this regard, see the papers [184, 185, 186, 1871; in the last of these a classification of such a.s.s. in the "plane of boundary value problems" is made.
§ 2 Approximate self-similar solutions in the degenerate case I Statement of the problem Let us consider the first boundary value problem: 1I,=(k(II)II,),. (I.x)EWj::=(O,TlxR,:
II(().
x) ::= 1I0(.~) ?:: 0, x E R"
110 E C(R+), SUPl/o < 00;
(((1.0)::= ((1(1):> O. () S 1 <
(I) (2)
T; (3)
We shall take the function III in (1) to be monotonc increasing. In this section we construct a.s.s. for a large class of equations (I) with nonpower law nonlinearities. These O. lk(.I)/k'(s)1' ->
00 . .I~> 00,
(4)
,
,.:;:
I
j
'N
k(TJ)
+I
TJ
.0
= 00.
~~dTJ
(5)
Condition (5) places a restriction from below on tbe behaviour of k(lI) for large u, while condition (4) restricts its behaviour from both above and below. In particular, it follows from (4) that for any a > 0 and all sufficiently large .I' > 0, we have the estimates (4') s'" < k(s) < .1'''. In the following we shall need the function £ defined by the equality
j
'EIII
.0
k(1)) - - d1) =.1', 7/
+
1
.I'
> O.
=
I I
Below we shall demonstrate that under the assumptions made above the solution of the problem (I )-(3') converges asymptotically in a special norm to the exact invariant solution 11., of the following first order equation (a Hamilton-Jacobi type equation): II,
-
k(lI) = ---(11.,)-.
The function
11.,
is an a.s.s. of the original cquation
(9):
1+ n
,
~tfl,
(8)
(we shall need this result below). Some typical coefflcients k(lI), which satisfy conditions (4), (5), are shown in Table I. There we also give the leading terms of the asymptotic expansions of the functions £(11) of (6) for large II. These are needed in the determination of the form of a.s.s.
== E(II) "" .....--......---....----:-:..~~---~~--.+-.-------~--'--~----.--__1 exp {In l/ " II (I + ,,)~)} exp!ln"( 1+ (1)1.0 < a < I expll(l + a)lIjl/11 t"ll explll/ In III ell
We shall show that from the point of view of study of the localization phenomenon, the most interesting boundary regimes with blOW-Up are of the form 11(1,0)
= lid!) = £1(1' -. {)"],
0 < ( < 1'.
= I.
> 0,
8,(00)
(II)
= O.
Existcnce and uniquencss of solutions of the problem (II) were established in Ill. Therc wc also studied its propcrtics. and. in particular, obtained the following estimates:
*4, Ch.
f<'(t) ::: o. t (j/l
k(lI)
+ nfl, = 0, t
8,(0)
Table I
I
and has the form
The non-negative function 8, is the solution of the boundary value problem for the ordinary differcntial equation obtained by substituting thc exprcssion (J 0) into
from (4) it follows that
II +Inll +In(1 +11)]1 I {I + In"(1 +1111 1.0 <: a <: 1 II + In( I + II)) I l [I +In(1 +1I)j ... _.. .. ll +Inll +In(1 +II111
(I)
(10)
,1
In"(1 +1I).1l' 0 Inll + In(1 + lI)j
(9)
(1,.1) E lur.
lI+l
(fly -
0, 11--+ 00
the function (3') describes a
1
max k(E(TJ));
--'>
00,
377
2 Formal determination of a.s.s.
1)c[O·1I1
pdll)/lI
=
where n < 0 is a fixed constant. As E(oo) regime with blow-up.
(6)
The function £ is positive and strictly increasing in R I , £ E C'«O,oo)) n C([O,oo), £(0) == 0, and £(00) = 00 (the laller IS assured by condition (5)). Therefore £ is a one-to-one mapping R 1 -> R+ and there exists £ 1 R j --'> Rt , a monotone function inverse to £. For all II > 0 let us define the function pdll)
* 2 Approximate self-similar solutions in the degenerate case
I
VI Approximate self-similar solutions
376
•
=
max O':(f;)
L'EI().~!I)
where to = meas supp 0,; moreover. R t ), to = 2 for 11 = -I and
for
11
< -I.
For
11
E
Ill. go):
= (I
to -
- n)/4 <
CX)
for
(12) 00,
11 E (-1,0)
(that is, 0, > 0 in
== -I the solution of thc problem (II) is the function 0,(.1)
== (I - x/2)~. x::: 0
( 13)
(in this case t == x). By thc condition 0,(0) = I, the function (10) satisfies (3'). Let us denote by 0(1, t) the similarity representation of thc solution of thc problem (I )-(3'), dcfincd by thc spatio-tempnral structurc of the a.s.s. (10): fI(I,~)
* 2 Approximate self-similar solutions in the degenerate case
VI Approximate self-similar solutions
37K
3 The convergence to a.s.s. theorem Let us show that the similarity representation of the solution of the problem under consideration converges as t ---> T to (I,. which ensures that the solution u(1. x) is close to the a.s.s. (10). Thus we establish asymptotic convergence of nonstationary solutions of equations of different orders: a parabolic one (I). and one of the Hamilton-Jacobi type (9). The physical reason for this sort of degeneration of the original equation in the case of k(u) == I was discussed in 4. Ch. Ill. It is not hard to present the same kind of analysis for k(u) of general form.
*
Theorem 1. Let cOllditiollS (4). (5) he satisfied. Theil the silllilarity represellllltioll (14) of the solutioll of the prohlelll (I )-c:n jill" 11 E 1-1.0) CO/1\'crj.;cs liS t ---> rto the jill/ctioll 0, (1:;). the solutioll of the prahlelll (II). Morcol'l'I". \I'e hlll'e the estill/lite \10(1.') - O,(·)llcm.1 == sup 10(1. 1:;) - O,(~)I
=
~c1L
= ()
(cr -
1)-" ('
Jo
I)d~~' I -
T)"I
dT)
--->
O.t
--->
Let us introduce in WI new functions V. IJ, defined by V = Substituting u = E(V) and II, = E(U,). into, respectively, (9), we obtain the equations
ProIJ(
U,
= CI(II,).
T .
T
(15)
E-I(u),
(I)
379
For Il = -I the function U, == (T - t) I (I - xI2)~ also does not have the requisite smoothness: U, ~ C!;2(WT) (but. which is very important, V.I E C'(WT). Therefore we shall be using the fact that the non-negative generalized solution of the first order equation ( 17) can be obtained as the limit as € - . Of of a sequence of classical positive solutions of parabolic equations
V:
(20)
satisfying the same boundary conditions as U, 1257. 2601. Here. since we have that V, E CI(wd. we have the uniform in € E (0. I) estimate jW:),,1 ::: const in (8. T) x R+: 0 < e5 < T < T are constants (this is important in the proof of convergence to a.s.s.). The sequence of functions ;:;: = V" - V: E C;;2(Wr) converges uniformly as k ---> 00, € ---> 0', to a function;: on each bounded set IV~ C W r = (0, T) X R+. T E (0.1'). From the argument above. we shall formally assume that the function ;:(1. x) is sufficiently smooth. Here we are implicitly assuming that the necessary estimates are first derived for the smooth functions ;:;:(1. x). and the final result is obtained by passing to the limit as k ---> 00. € ---> 0+. Let us note that if equation (20) is used instead of (17). the equation for ;:;: includes an additional term. which is not essential for the final estimate (15) as € ---> 0+. Thus, let;: E C!;2(wr) n C(Iii·Fl. Then from equation (18) by the comparison theorem we conclude that
and max ;:(1.
x) ::: ;:
,
(16)
j
(1).
min ;:(1.
x) :::;: (f).
\
where the sl1100th functions;: + (r) satisfy the inequalities
(\7)
Let us setlJ(1 ..1) -U,(t. x) = ;:(f. ;: satisfies the parabolic equation
x).
d;:+jdt::: supk[E·W(1. x)]supW,),,(1. x).
(21 )
o. ()
(22)
As follows from (16). (17), the function d;:- Idt ::: ( 18)
and the conditions
Below we shall analyse the solution of equation (I X) with the help of the Maximum Principle. To justify its usc, we make the following remark. The generalized solution u(1. x) (and therefore V (1. x» of the degenerate equation (I) does not necessarily have the smoothness required for the formal appliI. Ch. I). However the function u(1. x) cation of the Maximulll Principle (see (V(1. x» can be represented as the limit as k ---> 00 of a sequence of smooth. positive in WI solutions Uk E C!;2(ltJr) (Uk E C!;2(wr)).
*
< 1 < T.
and moreover ;:"«l) = max ;:(0. x) < 00, CeO) = min ;:(0. x) > -00. [n the derivation of (22) we take into account the lirst of the inequalities ([ 2). Using the notation (7) and the explicit form of the function lJ, == (T - I ) "(l,(~), we obtain supk[EWli = sup kIE(s)] = pd(T - 1)"1. ,'HO.II-II"I
sup(U,) I I \
= (T -
1) 'max(":(~) ~,:Il
= (T -'-lr1q",
Substituting these equalities into (21), (22). we derive as d;:f
dt
s: q"
1 --->
1 --->
pd(T-1)"j d;: - 0 T - t . dl::: .
T
r. the estimates
VI Approximate self-similar solutions
380
§ 2
Approximate self-similar solutions in the degenerate case
381
Hence by the inequality 118(1,,) - 8,(·)lIc
s:
Theorem I'. Let k(lI) == I. Then j(iI' all.\' n < 0 the following estimate of the rate (I{ cOl/vergel/ce 10 a.s.s. is valid:
(1' -I)" maxlz+(r), Iz'(t)I), 0 < t < 1',
we deduce the validity of the estimate (15). We need only to demonstrate convergence of 8 to 8, as t ---+ 1'-. Resolving the indeterminate in the right-hand side of (15), and taking (8) into account, we obtain .
.
,~.I]l118(t,·) -' O.,(o)lIc
.
q" pd(T - t)"l (1' __ t)"
s: ,'.~~~ =:-;;
118(1,·) - O,(·)lIciR.I
= OJ(T -
1)"lln(T -
1)11, I .......
r.
(15')
The case n E [-1.0) was consiuereu in Theorem I. Let us note for k == I is the same as (15'). since then pdll) == I (see (7). Thus, let 1/ < -I and without loss of generality 110(.1') == O. Then the solution of the problem (I )-(3') has the form pro(~r
= o.
(15)
o Theorem I allows us to determine asymptotically exactly the dependence of the depth of penetration of the thermal wave on time. From the convergence estimate (15) it follows that xcf U) satisfies as I ....... 1'- the equality 11,(1. xc/(I»::::
11(1.
x)
= ~/" f' 27T
-.10
exp {-
.1'2
}
4(1 - r)
lexp!(T - r)") -
II (t
- r) :'/2 dr.
1/(2£[(1' - I)"]).
Hence, using the specific form of a.s.s.
~ r t --
'. ( )
·\c
11,(1, x)
(see (10», we obtain
(I' __ )III·"I(1lJ.···· 1 EIIEI ~(1'( l'_ -t)"t t I, [
Here 0; I is the function inverse to 0, (0; the monotonicity of OJ. Let us demand in addition that .
1
,
t .......
exists on the interval
k(s/2)
11m - - -
,.,"0 k(s)
)"1/ 7- 1]
r
.
(0, I)
(23)
in view of
(24)
I.
(25)
which holds for all I sufllciently close to r. Let us consider separately the case 1/ < -I. For 1/ < -I the function (U,), has a jump discontinuity of the first kinu at the "front" point xo(l) = ~0(T _1)111"1(1, so that U., E C(wr). Therefore auditional difficulties arise in the proof of convergence to a.s.s. for 1/ < -I. Below we obtain an estimate of the rate of convergence (15) for 11 < -I for the case k(lI) == I, when E(s) = exps - I, so that the similarity representation (14) has the form
= (1' -
I)'"
. ~o 1\,,,/2 11(1, .\0(1)) S:~/"
27T -
Then it is easily verified that E 1(E(.I)/2)/s ---+ I, .I ....... 00. Since O;I(~) :::: (1 -- t) / (-1/) 1/2 for small ~ > 0, we deri ve from (23) the followi ng estimate for the penetration depth of the thermal wave:
O(l,~)
The main problem is to estimate 11(1. x) on the weak uiscontinuity surface of a.s.s. (10), xo(l) = ~o(T -- 1)11t"I/2, on which the a.s.s. II, == 0 and does not have the requisite smoothness. Setting in the last equality x = ~o( l' ,- 1)111 "1/2, and introuucing a new variable of integration by (1' -- 1)(1 -- r ) I = .1', we have the following estimate:
In[ 1+ 11(1. ~(T --
I)I'I"II!)I.
(14')
["" exp! -- A" P" (.I) (s' \
.0
II" . -
I
I .I.
where we have introduced the notation A = T --I ---+ ot , I ....... T , and P,,(s) stands for the function ~6s/ 4 -- (I + .I)" .I ". It is easily checked that PIllS) is non-negative exactly for ~o = 2(-11)"/2(-1II) ·1"+ III!. Therefore the above intcgral converges anu goes to zero as A == T - I ....... 0+. Hence 11(1, xo(l» = 0«1' - 1)"/2) anu thus U(I, xo(l) = 0llln(T -- t)ll as 1---+ T~. Since 11,(1, x) s: 0 (see ~ 2. eh. V), Ihis estimate holds everywhere in (x::: xo(l)}. In the uomain (0 < x < xo(l)}, where U., E C'c, the mcthod of proof of Theorem I can be applied, which givcs us as a result the estimate (15') of the rate of convergence to a.s.s. 0
4 Sufficient conditions for absence of localization Theorem 2.
ASSlIlIle thaI conditions (4), (5) hold. Then there is 110 lomlizalion in Ihe prohlelll (I )-(3') .fill' n < - I. The sollllion grows Il'itholll hOlllld liS I ---+ T~ everyll'here in R I . Furthermore,
odmils .fillile speed or proPO!:llliOIl 01" perlurholions Illld < 00, Ihell .fil/· Ihe size or Ihe supporl or Ihe solulioll we hllve Ihe
(I)
meas supp Uo eSlimate x.{/)
== meas suppu{/. x) 2:
=
(1' _1)(IIIIJ/2(~O - 1'(1)). 0 < 1< 1'.
2 Approximate self-similar solutions in the degenerate case
In the case k(u) == I, localization of the solution of (I )_(3') for II 2: - I is proved in § 4, Ch. III. by analyzing the heat potential. All the arguments above, as well as the results of numerical computations, indicate that for II 2: -I the solution of the problem (I )-(3') is localized.
(27)
= 2(--11)"12(-1 - II) <1+111/2 alld Ihe 1101I-lIe!:alil'ejilllcr. II" k == I, Ihell hy Theorem I' 11'1' hal'e equalily ill
5 Examples
where ~o meas suppH, lion 1'(1) --+ 0, I --+ (26).
Let us denote by O{/. x) the solution of problem (I )-Cn in (0. n x (0. xo(l)). xo(l) = ~o(1' _1)11111112. satisfying O{/. xo{/) == O. 0(0. x) ::: uo(x) in (0. xo(O)). Since 11(0 ..\0(1» 2: 0 in (0.1'). by the comparison theorem u 2: i/ in (0, 1') x (0. xo(l»). But 11., E C;}((O. n x (0 . .1'0(1»). 11,(1. xo(l» == O. Thercfore as in the proof of Theorem I . we sec that H{/. ~) --+ (}-<~) as I --+ l' in C«O. ~o) with the rate of convergence givcn by (15) (here fl{/.~) is the similarity representation (14) of the solution ill. Then the claim of Theorem 2 follows from the inequality u 2: ii in (0. 1') x (0, xo{/»); in the derivation of (27) we use the expansion
Proo(
(28)
o Theorem 2 provides sufficient conditions for the absence of localization in the problem (I )_(3'). Unfortunately. Theorem I cannot be used to establish the parallel result for presence of localization for II 2: - I. In the case II = - I we can prove the following assertion: Theorem 3. Assllme Ihal Clllldiliolls (4). (5) hold. Theil Ihe solulioll o(the prohlem (I
)-cn fiJI'
II
= -I
sali.l:!ie.l· Ihe relalioll . 11111
I:I(u(l, x))
,.... r
rr -- I)
:=:
(
.1')2 . x
I - -
1
2
E
(29)
R+.
I
Let us consider other examplcs. A Example 1 (compare with Example 2. § 5. Ch. V). Let k(u) = In ( 1+ u). where A > 0 is a constant. Conditions (4). (5) are satislied. The transformation £ in (6) has the form (sec Table 1) 1:'(11) :=: expll (I + A)IIII/il tAil - I and therefore. setting IIA = 11/( I + A) in (3') we obtain IIdl)
II
E (-I,
0) in
cn. then
E 1111(1.
x) I
:=:
0«1' _1)"). I --+
r.
in R+.
Relation (29). which follows from (15) and (13) for H-<~) in the case II = -I. means thatu(l,x) grows without bound for all x E (0.2) and u(l,x) :::: EI(1' -I) 1(I - x/2)~ I as I -, l' . If. on the other hand . .1'-:: 2, then u{/. x) :=: 0(£\(1' - I) I\). which. however, does not ensure uniform boundedness of the solution. At the same time it is clear that a, s. s. (10) (which correctly dcscribes the asymptotic behaviour of the solution of the problem) is localized for II -:: - I.
= expl (I + ,\)1/11 +,\1(1' -- 1)"' I
I, 0 < I < T.
(30)
To this boundary blOW-Up regime corresponds the a.s.s.
where ~:=: x/(T - 1)11111+Allld/ 2 From Theorem I it follows that for II E 1-1.0) the similarity representation (14) converges as I --+ l' to the function H,. and that we have the estimate IIH(I .. ) -
H,(-)
lie
0:=
()
«'I' - I)-II,) --+ O.
I --+ 1'.
From the struclUre of the a.s.s. (31) it can be seen that for II A < -- 1/( I + A) there is no localization in the problem. and that 11(1. x) ~ expl (1' - 1)"' l. I --+ 1'-, for any x E R'I' Equation (I) describes proccsses with finite specd of propagation of perturbations. Therefore for /1,\ < - 1/( I + A) the size of the support of the solution can be estimated using (27). For /1,\ 2: -1/( I + A) a.s.s. (31) are localized. In particular. in the case IIA :=: -1/ (I + A) (S-regime) the following equality holds asymptotically as I -+ 7'" (see Theorem 3): u{/. x) :::::
Remark. If
383
expl( I + ,\)1/11+'\1(1' - I) l/tIHI( I - .r/2)~/111'\11 - I.
Penetration depth of the thermal wave depcnds on time thus:
Hence it follows that in the case A > I for /1,\ E [-(A - I) 1.0) in (30) penetration depth decreases to zero as I --+ l' . This conclusion also holds true for the
§ 2 Approximate self-similar solutions in the degenerate case
VI Approximate self-similar solutions
384
boundary HS-regime, which heats up to infinite temperature the whole half-space R+ (see § 4, Ch. III). For A = I the behaviour of x,./(t) is practically independent of the parameter II: x,,/(t) = O((T~· 1)1/2), I -->- T. The above conclusions confirm that, in general, shrinking of the half-width is not sufficient for localization.
Conditions (4), (5) are then satisfied and the function £ defined by (6) has the form £(11) :::: explexp!II'f1) - II - II. Therefore in this problem the boundary blOW-Up regime 111(1)::::
Example 2 (compare with Example 3, § 5, Ch. V). Let us consider equation (I) with coefficient k(lI) = Inll +·In(1 + 11)1. Since here £(11)::::: explll/lnll} as II --> 00, from Theorem 2 we deduce that the boundary blOW-Up regime 11,(1) :::::
leads for
II
expl(T -1)"/lllln(T
~·l)ll. 1->
r,
(32)
< -I to absence of localization. Here
meas SUppII(l,
x)
Z. (()(1' - 1)11+"112
--> 00. 1-->
385
exp(exp!(T - I)"}- I} - I)
will produce no localization for
II
< -1/3. For II ::: -1/3 the a.s.s. are localized.
6 Localization conditions for arbitrary boundary blOW-Up regimes In the construction of a.s.s. in the degenerate case we made substantial use of condition (5). If the integral in (5) converges:
T'.
If on the other hand II ?: -I, then the a.s.s. are localized. To the boundary S-regime (II = - I) corresponds an a.s.s. of the following form:
l/k ::::
l
. ()
"-
k(ry) --
ry
+I
dry <
(35)
00,
then the function £ in (6) is defined on the finite interval (0. (/k), and therefore £~ I is uniformly bounded in R+. Thus it makes no sense to consider the family of boundary blow-up regimes (3'), and no a.S.s. of the form (10) exist here. To clarify the meaning of the restriction (5), let us consider I
---'f
T : 0 < x < 2.
Example 5. Let the coefficient in equation (I) have the form
From the relation (25) it follows that in this problem decreases as I -+ T: x,,/(I) ::::: (-II)
tn(T -- 1)11")/2 In Iln(T - 1)1
Example 3. Let k(lI)
= II + Inri
-111)1
(II
?: - I) the half-width k(u)
-->
o.
1-->
I.
T' .
(33)
ensures for II < - I unbounded growth of the solution of problem (I), (2), (34) everywhere in R I.. On the other hand, if II ?: -I, then we should expect heat localization in the problem (at least the corresponding a.S.s. have this property).
k(lI)
= 31n 2 11
-I In( I +
1+ Inri
+ II)
1I_~1.
+ In(1 + II)!III + In(l + In(1 -I It))!
I
x ...
(36)
... xll+ln(I+ ... +ln(I+II)) ... )II.II>O.
In this case £(11) :::: exp(e" - II -- I, therefore it follows from Theorem 2 that the boundary regime (34) 111(1):::: explexp{(T - 1)"1- 11- I
Example 4. Let the thermal conductivity coefficient have the form
= II
In each successive bracket the number of logarithms is increased by one. Let the last bracket contain M logarithms, that is, (36) has M factors (for M = I the coefficient (36) coincides with (33)). The transformation £ is determined from (6): E(II) ::::
explexpl···
k" - II .. ·) -
II - II,
where in the right-hand side M + I exponents are used. Therefore from Theorem 2 we conclude that the boundary regime LlJfI) ::::
£I(T - 1)"1
=
:::: explexp ... (exp((T -1)/1) -
(37)
ll ... -
I} - I. 0 < 1< T,
is not localized for II < -I. Comparison of (36) and (37) shows that increase in the number of logarithms in the last square bracket of (36) leads to the following
situation: to bring about the HS-regime without localization, faster and faster (more rapidly increasing as I -'> TO) blow-up regimes arc required. As M -'> 00 the intensity of these regimes does not have an upper bound (in a certain sense). At the same time as M is increased, the "rate of divergence" of the integral (5) becomes lower. Therefore as condition (35) becomes satisfied, the intensity of minimal boundary regimes that lead to absence of localization and unbounded growth of the solution in the whole space, becomes infinite. If condition (5) holds, all the types of boundary blOW-Up regimes arc possible: HS-, S-, and LS- regimes (under the restriction (4) this was practically established in Theorems I. 2). On the other hand, if k(lI) satisfies (35). then, apparently, there are no non-localized HS-regimes. Let us state again the result which can be proved by comparison with travelling wave solutions (see ~ 3. Ch. I).
k(oo) = 00. The function k defines a bijective mapping R j we can define k I : R, -'> Rjo the inverse function to k. Let us set (I>dll)
= Ik(s)/k'(.I')I'l-k
-'>
R+ and therefore
(4)
'1111 .
The main condition on k is as follows: there exists a constant that l
(T
E
R+, such
(5)
In Table 2 we list some coefficients k, which satisfy condition (5). In all the cases the constant 'Y > 0 is chosen sufficiently large. In the right column of the table we list the principal terms of the expansion of the function k I (II) as II -'>x. Table 2
If Ihe colldilioll (""
./1
~i1])
d1] < 00
k(lI)
(38)
Condition (38) is satisfied, for example, by the following coefllcients: k(lI) e- II , k(lI) = (I + II)", (T > O. k(lI) = II + In(1 + 11)1 A and
holds. allY hOlmda,.." hllJlv-lIp regillle ill Ihe prohlelll (I )-(3') leads /0 lo("ali:alioll.
k(lI)
387
§ 3 Pointwise estimates of the rate of convergence
VI Approximate self-similar solutions
386
k-- I (II) "" 1/" In-- It / IT In II (1'''''' II "In-- It / " II exp( _oIT"("+ II In" II,If
exp
{
..
III
In } In"lnll
0 II -IT '--
It will be convenient for us to write the boundary condition (3) in the following
way: (3')
§ 3 Approximate self-similar solutions in the non-degenerate
case. Pointwise estimates of the rate of convergence In this section we consider non-degenerate a.s.s. of the nonlinear heat equation, which correspond to given blow-up regimes on the boundary x = O. Unlike the degenerate a.s.s. of ~ 2. they satisfy (under other conditions on the thermal conductivity coefficient) a second order parabolic equation. Maill asslIlIlpliollS will FJrlllal defillilioll of lU.S. As before, we shall consider in WI" = (0. T) x R I the (irst boundary value problem II,
11(0. x)
= 110(.1')
11(1.0)
= A(II) = (k(II)U,),;
~ O. x E R, : 110 E C(R I
= III (I)
> o. IE (0, T).
III E
where 1/1(1) is a smooth monotone increasing function, '/1(1) --;. We shall seek a.s.s. of the problem (I )-(3') in the form
00
as
I -'>
T-·.
(6)
where the unknown funetions 0,(0,
(I) ).
sup 110 < 2
c qO. T)).
00;
(2)
The function V, satisfies the equation
(3)
where the function III of (3), which blows up in finite time, is taken to be monotone increasing. Let us introduce the necessary restrictions on the coefficient k(lI) E C'((O. (0)) n cqO. (0)). We shall assume that k'(u) > 0 for II > 0, k(O) = 0,
(V ,),
. 1,"0 I ; ' )(1) f)".\ (s) L = IT(lp 1/
(/" 1/
. d{i',' (ij)
(I)~-;i"{-'
(8)
_ _ _ _ _ _ _ _ _ _..0._...._ I Fllf IT =oc ()+ relation (5) is equivalent to the equality (I'd X! ) analyscJ in § 1.
x, This case was
------. *:1 Pointwise estimates of the rate of convergem:e
VI Approximate self-similar solutions
388
389
Here the functions O,(~) ::: 0 satisfy ordinary differential equations obtained by substituting IIs(l. x) into (14):
or, equivalently,
(UJ,
= (T ( -'1/ )
au,
( -¢' )
(I)U, -
o
(I)x-,-. dx
¢
(8')
Let us introduce another piece of notation: U(t,
x)
= /.:(11(1, x)),
x) E
(I,
I
+ IIlf
I.
" " (0, e,)
-
II.
(0"0')' \ ..\
+ -0' 2 sSt
,
--2-0,~ (T
+ flO, = O.
- 0.=\n '
~ > (;
.
)
(17)
n.
( 18)
...,J_~
>
(9)
(Vr.
as well as the boundary conditions For U we obtain the equation
0,(0)=1.0,(00)=0. (10)
where D,,(U)
:=:
UU"
I
1
+ -U~.
(19)
Under these assumptions on /.:(11) the method of constructing a.s.s. specific case depends on the form of the boundary regime.
in each
(II)
(T
Let us observe that U,(I.O) == U(I.O) = 0"(1) for t E (0.1'). Below we shall analyse the equation satisfied by the difference 2 = U - U, using the Maximum Principle, and we shall derive conditions such that 110"(1,') - O:(·)lIcm,) ---+ O. t ---+
where we have denoted by the problem (I )-(3'):
0(1,
0
r.
( 12)
the similarity representation of the solution of
1 Approximate self-similar solutions of type I In this subsection we construct a,s.s. of the problem (I )-(3') with function Os satisfying equation (17). The problem (17). (19) was studied in detail in 2, Ch. Ill, where it was shown that for any 11 < n its solution exists and is unique. For 11 < -I I (T the function 0, has compact support: ~o = meas supp 0, < 00, while for fl E (-I I IT, 0) we have 0, (~) > 0 everywhere in R,. Moreover,
*
e,(~)
In the case The convergence condition (12) ensures that as t ---+ l' . . the solution of the problem under consideration and a.s.s. (6) have the same properties. Then the function ¢(t) determines, modulo a constant factor, the dependence of the depth of penetration of the thermal wave x,.r(l) on time. Under the above assumptions, the construction of a.s.s. uses exact self-similar solutions of the equation with power type nonlinearity: II,
= (/I"u,),
*
II.
IISU, x)
= (1' -I)I/O,(t),~ = .r/(T - n,I+III,)!:. t E (0.1'); 11 = const < ~
= xlexp{(rtI21. t
= - II
(T
j,"f)
+ .... t
00;
---+
e = C(II, (T) = const
>
n.
" = [( I -
O,(l;)
~) -:-.
to
]:!" . t
to. =
> 0;
[2«(T(;2)JI!:
(21)
I
To prove convergence to a.s.s. we shall need
Lemma 1. Let 0., be the soilltioll o{ the problem (17), (19). nlell (22)
Remark. For Proof
0;
> O.
(20)
the solution can be written down explicitly:
( 14)
(here the constant (r > 0 is the same as in condition (5)). We shall need two types of self-similar solutions of equation (14) (sec 3, Ch. II): I.
Let us assume that there exists a point ~ = ~. E R f , such that .r;;'(~·) -IlIT. Let us show that this leads to a contradiction. Let us consider first the case 11 < - 1/ IT. Then (i, (0 == 0 for ~ ::: ~o = meas supp g,. and (g~/"g',)(~o) O. It is not hard to show using the Banach contraction mapping theorem that as ~ --'r ~()' we have the expansion
Theil the prohlel/l (1)-(3') has a.s.s. (6). where IP(l)
(25)
2'
(0,
n.
(29)
( 12) holds. Let us make some preliminary computations and sec what cquation is satisfied by a.s.s. (6) undcr thc conditions of Theorcm 1. From (17) it follows that
0" I
+
= !_Il(¢/'t 1/tII')(I)1 II"-. IE
Ihe fUllctioll (J,(~) is the sollltioll of'Ihe prohlem ( 17), (19) alld the limitillg equality
Thereforef' < ~o.i.e. g,(~') > O.
I IlIT IT--C.
391
3 Pointwise estimates of the rate of convergence
I
dCO"
= - -0" --~ lilT d~c
(30)
I
Substilllting this equality into
(8).
we derive the following equation for the
function U,:
Let us rewrite (23) in an equivalcnt form: 1+//(T
liT" (" .( g ) ,"'l,\'.\
2
/ ' \"' I: " 1". 1"'J\S
,""t,\
+ //(T ,-'sII = 0 ,SI: '
(1
"
>
0.
(26)
Integrating both sides of the equation over the interval (C. ~o) we obtain
_g~/
_
Setting now fJ':
IT~2//(f ~.] + + IT (I
+21l1~ + 11)
.t" g~/"(1))d1) =
= u, N", ~ = X Ic/J.
em o. where G(I)
= IIn(V"/I/JlI'(t).
I
Hence, taking (25) into account. we arrive at a contradiction. since
.t" g~/
>
o.
+ 7) -
( 11
.
I)
+- I'~' If + 2 . 0
Theorem 1. LeI cOlldilioll
t: 1/ "(1))d1)
(I/I/I//)'(t)
I --'r
-.
11
I
--'r
T
:
n
(I)
==
I. Therefore the
= g'(O).
(U,),
= D,,(U, ) + G(t)(U,) ,x.
(31 )
=
Selling in accordance with (7) U., k(u,), we deduce that under the conditions of the theorem a.s.s. (6) satisfies the equation (U,)I
= Aftt') + [~ -
k'(uI)(III)~ + G(t)(u,),x,
(32)
which differs from the original equation (I) by two additional terms in the right-
1
11 :::
-I
/(IT
+ 2),
hand side. g',(O) < O.
o
(5) hold. alld leI Ihe ./illlClioll 1/1 ill (3') satisfv the
conditioll
---1// T + 1
equation for U, has the form
•.1
which cannot possibly hold in view of thc conditions
and D" is the operator (II).
I(11/4/)
11
=
2
= (I + IUT)/(21l)
Howcver. by the choice of cP (see (29)) -_.
Now let 11 > -I/IT. Thcn g, > 0 in R+ and g':(oo) O. We havc to considcr two cascs. The first casc is 11 E (-I/If. -I/(IT + 2)), when g~/
we have
= COIlSt E I-I/IT,O).
Remark. It is interesting to note that a different equation can be derived for an equivalent a.s.s. For example, from (17) we have the equality (here 1/ # -1/ (T)
Substituting it into (8), we derive for U, the equation
H(u) > 0
and H(II) is non-increasing for
II >
n,
(39)
(40)
H(II) "'* 0, II -} 00.
Let us obtain an upper bound for :.(1, .). A lower bound will have the same form. It follows from the form of the operator N that the function:. satisfies the inequality
If instead of (29) we detlne the function cP(r) so that
then, setting in (33) U, :::; k(u,) we have for a.s.s. (6) another equation,
Using the fact that is not hard to show that:. is a subsolution of the equation
(u,),:::; A(II,)+
+ [1_ (!).(k(1I »] .IT ,.,
k'(11 )(11)" ., I,
+
k(lI,.) 2/l(T (In k'(u..}I+/l(T
1//)' r/J (I).
which differs from both (32) and the original equation ( I). In the sequel we shall only use equation (32).
Proot' (~t' Theorem I. Let us set? = U - U., in U)], ? E C(UJ'll. Then it follows from (10), (31) that? satisfies in w ] the parabolic equation
in the domain (:.+ ::: O}. Therefore:. ::: :.' (:.+ ::: 0) in (0] if this inequality holds for r = 0 and on the boundary (0.1') x (x = OJ. lt is obvious that the function:. + is, in its turn, a subsolution of the parabolic equation wi
(34) Here .'E I is the linear elliptic operator obtained by transforming the difference D,,(U) - D,,(U,): (35)
(36)
.N is the nonlinear operator N(t, x;:.)
dw
I/(T!«({!,) ,
+ :.,)2
(37)
2
-- H (II)
:s <1) d II) -
I/ (T
:s
H (II). II > O.
(38)
in
w(l)
dr
(41 )
(0],
== wsup(U,}" + sup I.MI + H(w)
Taking into account the specific form of U, (see hard to see that for II E [ -1/ (T. 0) sup(U-!"
1/1"(1) == -/-0-sup I[0,
,()-(I)
0
sup(U,)~, r E ({l,
= M o == sup :.(0, x)
11'(0) Here :.(1,0) == 0, M o = sup 1:.(0, x)1 < 00. Below we shall derive upper and lower bounds for:. in (v] by constructing spatially homogeneous sub- and supersolutions for equation (34). Thus, let condition (5) hold. Then there exists a continuous function H(lt):
> Oin(o].
where the function 11' satisfies the following boundary value problem for an ordinary differential equation:
-
= 1(!JdU, +:.) ,-
+ 2(U-!,)w;+
Appealing to the usual comparison theorem (with respect to boundary data; sce I), solution of this equation can be estimated in terms of the spatially homogencous solution w(l). Summarizing all the above, we have the following estimate: 1:.(1, x)1
dw if" 1// . 1'" J I -,= _-11111 -w + p"II/JIGI + -II/ ' II/H(w). (I -II
" lI"iJ
,
lim 1,(1)
\
w(/)
-
r
-
-/IIT
and therefore
IIJ" (I )
w(l)!tI/' U) --->
=C
lim t//IGI fm,
r IV
=
,=C lim -1111t//
(44)
< --.
to prove the limit equalilY (12), il suffices lo check that 1-+
+ h(l) + 1)(1).
Let us consider the expression for h(l). Consecutively resolving all the indeterminacies that arise as I -+ T. and using the exact form of G(t). as well as conditions (28). (29). we obtain (here C > 0 are some constants)
f·.
(·)lle = -..---- sup 1:(1. x)1 I(J" (I )
(43)
lhal
" I
(I,,) - (I
+ 1,(I)lr"(I) =
< 00.
has the form
(44)
"(I)
ll
= sup [(H':lr
Since for aliI E (0.1') we have by
= II (I)lr"(I)+/2(1)I!1
< 00;
f.·/
11/
1(1
- -(1/1 = C" hmr (I) f-·
=C lim
0 as
f-.r
l' .
From (43) il is not hard lo obtain an estimate of w(l). for example. on the interval (1'/2.1'). To estimate the last (nonlinear) term in the right-hand side of (43), lel us use the inequality
E
111(1)
~
E
1)(1)
=C
lim f,~ ..
T
H(MIt//J,,/I-IIJ(I))
=C
lim
H(lI)
II-'X
= O.
+ fllIll/,,,!1
'''1(1)
{'
./1/2
II/"/"!!
Remark. For II < -I/(T. by (24) the function wn~ experiences a jump at the point ~ = ~II. that is U, docs not have the smoothness required for the proof by the above method. In this case convergence lo the a.s.s. is easily proved for fI(I. x), the solution of the problem (I). (3') in the domain (0, 1') x (0, xo(t)) (there Us E C:/), satisfying flU, XII(t)) == 0 on the moving right boundary XII(I) ~() T' in C«{),~())), From this estimate we conclude that there is no localization in the original problem (I )-(3') for II < ... 1/ (T and derive, in particular, a lower bound on the size of the support of the non-localized solution (see proof of Theorem 2. ~ 2). Table 3 shows functions cp(l) corresponding to various boundary regimes (3'), in which t/I satisfies condition (28). Properties of the resulting a.s.s. are largely dependent on the relation between the quantities II and CT. Thus, for II < -1/ (T the half-width of the a.s.s. grows without bound as I -+ T', that is, the solution of the problem (1)-(3') is not localized (HS-regime). On the other hand. if II > -I/er
=
(1'/2,1')
M1Ifl,,,/1 "1(1)
T
(1'/2.1')
Therefore in view of (39). (40) we have the inequality
Consequently. for all I
lim f--..
Thus w(l) jt//' (I) ---> 0 as I .--> T , which. as shown by (44), ensures that (12) holds. This concludes the proof. 0
dw (/" 1(/ (t) > ---w. dl - -II 1(1(1)
-
from which it follows that for all I
Similarly. by (40)
"I(T)I(;(T)I ciT
+
~
VI Approximate self-similar solutions
396
a.s.s. becomes infinite at the single point x c.:: 0, which indicates localization of the solution (LS-regime). For II = -I /IT both localized and non-localized solutions can exist. Let us consider the case II = -1/ IT in more detail. It is easy to see here that condition (2S) is satisfied by the function 111(1)
Then in (29)
4)(1)
= (I' -
which satisfies condition (28). Therefore, by Theorem I, to the regime (48) there corresponds the a.s.s. with half-width satisfying
In particular, for
II
=-
I / IT this dependence has the form"
== I, so that the half-width is constant (S-regime). Hence for (l' > is/IT we have that If)(1) -> CXJ as I -'> T", which means that the solution graws without bound on the whole space (HS-regime). If (X < 8/IT, then cPU) -'> as I -+ T ,and the a.s.s. is localized (LS-regime); if a = 8/IT then the S-regirne obtains: as I -'> I' the solution has a constant (non-I.era) half-width. Substituting into (48) II = -1/IT, a = 8/IT. we see that in the case of equation (47) the boundary blOW-Up S-regime is
°
In ewn Iln(1"·- 1)1 1)11+,,"lnlln(T I)I,w;2
(T·-·I)"ln"lnI2+(1' I) II (T--I)"ln"12+(1' I) II
iT
1)1 I + lilT 1/2
II.
IT
1)(1
(I' ... I)" explln"llT
1)'\
cr
t""I;2 exp { } IlnCr- I)I"}
O
a>
397
(45)
I) I."T
Table 3
(I' - I)" exp {
3 Pointwise estimates of the rate of convergence
i;;a 11~lli~t-;T~~~};"il },
(1'
_I)II+IIITI/"ex P
{~~(T '_I)_I_} 2In"lln(T
1)1
2 Approximate self-similar solutions of type II
0
Thus, if condition (5) holds, to the S-regime in the original problem there corresponds the boundary condition (46)
Below we construct a.s.s. of problem (I). (3 ' ) considered in a domain (0. T) x (0, xo(l}), xo(l) == ~(4)(I) > 0, with an additional boundary condition on the moving right boundary 1/(1. xo(l)) = O. In this case in the a.s.s. (6) the function (J, satisfies the problem (18), ( 19) and has compact support: meas supp 0, = to < 00 (sec ~ 3, eh. 11).
''Then a.s.s. can be written down explicitly: Lemma 2. LeI fl, be lile .whilioll
or problem (18), (19).
Theil
Example I. Let equation (I) be (47)
where 8 f:. 0, 'Y > I, and let in problem (I )·-(3) lIill)
= (I' -
I)"
In"l J
+ (I'
- 1),1 J, I E (0, T);
The proof of this lemma, which is similar to the proof of the previous one, uses the asymptotic expansion (J
(l'
= const f:.
0.
(r
,
(r2~O = ------(1'(1 2 S
(48)
Then it follows from O') that to this boundary regime there corresponds the function
In particular. we obtain
-
C) S
IT:'' . .. + -4(IT ---(l;o + I) -
. 2
-~) -
(O':nt:o) = IT"/12(IT + 1)1
+ ... ,1;-, I;(}. .-
< IT.
"Let us note that only in the case Ii co () (when (47) hecomes an equation with power law nonlinearity) and n = 2/(T can the expression r/)(I) :c: Iln(T -1)[. I --, I' ,he ohtained from an analysis or the exact self-similar solution 1184. 3211·
Theorem 2. Assume that condition (5) holds and that the function I/J in (3') sat-
399
Approximate self-similar solutions of nearly linear equations
i.ljies rhe condition
[I/JN']'(I) -, 0. t
-'>
T
(49)
Then there exists w/ a.s.s. (6), such that
In this subsection we shall construct a.s.s. of equation (I) with a coefficient k, which satisfies k E
the /illlctiol/ O\(~) is the solwiol/ or (he prahlI'm (18), (19) (ll/d as t -'> Tw in C((O, ~o»).
C 2 ((().00)
(I"(I.~) -> (I~T(~)
n C(iO. 00);
(klk')'(I/) -> 00.
Some functions k satisfying
The proof of this theorem is identical to the proof of the previous one. il is easily checked that if condition (49) holds, (p(l) -> 00 as t -> 1'-. Therefore under the conditions of the theorem, boundary blOW-Up regimes in (3') are HS-regimes.
t E (0, T);
1/
+ II),
satisfies condition
lY
lY
> ()
+ II) In" In(3 + II). lY > cxp!ln"( I + Ill. () < It < I In(3
(/J(t):::::(-I/)
+ II)
::: (1' .._I) "~/" exp{(T
1
(-'>1',
li, :::::
§ 4 Approximate self-similar solutions in the non-degenerate
case. Integral estimates of the rate of convergence
= 1I0(X) 11(1,0)
= lI,(I)
(I)
(I. x) E WI;
~ 0, x E R+; 110 E C(R
j ),
> ()
sup 110
> O,lI',(1) > 0, t E (0,1').
<00;
(2)
(6)
iI,\\"
This approach is different from that of ~ 2, where, under condition (5), a.s.s. were determined using invariant solutions of the tirst order equation lI, ::: k(II)II~/(u+ I). Let us observe that in this section we are considering more general coefficients k(lI), since there is no need to impose the restriction (2.5), which was essential in ~ 2. We shall need two types of self-similar solutions of equation (6):
In this section we present a different method of constructing a.s.s. for the problem
II((). x)
(l'
..__..
can be constructed using the self-similar solutions of the linear heat equation
"(~ililj2exP{~(T-I)"}'
= (k(lI)lI, ),.
.
_-------It wi II be shown that for a particular choice of II d I) a.s.s. of the problem (\ )-(3)
- t)"}
for the half-width of the solution.
lit
}
~-----_
Therefore we have the estimate 1 / 2(1'_1)1
0
< 0. In( I
(49).
4.
> ()
cxp { I~" In(3 + II)
= k1/fT[ul (t)]
(5)
=
k(lI)
In" In(3
Then the function 11/(1)
(4)
/I -> 00.
arc shown in Table
(4). (5),
In"(3 + II).
= expl(T -1)"1.
> 0, k'(u) > 0. II > 0;
Table 4
Example 2. Assume that in the problem for equation (47) 11,(1)
k(lI)
I,
x) :::
cr -I)"II((), (::: xlC!' _1)1;2,
I.
lIs(l.
II.
I(s{(, x)::: c'h(x), r >
r E (0.7');
II
< 0;
O.
(7)
(8)
It is assumed that f,(O) ::: I. f,( (0) ::: 0 , i ::: I. 2. Equations for the functions are obtained by substituting (7), (8) into (6). It is easy to verify that
(3)
In all the cases considered below, the proofs proceed by deriving some integral estimates of the difference of 11(1, x) and the corresponding a,s.s. This analysis in fact establishes "transformation rules" for known invariant solutions of (I) for k ::: lifT, (T ::: const ~ 0, under the transformation of the thermal conductivity coefficient u fT -'> k(II).
...
I, ( I/J.... _. Il ) /."" ~- ""11-1 . 1I~"ln exp {1::2 _._--.} s (1+.\) 7T'/-I (-n) ,II 4
fl(~)=-,--'-"
.
In each ease an a.s.s.
II,
_.
.
.
_
)-,\
d.l·h(.t)-(·
of the problem (I )-(3) is sought in the form (9)
*4 Integral estimates of the rate of convergence
VI Approximate self-similar solutions
400
where the non-negative and sufficiently smooth functions 0,(0, cP(t), have to be determined. Setting e., (0) :::: I, we have that the a.s.s. (9) satisfies the boundary condition (3). Let us denote by e(t,~) the similarity representation of the solution of the problem: I OU, t;) :::: -.-IIU, t;qJ(r)). t; E R.;. II Jl r ) '
.
(10)
.
We shall show that for a particular choice of the functions II I. {}" cP, the function r. which ensures asymptotic closeness of the solution 1I{r. x) and the a.s.s. 1I,{r, x). In the proof in the sequel it is assumed that 1I{r, .) E L 1(R t ) for all t E [0. 1') (the norm in L 2(R,) is denoted by II . 112). Moreover, without loss of generality. we shall take the initial function 1I0(X) to be non-increasing in R+. Then by monotonicity of the boundary regime, this property will also apply to the solution 1I(t, x) (see 2, Ch. V).
I Approximate self-similar sollllions (~ltype I
Theorem I. Let cOllditions (4), (5) hold, and aSSlIIlle that jilr all r sldliciently close To T 1I1(t)=(T-fL
I) as II ._, 00, k (II) grows slower thall allY power: FJr II
WI."
a
:>
k(lI) < II", k'(II) < II" I. 2) k!l (u) <
0 Fir all
> 0 we hal'c rhe illcqllaliries
0 for all sl((!icicllrl\' large
II :>
(I I)
0:
3) jill' any ~: E (0, I)
flU)
=T
-
[' • I
Then rhere exisrs
dT --) T', t -- T k!(T - T)"I
(k/k')'(II)
==
alld
lim lIe(l.·) - 0,(·)112 '····r
k(~II)/k(lI)
::::
II + k'(?)I1( I
However. by 2) k'(n < k'(gll) for large
II
_1:]-1
k(~II)
- ~)/k(~II)1
k(lI)
k(~II)
The exact form of the rate of convergence of flU. ~) to II, (~) will come out of the proof of the theorem. Unboundedness and monotonicity of the function (12), (12') as r --) T follow from Lemma I. In a number of cases we can write down asymptotically exact expressions for the boundary regime (12). (12'). For example. if .
1/4 in (12) can be rclated to the need for the inclusion
-
l.
[(_kJ.::L)' J \ ...k'(s)
lcr -
E L'(R, ). Since
it does not hold for lim
r)"k"I(T - 1)"1.
c!I(I) :::::: (T - 1)1/1k l12
which completes the proof of ( I I'), since
I/-·OJ
(14)
(this condition is satisfied by coefficients of lines one to three of Table 4. and by the coefficient of line four. if a E (0. 1/2)). then it is not hard to show that
> 0 and therefore
I 1+ k'(~I.,)t;1I .. -'
lim k'(gll) (II
= O.
J - k"(II)k(II)/lk'(II}\' -, 00. II __ Xi.
To prove 3), we shall use the finite increment formula: k'(?)I1( I - ~). where? E (~II, u). Hence
[
(\2')
(13)
,. k(s/k"(s))
The claim I) follows immediately from (5). Concavity of k for II --) 00 (claim 2)) follows from the relation
.
(91, where
UII (1..\ . .1'.
(II' ) Pro(~l
(12)
(t))",
monotone ./ill1crirm
Lemma I. Let the coetJicicllr k satis!)' cOlldiriolls (4), (5). Theil: sl~f1iciellrly largc
1
where Il < -1/4 is a COllstallr, and fL 1 : 10.1') --) [0,1') is rhe i/l\'erse of the
OU.~) converges to O,(~) as r . --+
*
401
II
E
00.
1- 1/4.0).
Proo(o(ThcorclII I. Let us define a smooth monotone function fL : !O. 1') --) !O. T)
Let us show that the right-hand side of this inequality goes to zero as ( ~ TO". The first term goes to zero hy the assumption II < -1/4. Let us consider the second term. Since 1-/-'(1) == 1/lk«(T - t)")\ as { -, T, and resolving indeterminacies in (20), we have
( 17)
WI)'
- T) ""
. I.
lim IIIJ
I
(1' - T),,·1/2G«T - T)"; T)
1
, ..·T
-(j.L (t)k(II)II, -
I
')II~ + 2q,
n. where t. <
E (I ..
we obtain
and integrating hy parts, we obtain 0
Then for all t
Hence. using the easily verifiahle equality
- t) i2).
(16)
I c/ L ( t
00.
.
Ohserve that u,(j.L(I), x) is precisely the self-similar solution (7) of equation (6). Let us set w(l. x) = 11(1-/-(1), x) - u,(j.L(I). x). Then under the ahove assumptions w(l.O) = 0, w(r, x) ~ () as x ~ 00 aIllI w(l . . ) E e(R I l as { ~ T . Taking the scalar product with lJI of hoth sides of the equation -
is increasing. the integrand is hounded uniformly in ?
E
(0, I) as
Under the conditions of Theorem I. we have that (/)(t) -, 0 as ( --> I' . Therefore the struelllre of the a.s.s. (9) indicates that the solution u(f, x) grows without bound only at the one point .\ = O. This points to localization in this prohlem. with (12) an LS hlow-up regime. Example \. Take in (I) k(lI) = In"(2 + ,i). from ( 12). (13) that the houndary regimc
==
n";
k(lI)
/.\
. II
( ~ T. and hy (II') goes to zero as t-> l' for any t" E (0, I). This proves (14), while (20) provides an estimate of the rate of convergencc to the a.s.s. 0
we derive from (17) the estimate
- sup
.
- 2/-;-+ 1/2 ,'I.IP
Using the identity
111(1) :::::: (-11)""(1'·
(¥
>
0 (sec Tahle 4). Then it follows
1)"\ In(1' - 1)1"",
t~-> l' .
404
*4 Integral estimates of the rate of convergence
VI Approximate self-similar solutions
leads to the appearance of a thermal wave, the half-width of which decreases as T- according to
I -'>
The constant ~* E R+ is such that
f
df') :;;: 1/2.
2 Approximate selfsimilar soliliiolls of Iype /I Theorem 2. Lei cOlldiliolls
(4), (5)
I sl~t.riciel1l/y
and u,(I, x) --+ 00 in R+ as I -+ TO,. It follows from Theorem 2 that the solution 11(1. x) will have the same properties, i.e. in this problem there will be no heat localization and the boundary blOW-Up regime (24) is an HS-regime. On the other hand, the boundary condition (24) leads to the appearance of a thermal wave with a constant (as I --+ T ) half-width eJ>(I) == I. Condition (23) ensures that such boundary regimes belong in the class of blOW-Up regimes. In l I 87] it is shown that divergence of the integral
[o.c dry
hold, alld assllme Ihal, moreo\,{'/',
.10 (23)
For all
405
close 10 T " lei
k(e")
=
00
leads to solutions with constant half-width being generated by boundary regimes without blow-up, which are defined for all I > O. In this case a.s.s. arc defined by the same formula (9). This will be the case for IIj(l) = exp{f.L I(ll}. where we have denoted by f.L I the inverse of
(24)
R; is Ihe ill\'erse otlhe mOllotolle illcreasillg fllllClioll (!£:filledfiir .1'IIf.ficielllly I(//:I;e 7 hI'
where f.L'1 :
(0, T) ._>
f.L : R+ -> (0,
n,
f.L(7) = 7· -
l~ -dry -.
Theil
(hI)
==
l, H,(x)
(25)
k(e'l)
,T
== h(x) ::: e • alld Ihe eqllalilv
(/4)
We establish the
(26)
Hence, by Lemma I.
is a necessary and sufficient condition for a thermal wave with constant penetration depth to be generated by a boundary blOW-Up regime. The precise form of this regime is determined by (24), (25). Condition (23) (or. equiv I, also in the second and third lines. To illustrate the possibilities of Theorem 2, let us consider
Example 2. Let
linl 110(1.')',0,(')11; Slim /,1 ..
H r . II
{[~J~I..'y)t)] In _ 1}2 dt = O. k (I ))
(27)
(III
Under the conditions of Theorem 2, the a.s.s. (I) has the form
= explf.L
I
(I) -
xl.
=
In( I and it follows from (25) that
(To obtain (27) from (26), it is sufficient to take (25), and, taking (24) into account. transform back, f.L(I) -> I.) 0
11,(1. x)
I -+ 00.
(28)
holds.
Proof The proof is essentially identical to thaI of Theorem I following estimate:
/. /
Then. if eJ> == I and H, == ('''' in (9), we have 0(1,') -,0, (') in L"(R , ) as where OU. f) is the similarity representation of (10). Therefore the inequality (23), which is equivalent to
k(lI)
+ II) In 2 1n(3 + II).
f.L(7):::::
T - 1/ln7. 7
Then condition (23) is satisfied
->
00:
f.L IU)::::: exp{(T - I ) I}. 1---> T
Therefore the boundary blOW-Up regime IIj(tl:::::
exp!exp{(T
-,I)
III
generates a solution with constant (as / ---> T' ) and non-zero half-width.
2 Approximate self-similar solutions of equations with nearly power law coefficients
where we have denoted by V = (-il 2 /ih 2 ).I V the solution of the problem
a2 v/iJx 2 = '-v.
In this subsection we consider equations (I) with coefficients k, which satisfy the condition (k/k')'(II) -> I/(T. II 00; (T = const > (J (29)
which satisfies
(for (T = (J'( (29) coincides with (5)), We shall also assume that the following conditions are satisfied:
h-I(R.).
--)0
k(~II) ,
- E,"
IS
non-Illcreasing in ~
k(tll) - ' -
--)0
g"k(ll)
I as
II
--)0
00
E
(0. I) for any II > (J.
for any
~ E (0.
I).
(30)
All the above requirements are fullilled, for example. by the coefficients k(ll) = In"(1 +ll), lr > 0; k(lI) = lI"explln"(1 +11)1, 0 < lr < I; k(lI) = lI"tl/lnlnISt"I, In this case the a.s.s. of thc problem (I )-(3) arc constructed using two types of invariant solutions of the equation with a power law nonlinearity, II, (T
> (J
A,
= (II"U,),
11.1'(1.
(2l))),
x) = (T - n"gl(?). ? = .1'/(1' -
which have the form
nil i
"'riO. (32)
I E (0. 7); n < 0;
B,
us(l,x)=e'g2(?). ?=x/exp(lTI/21. 1>0,
(33)
The functions gl, g2, which satisfy the boundary conditions g,(O) = I, g,(oo) = O. i = 1. 2. are determined from ordinary differential equations obtained by substituting the expressions for 11.1' into (31) (see ~ 3). In each of the cases under consideration, we shall seek a.s.s, in the form (9). and will denote by O(t. g) the corresponding similarity representation (10) of the solution 11(1, x) of the original problem, We shall establish convergence of fI(I . .) to 0,(') in the norm of the space 17 I (R I)' All the functions II in L I (R( ), which satisfy the conditions
'''- lI(y)dy E L2(R I)'
./ \
1/'. ' ' dx ./.'" lI(r)dr! ()
hl(R+):
1;2
= ,0 /''''
II(X)
'IIC J'
Below we shall assume that 110 E h -I (Rt.), u(l. ,) E 17 I (R t ) for all I E (0. n. Note that the second condition holds for a generalized solution with compact support u
E
C~;2(PrlUj).
Prlul
= {I
E
(0.
n. x E R+ lu(t, x) > OJ,
with a continuous derivative k (u)u" The following easily verified assertion will be used in the sequel.
Lemma 2. Lei a .limclion k E C 2 «0. (0)) SlIli.lj\· condition (29). and lei a E (0. (TI be WI arbilrar" COl1stanl. Then .lill' all sllfficienlly large u > 0 11'1' IU1\'e the inequalilies
(34)
I Approximlile sclr.limilar sollllio/1S or Iype A
Theorem 3. Assllme Ihal condilio/1s (29), (30), (30') hold, a/1d Ihal.!ilr all I slIr
.licimllr dosc to 1', 11'171'1'1'
/I
i/1\'erse
< -3/(3(T
uJ
+ 4)
is
£/
= l' -
Theil Ihe problem
= cr-JL"I(t))Il.
T ( ,/
(1' .. , r)""dr
--~_...~
q(T-r)"j
(I )-(3) 17(/.1' 1171' (1,.1',.1', (l)). c!1(t)
(35)
cO/lslwl1 (//Id JLI : [0. T) -.
ll!
< 00.
\
.iJ.2o) 1v] [( _
= (II. u)"
I
It is not hard to see that
JL(t)
belong in the Hilbert space 17 I (R I ), The scalar product in 17 I (R ( ) has the form
(II. tt)" 'lit, I
W(oo)1 < 00
(it is easily checked that under the assumptions we have made on functions in a solution of this problem exists), We denote by 111111 1 the norm 111
Monotonicity and unboundedness as I ----'> r- of the function til (I) in (35) follow from Lemma 2. The restriction II < -3/(3(T + 4) has to do with the condition 0, == gl E h-I(R+). Since (see § 2, Ch. Ill)
giCg) :::
II E
[-3/(3(T
Using now the fact that (F(II) - F(u,), 11- II,) ?: 0 and appealing to the CauchySchwarz inequality to estimate the last term in (40), we obtain from (39)
C> 0,
Ce"!III,,,n, (; -, 00:
this inclusion does not hold for
409
§ 4 Integral estimates of the rate of convergence
VI Approximate self-similar solutions
Hence
+ 4), 0).
d IIwll-.1 -,
(4\ )
k'(u,) - II,ITIII 2· :'C lI(u,).Ij-L'(I)
( I
Pr()(!1' oj Theorem 3. Let us change the variable I to j-L(I), where the smooth. monotone function j-L is defined by (15). Then the function u(j-L(I). x) satisfies equation (I') and the boundary condition (15'). By (15l. (35), (36) we have (/>(j-L(I)) ::::
and. as is easily seen. for all u,(j-L(t) , x) :::: O(j-L(t),
By the equality 0, == gl, as I -, r . the a.s.s. u,(j-L(t), x) becomes exactly (32), the self-similar solution of equation (31). The function 10(1, x) = 11(j-L(I), x) - u,(j-L(I), x) satisfies the equation W,
= [j-L'(t)k(U)II,
°
- 1I~(u,).L.,
-
(j-L'(t)[F(lI l1
where F(u)
••
j'"
=
-
IT _1_(u + 1) (T
+1
'
.•., ,
(1J(.I: I)::::
)
as
I ----'>
(_~~_) iJ x:'
1'-. Taking
1 10 ) ,
:'C{-sUPIII~(II,),1 ('''I(II(u,; .\ ./0
10':
dfl'; dgl < 00. Then we obtain from (41) the following where q, = max estimate. which is valid for all I E (I" 7') which are sufficiently close to I' k(T))dT), u?:
O.
Ilw(l,
(-i~)
·)11
I :::
111/1(1" ·)11
'",) +
( j-L I
=-
1
ITtl (T +-I(u, - )",
. (II,) " -
j-L'(I)(F(II) - F(u,), u - u,)
+ (II,),!j-L'(I)k(U,)
-
I(rl,
+
CiJ 1II) .
j' (I' _
I)I"IIT 121 11/-1<11112( (I'
- r)"; r) dr.
. I.
(iJ:')-I) (Jr2 --
1+
+ (/~/2 Since
+
II"
rJl,dX}"
(39)
We rewrite the right-hand side of (39) in the equivalent form
'()If'l
(0,7').
IE
T)
Since the function u, is monotone in x. we have finally
.Il
,lUI (IFIUI- Flu,il".
t [j-L'(r)k(~~ - 1]:' T(dT) . .I> 0,
./0
(38)
and w(l, 0) :::: 0, w(t, x) ----'> as x -, 00, 10(1, .) E h I (R t the scalar product of (38) with (-iJ"/iJx 2 ) 1 10 , we obtain
~«"llIw(t")1121 =
Let us estimate the right-hand side of this inequality:
1/1
111/1(1,,)11
(40)
I
= (T_I)"I.llll"ITI/-11I1J(j-L(I),·)-O,(·)1I
I,
from that inequality we derive the estimate 1I0(j-L(I),') -0,(·)11
or. which is the same. From (35') it follows that ,.
=
fL(I)
(1' - I)"" kl(T-I)"1
and therefore lim IIH - 0,11"
,_.1
~ l' .
1 1
q
I ::: - - - ' 111 3(1
+
I
,
+ II(T)/4]-
lim
{
HI' .0
where we denote by k I the function inverse to k. Asymptotic equivalence of these two expressions follows from condition (29). In this case the a.s.s. has the relatively simple form
k«((T - I)") }" . - 1 (" de· ("k«7 _I)")
11,(1. x)
= 111(1)
[ ~eT+2)111"
'1"
~) -
( I -
·\0
By (30), the integrand is bounded uniformly in ( E (0. I) as I ~ T. Then 0 (37) follows from (30').
.10 :=
,
Example 3. Let k(u) = u" In(2 + II). Then from (35) we obtain that the limiting localized S-regime in this case is
*
Remark. In 3 we used a different method to construct a.s.s. of equation (I) for coefficients k and boundary regimes covered by Theorem 3. It is not hard to derive from (35), (36) sharp estimates for the spatio-temporal structure of a.s.s. Let k(lI) = U"K(II), where the function K(U) > 0 grows slower than any power as II -)0 00. Then if the condition
which generates a solution with half-width which becomes constant as t -+ 1'-. 2 Approximate seiFsilllilal' soilltions
.
I lin
K(.I)
.\.""" K(sIK"(S))
=I
ill
(1),41(1)
e'TTJ
in (35), (36) admit the asymptotic estimates
Suppose Ihal jill' 01/ I sullicielll/r close 111(1)
lI'here fL(T)
fL :
I
RI
:
((I
(0,
n,
which
./ill'
fL(T)
= exp(fL
(43)
.I'll!ficiellll\: large
!"""-k(e e"'1 = 7.. _ /)
e/J( I)
= exp { ~ fL
g"(~) alld Ihe equalily
T
> 0 is delerlllilled liy the
(44)
dYJ.
'
. 1
==
1(l1I
l'
(0. 1') -+ K t is the illl'erse or the lIlonotone increasing jill/ctioll
-+
./i1r/1Il Iia
H,(~)
(42)
k(e1/)
l(T - 1)11 ""'kl(T - 1)"11 111 ""11", I ~ l'
Let us use the above theorem to study the heat localization phenomenon. The spatio-temporal structure of the a.s.s. (9), defined by (35), (36), indicates that the properties of the solution of the problem depend on the relation between the quantities (T and 11. If n < -I/(T in (35), then 4>(1) ~ 00, I -)0 1', and therefore the solution grows without bound as t --> r everywhere in the half-space (x > OJ (HS-regime: no localization). If 11 > -I ler, then e/>(I) -+ 0 as I - , T (LS-regime), which indicates that the process is localized. The value n := -I I (T corresponds to the limiting localized a.s.s. (S-regime): in view of the fact thai (/> == I, as I -+ l' it has constant half-width, which is
In the course of the proof of this theorem, which is similar to the proof of the previous one, we cstablish the following estimate: 116(J.L(I) , .) - 6,(')11_1 :::
Ilw(l,. ·)11
+ q~/1 exp { -I
{-I (I + 3:) } + (I + 3:) } I exp {i (I + j)}x I
exp
x { (' [J.L'(r/(rll _
./0
rl"
I]
2 r( dr } li2 dr, I
/
~ oo.
Hence by the cquality e(rr
= --. k(c')
J.L'(I)
1->00.
413
Remarks and comments on the literature Our exposition of thc rcsults of ~ 2 follows mainly 1119. I!lSI. First studies of thc hcat cquation III = 11.,., using degcneratc a.s.s. arc rcportcd in 1149. 347. 348] (sec ~ 4. Ch. III). Applications of this theory to the study of hcat localization (Theorcm 2. Section 4) are containcd in 1154[. Thc thcorems of ~ 3 are proved in [1841; partial rcsults werc obtained carlicr in 1119 [. Rcsults of ~ 4 are given in Il!l7. 1191· It must be said that at prescnt there are few examples of really non-trivial a.s.s. of nonlinear heat equations. In this regard. let us mention 1234 [, where a.s.s. of the Cauchy problem for III := (k(II)II,)" I > 0, x E R. with cocfficicnt not of powcr type. are constructed. A rclatcd result has bcen cstablishcd in II !l71 by a different method; an estimatc of the ratc of convergcnce to a.s.s. was also derived there (this cstimatc could not be obtaincd in the framcwork of thc mcthods of [234]).
and assumptions (30), (30') wc have the desired result
Open problems I. (~ 2) Provc that localization of dcgcnerate a.s.s. (10) for II .:: -I implics localization of solutions of the original problcm (I )-(3') (for thc casc k == I this is proved in 1149. 347. 348[; sce ~ 4. eh. 1II). 2. (~ 3) Prove that undcr the conditions of Theorem I for II .:: - 1/ (T solutions of the problcm (I )-(.1') are localizcd (for II < -II(T a.s.s. arc not localizcd. which implics thaI therc is no localization in thc original problem).
Remark. If the condition contrary to (42) holds: r~
./0
1'''" k(£"I) dT}
= x,
the solution obtained from a sclf-similar lYpC B solution by thc "transformation" u" -+ k(II), is dcfincd for all I > 0, that is, it does not blow up in finite time Il!l71. From Theorcm 4 it follows immediatcly that undcr thc assumptions we havc made, thc boundary regimc (43) Icads to abscncc of localization. i.c., it is an HS-regimc. Example 4. Lct us consider thc coeflicicnt k(u) := u"ln 2 ( I + II). Condition (42) is satisficd, and thcrcforc wc deduce from (44) that
Hencc thc boundary rcgimc III (I) ::::.:
cxp{(T -
I)
II.
1--'
T
gcncrates a thcrmal wavc that moves according to x,./(I)::: f'(!J(I):=
I Method of stationary states for quasilinearparaholic equations
Chapter VII til
Some other methods of study of unbounded solutions
= 'V. (k(tI)'VtI) + Q(II). I> O. x ERN. N 11(0. x) = tlo(x) ::: O. x E R •
415
(I)
(2)
where. as usual. k ::: o. Q ::: 0 are known sufficiently smooth functions, 'V(.) = grad \ (.). The main question we are considering here is to find conditions on the coefficients k. Q. under which the solutions of the problem ure not localized. An unbounded solution of the problem (1). (2) is said to be locali;,ed. if it grows to infinity us I -+ T(; < x on a bounded set in R N • that is the localization domai/1 In this chapter we conduct a study of unbounded solutions of various nonlinear parabolic problems. In I we study the character of the asymptotics of unbounded solutions of a quasilinear parabolic equation with a source close to the blow-up time. We obtain a nearly optimal condition for the absence of localization. In 2 we investigate boundary value problems in bounded domains. In 3. 4 we consider parabolic systems of quasi linear equations which admit unbounded solutions. Most results are obtained using the same approach. Ihe melhod or sll/lio/1arY sll/Ies. It is based on an analysis of a special family of stationary solutions. which satisfy the equation or system of equations almost everywhere (this is why it is convenient to call them states. thus cmphasizing thc fact that thcy are not stationary solutions in the usual sense). Wc find that the family of stationary states contains in a certain parametrized form several important properties of the evolution of the non-stationary problem. We stress that the method is applicable to problems with nonlinearities of a sufficiently general form. when the problem admits no appropriate stable similarity or invuriunt solutions. In 5 we study for the most part a nonlinear (implicit) difference scheme for the equation til = (tI"! I ) IX + 1I f3 The most interesting case here is f3 ::: IT + I. when the discretized problem admits unbounded solutions.
*
* *
*
(VI.
= {x E R
N
III(T;).
x)
= lln1 f
''''>
7 11
tI(t.
x) ,,;::
x}
(3)
is bounded. The function tI(T o' x) is culled the limili/1g dislrilJlltio/1 (I.d.) of the N solution. If. on the other hand, IV,- is unbounded (for example. w,- = R ) we say that Ihere is IW locali::.atio/1 i/1 the f1rohlem. Earlier. in Ch. IV, we studied the localization phenomenon in detail using as an example equations with power type nonlinearities. (4)
by analyzing in an appropriate way their self-similar solutions. It was shown that for f3 < (]" + I there is no localization. while for f3 ::: IT + I all unbounded solutions are loculized. Equation (I) of generul form does not admit such selfsimilar solutions, and therefore the compurison methods developed in Ch. IV are not applicable here. In this section we propose an approach to determining sufficient conditions for the absence of localization in the case of equation (I) with arbitrary (not power type) coefficients k. Q. We also study the structure of I.d. of unbounded solutions. This method encompasses also the case To = 00. when suP., l/(t. x) -+ 00 as I -+ 00. Therefore we do not pay any special attention to the condition J'XJ dri/Q(Tj) < 00, which. as is well known, is a necessary condition for existence of unbounded solutions (see 2. Ch. I). The method we employ is based on the construction of a one-parameter family IU} of stationary solutions of equation (I):
*
§ 1 Method of stationary states for quasilinear
parabolic equations This section is entirely devoted to the study of the phenomenon of localization of unbounded solutions of the Cauchy problem for a parabolic equation with a source of general form:
'V. (kW)'VU)
+ QW)
== O.
(5)
In essence. we shall show that the family (Uj contains information about several properties of the evolution of solutions of the non-station,uy equation (I), which are parametrized in a special form. Actually this assertion is quite natural. since the main part of equution (I) is exactly the stationary operator, which contains all the nonlinear terms responsible for the evolution of a solution.
416
~ I Method of stationary states for ljuasilincarparabolic equations
VII Some other methods of study of unbounded solutions
In ~ 3 (and partially in § 4) we use this method to analyse a parabolic system of quasi linear equations, and, while studying localization of unbounded solutions of the Cauchy problem, we at the same time determine conditions for global solvability of the problem in a bounded domain. The concepts we present make it possible to give a general formulation of the method of stationary states for the study of nonlinear parabolic problems satisfying the Maximum Principle. Below we shall lake the function Q(II) to be monotone increasing and we shall also assume thaI (11) --" 00 as II -+ 00, where (/J(u) == J;' k(1]) £11]. We shall denote by (/I" I the function inverse to . As far as the solution of problem (I), (2) is concerned, we shall assume that it exists, is unique and for all 0 < I < To = To(uo) ~ 00 belongs to C~;] wherever u > 0, and that in addition k(II)'Vu is continuous in x in R N for all I E (0. To).
417
Integrating it from 0 to r. we obtain the following estimate: U(r;U o )
where G
I
~G
1
U:
+G(UO)) < G
1
(~':).
I'
> O.
(\0)
is the function inverse to G(lI) ::::
' f
,
In the particular case Q{l1) written down in explicit form:
1/
k(1]) £11] ----. Q(1])
:::: /-Hp(u).
/l
u> O.
= const
> 0, the solution lJ can be
( I I) I' < ;:\: ] /l I I]. where .::~ ] > 0 is the first root of the Bessel function and CN(UO) = (p(Uo)I'(N/2)(2/l I/C)IN ]1;2. The main properties of stationary solutions of equation (4) with power nonlin-
for 0 < luV.]I!C
1 Construction of the family of stationary solutions We shall be interested in the properties of bounded radially symmetric solutions U of equation (5). Each of those, at points of positivity, satisfies the problem
r~1
(r N I ((IJ(U))')'
+ Q(U) = O.
I'
= Ixl
> O.
(6)
U(O;U o ) = Uo. U;.(O;U o ) = 0; (7) while at all other points we assume for convenience U == U(lxl: U o) = O. Here lJ o > 0 is an arbitrary constant (the paramcter in the family IU(lxl: U o)}). Local solvability of the problcm (6). (7) for small I' > 0 follows from the analysis of the equivalent integral equation (/J(U(r; U o))
= c/J(Uo) -
I· r
.0
tiN £It
[~
1]N
IQ(U(1]; U o)) £11],
I'
> O.
.0
Hence we have that the solution can be extended in I' > 0 and is strictly monotonc in I' in the domain {I' > 0 I U(r) > OJ. Let us first derive a lower bound for U. Clearly, (r N 1 (c/J(U))')' :::: -Q(Uo)r N I for. I' > O. Integrating this inequality twice over (0.1') we obtain the following estimate: lJ(r;U o ):::: lJ (r;lJo):::: (/1 I [(P(UO)
(I -~);].
l2N c/J.(~~l}2] 1;2 1':::: o. Q(Uo) To derive an upper bound for lJ, let us usc the relation 1'0
=
earities,
~ (rN1UrTU')' +U fJ
=0.
I'
> O.
(12)
were discussed in ~ 3, Ch. IV. Let us emphasize that they crucially depend on the relation among the parameters (T, f3, N. In particular, for N :::: 3, f3 ::::: N ((T+ I)(N+2)/(N-2) the problem (12). (7) has strictly positive solutions in R (in the other cases the solutions have compact support). For k. Q not of power type it is also possible to have solutions of the problem (6), (7) that are defined and strictly positive OIJ the whole of R N This, for example, occurs if (sec [3321l N - 2l1Q(c/J1(lI)) > [" Q(4) 1(1]))£11]. 2N ./0
II
> 0; N> 2
(12')
(the proof of this fact is the same as in the case of power type coefficients; see § 3, Ch. IV).
We shall also need conditions on k, Q. under which the functions U(lxl; Uo) N have compact support in R N (that is, they are not stationary solutions in R ). As in § 3, Ch. IV, it can be shown that for N = lorN:::: 2 the functions U(lxl; Uo) have compact support. while in the case N :::: 3, under the assumption of non-negativity of U we derive the estimate from below (9')
Comparing (9') with (10) for large,. > 0 gives us the following suftlcielJt condition for an arbitrary function U(lxl; Uo) to be of compact support: c/JN/(N
N :::: I. 2 or lim
;-:0
]'(.1')
Q(.I')
-
= 0 for
N > 3. -
( 12")
••
•• ••• •• •• •• •• •• •• •• •• •• •• •
•• •• •• •• • •• ••
•• •• •• •• •• ••• •• •• ••
•• •
•• •• ••• •• •• •• •• •••
~
VII Some other methods of study of un hounded solutions
41H
For equation (12) this criterion imposes the restriction (3 < «(T + I)N/(N - 2)+ and is practically a condition for non-existence of solutions V > 0 in a neighbourhood of r :=: 00. In the one-dimensional case equation (6) can be integrated in quadratures, and the solution of the problem can be determined from the relation ( 13)
I Method of stationary states for quasilinearparaholic equations
419
Remark 3. Applied to equation (4), condition (14) (or (16» takes the form s"+ 1-/3 --'> 00 as s --'> 00. Therefore in the case (3 < (T + I unbounded solutions of the Cauchy problem are not localized (for N :=: I this result was obtained in 4, Ch. IV). . To prove the theorem we shall need the following Lemma
*
Lemma l. LeI Ihere exisl Vi)
>
0, sllch Ihal under Ihe cOlldilions o( Theorcm I
wc hal'c Ihc incqualitY
It is strictly positive for 0
~
( 17)
r < xo(V o ), where Theil Iherc exisls 10
:=:
lo(Ui)
E [0. To), such IlwI jllr all I E (10. To)
1I(r. x) :::. V([xl:
Below we shall assume that the following conditions are imposed on the initial function: Uo :=: uo(lxl) is a function with compact support, (P(llo) is uniformly Lipschitz continuous in R N , uo
2 Sufficient conditions for the absence of localization
Then u(l. x) is
1101
loca/i::.ed alld
11(1,
:=:
00.
( 14)
x) .._", 00 as I --> To el'er\'where ill R N .
Thus. if ( 14) holds, the Cauchy problem exhibits the HS blow-up regime. Remark 1. Condition (14) will necessarily hold if k(S)/Q'(.I) --> 00 . .I' -> 00.
ill
R
N
( 18)
*
We shall apply the comparison theorem of 4, Ch. IV (on the "I},0nincrease" of number cf spatial intersections of solutions of a parabolic equation). First of all let us note that from (17) it follows. in particular. that V is a function with compact support. In the opposite case, if V is a stationary solution in RN • by the Maximum Principle u ~ V in R+ x R N , that is. u(l. x) is necessarily bounded. Thus, supp U C R N is a bounded domain. By (17) the number of spatial intersections (in r) of uo(r) and U(r: Vtl) in W(UI~) :=: supp V(r: V~) is zero. Let N(t) be the number of intersections in r:=: Ixl in (v(Uo) of two different solutions u(t. r) and V(r: Vil) of equation (I), By the comparison theorem (sec 4, Ch. IV) N (I) does not exceed the number of changes of sign of the difference w :=: U - V on the parabolic boundary of the domain (0.1) x w(U o)' By assumption, N(O) = 0 N (w ~ 0 for I:=: 0 in W(U;'I))' If u > 0 in (0, To) x R , then II' > 0 on (0,1) x (lW(U(j) , and therefore N (I) ~ I. If. on the other hand, ( 1) admits finite speed of propagation of perturbations, then by a known property of parabolic equations with a source sUPPU(lI. r) c; sUPpu(l2. r) for II ~ '2, that is, N(I) ~ I for all f E 10, To)· Obviously, there exists I. E (0. To) such that 11(1 •• x) > () on iltu(lJo). In the opposite case, if II "'" 0 on ({). To)xilw(UI;l. (17) would imply uniform bO~l!ldedness of u in (0, To) x ilw(Ut'I): LI ~ V. Then u(l,x) > 0 on iJ1V(lJt'I) for all I E (I,. To)· Let us choose now 10 E II•. Tol. such that u(lo.O) > U;,} (this is always possible, since by assumption Iilllu(l. 0) :=: 00. I ...... T(I ). Let us show that NUo) = O. Indeed, u(lo, r)-· U(r:UI~) > 0 for x :=: 0 and x E iJw(lJ!}). Therefnre in the interval l'I(UI~) there can only be an even number of intersections of u(lo. r) and U(r: UI~)' However, we established earlier that N NUo) ~ I. Therefore N(lo) :=: n. so thaI u(to. r) :::. V(r:V~) in R . By the 0 Maximulll Principle this inequality will hold for aliI E (1o, To).
Proof
*
Theorem 1. LeI Ihe soll/lioll o( Ihe !Iroh/el/l (I). (2) he ullho//l/{!ed alld
lim I(s)/Q(s)! ,--.'X:
VI~}
( 15)
Remark 2. For the one-dimensional (N :=: I) case a suflicient condition for the absence of localization can be formulated as follows: ( 16)
* I Mcthod of stationary states for quasilincarparabolic equations
VII Somc olhcr methods of study of unbounded solutions
420
P!"Old' lit' Theorem I. From (9) it follows that if (14) holds, V(lxl; V o ) --..,. 00 in R N for a sequence V~ --..,. 00. Therefore we can make (17) hold for any compactly
passing in (lK) to the limit V'0-- V 0k supported function uo(lxl). Then. " conclude that for any x E RN
tim
{((I. x):::
{.,.• I (I
lim V(lxl:V~)
l-'~I ."'...-
that is. the solution grows without bound as in the whole space.
In this case the family IV) of stationary states allows us to determinc certain properties of the limiting temperature distribution II(T() , x). All the results below rely on the following lemma, which is proved exactly like the previous one. Lemma 2. Lcr
r
The ha(t'-Ividlh ut'lhe slr{(cl{(re call be ho{(nded .li'Om he/u\j' hr
IT
Thus, a necessary condition for localization of unbounded solutions of the problem (I). (2) is the following:
.limclioll Uo sllch Ihal jill' all slll.ficiellily large Vo > Vt) Ihe .limctiulls ((o(ix!) alld V(!xl: U II ) illiersect (ill Ixl) al 1110.1'1 III 0111' poilli. Theil Fir llll sl~fficiellll." .I'll/all
1;2 -'> 00. 1-' 1'(1'
Q({((I.O)).
In the case of equation (4), f3 <
3 Some properties of localized solutions
0
Corollary I. LeI (I) describe processes \Vilh a /illile speed ot'propagatioll ot'pcrlurbaliollS. Theil ullder Ihe cOlldilioJ1S oj' Theurem I Ihe diameler D(I) ut' Ihe suppurl supp, u(l. x) ut'lhe sulLlliulls .\'(/Ii.llies Ihe eSlimale DU) > 2...fiN [t/I(U(l,
there exists I' > 10 such that u(l', 0) > Vn, {(it', x) > 0 on (Jw(l/o), and lI(t', x) and V(ixl:V o) have at least two intersections in the interval w(Vo). This contradicts the condition N(I') :s I (see proof of Lemma I). N Thus 11(10, x) ::: V( 1.11: Vo) in RN . But then 11(1. x) ::: U(ixl; Vo) in (10, To) x R and therefore 11,(10.0) ::: O. Since V o ::: U was arbitrary, this proves the claim of the lemma. 0
To at the same time everywhere
From the method of proof of Theorem I we immediately derive the followin" corollaries (they show how the spatial structure of the family of stationary solution: IVI describes the features of the evolution of the solutions of the non-stationarv problem). -
421
= 1.1'1
> 0 we hlll'e Ihe eSlimate 11(1'11' x)
2
== lim
}li
1
~
11(1.
(20)
x)::: sup Vtlx!:V o }.
t(1
Uo'U;)
t/1({((I.0))
+ I,
this estimates have the form
Inequality (20) allows us to bound from below the size (diameter of (tIJ.l of the localization domain (3) and to describe detailed behaviour of H(T(> , x) in a neighbourhood of the singular point. In particular. from Lemma 2 and (II) we have Theorem 2. LeI QUt)
...fiN
r''l(l) ::: --~-I{(U. O)I{ il./iIi2( I - 2 lf
J7r+I
and agree well with self-similar behaviour (see
{,r+II)li 2
0/1
(I" To).
=
(21 )
* I. eh. IV).
Corollary 2. Assume Ihat {(Ilder Ihe cUlldiliulls ul'Theorem I Ihere exisls VII> O. s{(ch thai {(o(ix!) :s V( 1.11; Vol ill R N jill' all V o ::: VI~' Lei {((I,. 0) = V~. Then u,U, 0) ::: 0
=
J.Lt/)( II). where J.L 1'011.1'1 > O. Theil 1111 Hllbollllded sulUlioll of Ihe pmhlem (I). (2) Clill/lOI he 10Cll!i::etl ill II IJIIII uldillmelcr less Ihall
To prove the theorem. it suffices to check that the functions (II) satisfy conditions of Lemma 2. Therefore for the functions of (II) we obtain from (20)
'
1/(1'11' x):::
sup V(lxl; Vo)
= 00.
O:s 1.11 < ;,':~IJ.L
In
U{I,U;I
Proof: Let us fix an arbitrary Vo ::: Ull and set I, = inf \t E (0. To) I {((I. x) > 0 on iilu(Uo)} and 10 == inf{1 E (0, To) I {((I, 0) > Uol. By the Maximum Principle N II :s 10. Let us show that 11(10, x) ::: V(lx!;U o ) in R . Indeed, if that is not so,
As an example we quote a stronger result. which holds in the one-dimensional case.
VII Some other methods of study of unbounded solutions
Theorem 2'. AS,l'wlle that N
=
Method of stationary states for quasilinearparabolic equations
423
I and that the/(iiOlving conditions hold:
(21')
I. = 2xo(x)
== 2 lim xo(s) .. ;,
< x,
~
Then iF 11(1, x) is IIl/holll/ded, mellS lUI == //leas Ix E R 111(1'0' x) ProO( For N = I we obtain from (6), (7) that for any fixed U. .1= x. where U(lxl: Uo ) = U" we have the equality
[ ~I,!J{lJ(lX.I:UO))]2 d.1
= 2 lUll Q(1])k(1]) d1]
->
,u.
= x I ::: E (0,
x. U o ->
I"
U o ) at points
X.
Therefore in view of uniform Lipschitz continuity of l/i(lIo), for all sufficiently large U o the functions 110(1.11) and (J(lxl:Uo) intersect in r = Ixl in w((}o) = supp U(ixl: U o ) at most in one point. Then, as in the proof of Lemma I. we have that 1I(1'(). Ixl) ::: U(lxl: Uo ) in (tJ{lJo) for all sufficiently large U o > O. and, in particular. meas WI ::: 2xo(Uo). 0 Of course this theorem is also true when Let us now consider the case
,* = x
in (21 '). Fig. 78. Example of constructioo of the family of functions {U I and envelopes in the S-regime (0 < .10(%) < %), Thick line denotes the initial function 1I0Ld: N = I.
sup U(ixj: U o ) < x, 1.11
:>
n.
(22)
(1",0
This condition indicates that the localized solution becomes infinite as t -> T() only at one point. that is. it exhibits the LS blOW-Up regime. Then Lemma 2 allows us to bound from below the asymptotics of behaviour of II(T(). x) as 1.11 -> 0 and to determine the rate of change of the half-width of the localized struclUre as t -> T(l' Let us note that by (9) localization of the solution in LS-regime is possible when -lim ---I /"
k(1])d1]
,'-x. Q(s) ,II
= n.
(23)
In the one-dimensional (N = I) case the necessary condition for the occurrence of LS-regime has the following form: lim
\-.cx..
XIl(S)
= n.
(23')
If condition (23) (or (23')) holds, the family of curves {U = U(r: U o )I allows us to construct in the Ir. Uj plane. for all sufficiently large U o. monotone envelopes. tangent to the curves U = U(r: U o ).
Let us choose a maximal continuous branch r = FIl(l!) (see Figure 78). Two cases have to be considered. I) The envelope r = Fo{lJ) 2: 0 is tangent to the curve U(r: 1111(0)) at some point r O :> 0 (see Figure 78). In this case let us set
~
(,(x)
={
O
F o I (jxl).
1.11
U(lxl:
1.1'1 2: r
1I1l((»)),
< r , O
,
where F o I is the inverse function of F o (at points where F o I is not defined, we set F(II = x). 2) The envelope I' = F o((}) ::: 0 has no points in common with the curve U (I': 110(0), the curve U = F()·I (I') is defined for all () < I' < 1" and is tangent at the point I' = 1'* to some curve U = U(r; U~). Then we set G(x)
={
F()'(ixl),
Ixl
<
1"'.
U(lxl:U~).
1.11:::
1".
From Lemma 2 we immediately obtain
424
* I Method of stationary states for quasilinearparabolic equations
VII Some other methods of study of unbounded solutions
Theorem 3. Assume Ihal cOlldilion (23) (or (23'») is salisfied, and Ihat Ihe SOllllioll u(l. x) is Ullhoullded. For all Vo ::: V(') = uo(O) leI Ihe ./illlCliolls lIo(ixll alld V(lxl; Vo) inlersecl (ill r
= [xl)
almosl alone poilll. Theil
Then if G, = G,(r) is the envelope of the family IU . (r; Un)} of (25). we have clearly that L,(r) == c!)(G,(r) < Ur) for r > O. The function L(r) is determined from the system of equalities (28)
(24)
The estimate (24) makes it possible to determine the degree of singularity of the limiting distribution u( Til' x) in a neighbourhood of the point x = O. where the solution 1I(t, x) grows without bound as I -+ Til' Here it is convenient to use the estimate (9) for the stationary states V(r;Vn). which we rewrite as q)(l} (r; Un)) ::: q)(l} (r; Vn))
r:::
==
q)(l}o)( I - r' / r~); ,
0; ro
=
[2N
eP(l}O)] II! (l}o)
(25)
(lx[;u(I,O), [xl>
> 0
is a parameter. This system is equivalcnt to the following one:
Then, by eliminating the parameter U o we obtain a differential equation for the envelope L. = L(r): L,(r)
First let us present a simpler claim. which follows directly from the estimate
u(l,x)::: V
where V n
Q
(25) and Lemma 2. It has the following form: under Ihe cOlldiliollS r~f Lemma 2 for all I < To sl!t1icielllly close 10 Ihe blOW-lip lime I = To we have Ihe eSlimate
o.
(26)
425
(NL:(r))] ---
= e/J [ Q ,I
/', (r), + -L 2 "
r
(28')
This makes sense for all sufficiently small r > 0 (here Q I is the function inverse to Q), We are interested in monotone decreasing solutions of equation (28') which satisfy the condition L. (() ') = 00. Such a solution is especially easily computed when the coefficients q). Q in (4) are of power type, Then (2W) takes the form
Proof of (26) proceeds as in Lemmas I. 2; let us note that similar statements were frcquently used in § 4-6. Ch. IV, From the last inequality wc immediately deduce
= (rT + I)
L.(r)
NL:(r))I"III/fJ I
(
-----
r
r,
+ -L,(")' 2
r > O.
(29)
Theorem 5. LeI eT + I < {3 < (eT + I )(N + 2)/(N - 2) I alld leI To < 00 be Ihe
Theorem 4. Vllder Ihe condilions of Theorem 3. we hal'e the eslimale
blow-up lime/ill' a solUlion oflhe Cauchy problem (4), (2). Then/il" al/sufficielllly
IIq)(u(l. ,nll/'m") :::
'~
4(27TN)N/2
[r/J(u(l.On)IINI2
smal/
(27)
Ixl
> 0,
11'(:'
have Ihe eslimale
I/(T;;,x) > G,(r):= \(eT+ I)L,(rlll/I"111
N(N+2)l'(N/2) IQ(u(l,0)!N/2 . 1-+ 1'1';,
From this estimate we deduce. for example. thc condition for IltPUt(t. ,)) 11/, m') to grow to infinity as I --> I'll:
where
Of course. using (26) wc can obtain other integral estimates of unbounded solutions. However. for general ell. Q. they look too cumbersollle. For some particular cases sueh estimates will be obtained below. Let us show how to derive from (26) pointwise estimates of u(T II ..1). Set L(r) = c!)(G(r). where G(r) is the envelope of the family of curvcs IU(r;U o)} (see Theorem 3): G(lxll = sup U(ixl; Uo). III > o. 11",0
:'/Ifi 1"1111
(29')
c. = {,_!~ [f3 =J-'T +_11J/)/IIT+II}l/l/lI"tlll {3 - (eT
If {3 ::: (eT
= ('.Ixl
+
I )(N
+ 2)/(N
+ I) - 2)
{3 I.
Ihis eslimale is \'£I!id jill' 01/ crilicaljimcliolls N 1101 decreose ill I ill (0, Till X R ),
uo(x) (Ihal is, sl/ch Ihal u(t, x) do(:'.\'
Proof II is easy to check that in .the case of equation (4) we have the equality U(!xl; Uo) = UoUW){)
I"!
11!i2 lx j; I l, Uo >
n.
For f3 < (eT + I)(N + 2)/(N - 2) I the function U(ix!; 1) has compacl support, so that suppU(!xl;lJ o) -+ {OJ as U o -+ 00. Therefore in this case the
~ I Mcthod or slationary slatcs for quasilincarparabolic cquations
VII Somc othcr mClhods or study or unboundcd solutions
426
conditions of Lemma 2 are satisfied. Then. taking into account the fact that sUPu>u' U(lxl;U o) > (1.(1.11) in RN\{OI. and computing the precise form of the 0function L.(lxl) in (2tJ). we obtain (2tJ') from (2tJ). Now let (3 :::: (fr + I)(N + 2)/(N -·2)1' Then U > 0 in R N. Let us tix an arbitrary U o > uti = 2uo(0) and let us consider the ball B = II xl < ro I. where ro > 0 is such that U(ro; Uo) = U o/2. Then if N(t) is the number of intersections in r = 1.11 in the ball 13 of the functions u(t.lxl) and U(\xl; Uo). then N(O) = 0 and by monotonicity of u(t. ro) in I on rlB, we have that N(t) :::. I for all I E (0, To)· Therefore (20) obtains (see proof of Lemma I) and. defining L. by (2tJ), we arrive at (2tJ'). 0 1)
Hence we obtain - fT ( 2(TN )
L.(r) = [(3 -(3
-(3-fr
"/f3] /.l/I#"I r ·leT/ill'-fT} , r
--+ 0+.
It is easy to check that in this case supp [}(Ixl; U o) --+ IOj as U o --+ 00; therefore by Theorem :1 we have u(T o , x) > (I,(r)
I
== -lnlL.(r)j fr
= (3
2
[(3 - fT ( 2frN )fT//3]
= - 73::-;; In Ix I + rr (73=--;;) In '-73-- /3 Remark l. For (3:::: (fr + I)(N + 2)/(N - 2)1 the criticality condition on Uo can be replaced by requiring I/o to satisfy the condition of Lemma 2. Remark 2. For equation (4) we can write down the exact expression for the envelope G = G(lxl) of the family of functions lUI. In its dependence on [xl it is the same as (29'): G(ixl) = Clxl 2/11! ('Hili, where C = C(fr, (3, N) is a constan!. and, moreover, C > C •. Juxtaposition of the nature of the singularity of u( Til" x) with the upper bound derived in ~ 6, Ch. IV by comparing with the self-similar solution, testifies to the optimality of the estimate (2tJ'). Inequality (29') allows us to derive a number of other estimates for unbounded solutions of equation (4). Theorem 6. Assl/mc Ihal Ihc cOlldiliolls o( Thcorem 5 hold. ThclI fill' al/ J! :::: 1(3- (rr+ 1)IN/2
wherc Sf
= 11.11
<
r '/11, 1:1
ul'(t. x)t!x
--+.1 --+ To.
is a hall o( arhilral'\' radius
I:
>
(2tJ") O.
It follows from results of ~ 6. Ch. IV that the restriction J! :::: 1(3 is a necessary and sufficicnt condition for (2tJ") to hold for arbitrary if wc demand in addition that (3 < (fr + I)(N + 2)/(N -- 2) I' Let us consider anothcr simple example. Let
(tr
+ I) IN/2
LlI)
= 1I1)(lxl)
where fr ::::. 0, (3 > fr (for (3 < fr the solution cannot be localized. as can be seen from Theorem I). Here the equation of the envelope L.(r) has the form 1.• (")
NL~(r) ) "/fi
= ( - --,.--
1',
+ "'iL.(r),
r --+ 0' .
427
fr
for all sufficiently small 1.11 > O. The corresponding integral property of unbounded solutions has the form
r exp(yu(t .
.In,
.1)1
dx
->
00.1'--+ 1'1;'
for any y :::: ((3 - fr)N/2. Using the above approach. which is based on the analysis of singular solutions of the ordinary differential equation (28'), it is not hard to obtain lower bounds for the limiting profile u(T ,I , x) in the case of sufficiently general r!>(u). Q(u).
4 Necessary and sufficient conditions for localization In previous subsections we formulated a criterion of localization of unbounded solutions of the Cauchy problem. which determines conditions for the occurrence of HS. S, and LS blow-up regimes in general nonlinear media. For convenience. let us restate this criterion. Thus, everything depends on the quantity D.
= sup {r I sup U(,.;
UI))
= oo} .
l/():.{)
If D. = 00. then there is no localization (Theorem I. HS-regime). If D. E R+ then only localization in the S-regime (in a domain with a non-zero diameter) is possible. Finally, if D. = O. then. apparently. the LS-regime is to be expected. This localization critcrion has an especially simple form in the case N = I; .10(00) = 00 leads to the HS-regime. .1"1)(00) E R.i. to the S-regimc, and .10(00) = o to the LS-regime (see Figure 79); XI)(oo) is computcd from (21'). Here the dependence xo(U o ) also determines certain properties of cvolution of unbounded solutions (sce Corollary I of Theorem I).
428
I
VII Some other methods of study of unhounded solutions
i
I
-HS
I
-5
k
II + I {[" k(TJ) = k(u). Q = Q,,(u) == -_- dry }" k(lI) . 0 ry + I
= const >
_ LS
./0
-------0;
k(ry)
TJ
+I
= 00.
dry
lim [kk,(U)]' I I ' ""'"
k(v) = --I\7vl+ Q,,(v), J
V,
v+1
IT
> 0,
J-L.I'
Hence we deduce that for J-L > II there is no localization (HS-regime), while for J-L ::: I' it occurs, and meas lVI. ::: 7TJ(TTI if J-L = II (S-regime) and meas WI. = 0 if J-L < I) (LS-regime). IT :::
(31 )
0, {3 > O.
J-L. I'
are
/ > O. x
E
N
(32)
R .
These a.s.s. are localized for a ::: 2 (for a = 2 the S-regime obtains and diam lVI- = 27T), while for u < 2 there is no localization. At the same time it is not hard to check that the localization criterion D. < 00 is not applicable here. For example, if k = I, Q(u) = (I + u)lnll(1 + II). for (3 > I we have
*
= (I + u)"c li",
= 00.
(u)
[150, 160.3471
Numerous estimates, obtained by constructing approximate self-similar solutions. as well as results of numerical computations show that the condition D. < 00 does indeed entail localization of unbounded solutions for arbitrary coefficients k, Q. which do not belong to a certain class of weakly nonlinear functions (this class is discussed below). In the case of equation (4) the criterion leads to a correct result: for {3 ::: IT + I there is localization (if {3 = IT + I, D. E R I ' and we have the S-regime: for (3 > IT + I, the LS-regime), while if {3 < IT + I solutions arc not localized (see # 4. Ch. IV). Let us consider more complicated examples. We restrict ourselves to the analysis of the case N = I.
Example 2. Let k(u) = (I + II)!"!'''", Q(u) constants. In this case as U o -+ 00
•
A property of unbounded solutions of equation (I) of this class I is the fact that their structure is described by a.s.s., which satisfy the nonlinear first order equation
Fig. 79. Classification of unhounded solutions according to the criterion 0 ::: xo(oo) :::00 (N '" I)
Example l. Let k(lI) = II" In!"(2 + II). Q(II) = 11",-1 In"(2 + II). where are constants. Then it is not hard to sec that for large Uo
(30)
I and the coefficient k(u) satisfles the conditions
r~
I
I o------u~------~
429
Hence if IT > {3 or if IT = {3, J-L > P, HS-regime obtains, while if IT = {3, J-L = P, we have the S-regime. and if IT < {3 or IT = {3, J-L < p. the LS-regime. Conceming the class of weakly nonlinear functions k, Q, in which the localization criterion D. < 00 (or xo(oo) < 00 if N = I) is no longer valid, its typical representatives are the family of functions
where a
I I I
* I Method of stationary states for quasilinearparaholic equations
I
lim -.--.
_I-'~
[_'
Q(s).
k(ry) dry
0
= 0,
which indicates the presence of the LS-regime (the criterion 0 ::: xo( (0) ::: 00. N = I, leads to thc same conclusion). A correct analysis of such a case is given below. Example 3. Let k(u) = In"(1 + u), Q(II) = (IT + I) Y( I + II) Inll(l + u). IT ::: O. {3 > I. y = ({3 + IT)!(IT + I). Conditions (30), (31) arc satisfied if we set a = ({3 + IT)!(IT + I). Hence it follows that there is localization for f3 ::: IT + 2 (that is, a ::: 2). while for {3 < IT + 2 there is no localization (for IT = 0 the validity of this result is demonstrated in 7. eh. IV). To conclude, let us observe that there exists a direct connection between the localization property and global solvability of the boundary value problem for equation (I) in a bounded domain.
*
3
'Let us ohserve that equalion (I) with k(lI) I. Q(II) = (I + 1I)ln/ (1 + II) helongs to this class; it was considered in detail in 7. Ch, IV (thal section also conlains the method of construclion of a.s.s.).
VII Somc other Illcthods of study of unbounded solutions
§ 2 Boundary value problems in bounded domains
I Equation with power type nonlinearities
In this section we consider, in more detail than in Ch. V, blow-up regimes that occur in bounded domains. The need for such a study stems from physical considerations (for example, taking into account heat loss from the boundat)' of the domain where the process takes place). In the analysis of boundary value problems there arises a whole range of new phenomena not encountered in the Cauchy problem. Let n be a bounded domain in R N with a smooth boundary af!. For the quasi linear parabolic equation
In this subsection we consider the prohlem for equation (I) of the particular form
II,
= llqJ(II) + Q(II).
I >
n. x E n.
(I)
we consider the first houndary value problem
11(1. x)
= n.
I
?:
n. x
E
an.
(3)
II,
= !iIl
./,
dTJ Q(TJ) <x.
(4)
The boundary condition (3) models heat outflow from the domain n, in which diffusion and combustion processes take place. Of course, the magnitude of heat loss depends on the intensity of combustion inside fl. The following questions arise. Under what conditions on the coefficients k. Q. initial perturbation 110 ¥= 0 and the spatial structure of the domain! 2, will combustion in the problem (I )-(3) lead to finite time blOW-Up? Conversely, when will the problem have a global solution, defined for all I > 0') In other words, we want to find out under what conditions the heat loss at the boundary is able to "extinguish" the vigorous combustion process, and when this will not happen. For convenience in the exposition below, let us introduce in the space of initial functions 1110 ?: 0 I 110 E nIh ep(lIo) E H(I)(n)) two sets: the stahle sel 'lV and the IIllslahle sel Y. These sets are defined as follows: if 110 E 'W, then there exists a global solution with initial data 110; if, on the other hand. 110 E 'F, then the problem (I )-(3) is globally insolvable. Below we present methods of constructive description of the sets 'W', Y. The structure of the unstable set 'V was analysed by different means in 6. Ch. Y.
*
+
1
+ 11 13 .
I>
n. x
E
!2;
(5)
O. f3 > I.
eT>
= lIo(X) ?: O. x E f!; 11(1. x) = 0.1 ?: O. x E an. 110 E nIh 11;;+1 E H!)(f!). In (5) the heat conductivity
11(0. x)
(6)
It is assumed that coefficient k(lI) = lilT /(eT + 1) reduces to the familiar form k(lI) = lilT by the change of spatial coordinate .\ ---> x/(eT + 1)112. It will he shown that the properties of the solution of the problem (5). (6) significantly depend on the relation hetween the quantities (T and (3: as (3 passes through the value f3 = eT + 1. thc structurc of the sets Y, 'W changes drastically. The value f3 = eT + I is critical also for the Cauchy problem for (5) (but for a different reason having to do with the localization phenomenon). The relation between localization in the Cauchy problem and the structure of the sets '1", '1V for a houndary value prohlem for the same equation will be discussed in the following, In the statements of results below, we shall use the fact that thc solution Of the problem for a paraholic equation which admits negative values of temperature, II,
We impose the usual restrictions on the non-negative functions cP. Q. [n particular. we require the necessary condition for finite time hlow-up to hold: r~
IT
= ~(IIII"II) + q(II). q(lI)
= {
I>
O. x
O.
11<0,
III! .
II ?:
E
n.
(5')
0
«5') is the same as (5) if 1/ ?: 0) satisfies the weak Maximum Principle. Therefore 1/(1. x) ?: n almost everywhere (a.e.) in n for all admissihle I > n if 1I0(X) ::: 0 a.e. in f!. Analysis of the problcm (5), (6) separates naturally into three cases. 1 G!olwl solvabililY Fir (3 <
Theorem l. LeI f3 <
IT
+L
eT
+I
Theil Ihe pwhlelll (5), (6) ha.\' a global sollllioll alld (7) (8)
Remark l. From (7) it follows that the mapping III ,,,n : 10. Tj -> L 2 (0) is continuous (after. possibly, a modification on a set of measure zero), so that the initial condition (6) makcs sense. Remark 2. For f3 < (T + I the unstable set ''IF is cmpty, boundary docs not allow finite time hlow-up.
I.C ..
loss of heat at thc
432
433
~ 2 Boundary value problems in bounded domains
VII Some other methods of study of unbounded solutions
The theorem is proved by constructing a global solution by the Galerkin method, using a basis Iw;} in H !)(O), consisting of eigenfunctions of the problem
Tn
will be determined from the system where the unknown functions Xjll/ E C I (10, of equations (here and below, denotes differentiation in t) I
(9)
The eigenvalues Aj can be ordered in an increasing order. Then the eigenvalue AI > 0 is simple and the corresponding eigenfunction WI (x) > 0 in n. The magnitude of A1 can be bounded from below by [282. 2831 ( \0)
K(I)
\
= -, TT
K(2)
n
In the particular case when
= 2,
K(N)
=
2(N - \)
= (Ixl
is a ball. 0
III AI:::: [ ~N /
N - 2
(10')
N ~ 3.
RI.
<
] 2
(II)
R
Lemma 1 (1282,2831). Let the jilllction v(x) be such that IvllTv E H!)(!l). Theil we have the estimate
4((r+ I), n . 12 ___ 1I{lulI/(s)1 I TulI/(.I)III/,'llll d.1. (Ir + 2)- . II
+~
{t,'
=~
{t \\.:,a
2
' } 1/12(,nlll
N.
('I {
Local solvability in the problem (15), (16) follows from the theory of ordinary differential equations (it is not hard to show that the system (\5) can be always resolved with respect to the derivatives X',II/)' Below we shall obtain a priori estimates. which ensure existence of the approximate solution UII/ for any m on an interval [0. TJ of arbitrary length, Let us multiply each equation (15) by glll/' sum the resulting equalities in j from I to m, and then integrate the resulting expression in t. We have
i'
where z.~1 is the smallest positive root of the Bessel function .I IN .21/2·
IIvllo''''''llll :::
and the initial conditions
L 11~(Iv'ITV)II(h, 1
( 12)
Cilli
where f3 > I is arbitraryji}r N = I. 2 alld f3 E (I. (IT + I)(N ~ ~ 3. The cOllstant C 1 is de/ermined jiw/l the fiJl'lllu/a
+ 2)/(N
~
j,_1
i.1
II'
+
~~(IUII/(t)IITUII/(I))\\2 } !lx, t'llll 'LI,
(\UOII/IITUOII/)\\2 } l'llll
-
f3 IT
+I
+ iT +
+~
I IT,
f3' :
I
r
(1)(ulI/(I. xl) dx
.Ill
l
<[)(1I01l/(X))
=
dx,
II
(17 )
- 2)) ji}r
where
3/2 fiJI' f3 E where K(N. IT. (3) = K(N) ji}r N = I and N ~ 3. K(2, IT, (3) (1,2«(r+ I)] alld K(2,IT,f3) = (f3+ IT+ 1)/12(IT + I)lti)!' f3 > 2(IT + I). If f3 :::: IT + I, thell we have the inequality ~
IIvll/'I"'llllll:::
N AI
'
iJ
1/12(ITIIII {
L II-,.(Ivl I
d,l,
IT
..(\1111/1" II,,,) \\2} -< ----, r \\t\lIl1/! "/2 1111/1 II,1.·(111 dol + 1~I L1 II (d l.'lll)
4(IT + I) , (IT + 2)- .In
V)II Cilli
.
( 13) ::: l'
For every integer m > 0 we shall seek an approximate solution of the problem. 1111/. in the form
Theorem 2. LeI f3 = (T + I, Then if Ihe domain n is such Ihol 11.\ > I, Ihe problem (5), (6) has a glob(d sollilion, which sOli,ljie,l' (7). (8), Fllrlherlllore (20)
u:
on Ihe olher hand, AI < I, Ihenjiir any uo(x) ct 0 Ihe (il'Ohlem (5), (6) has global SO/lIlioflS 2 and Ihere exisls To E (0,1'.1, lI'here
+
flO
(21)
( 18)
2
a
II -,-(IIIIIII"l/III) 11
f3 = (T + I
2 The ('ose
2
Hence it follows that the functions (1II m l,,/2 11111 )' are bounded in L 2 (0, T; L (U»; 11I1II1"/2 ulII are bounded in L""(O, T;L 2 (U»; lulIIl"u lII in L"'(O, T;He'I(U»; (lulIIl"u lII )' in L 2 (0, T; L 1 (U», In particular, we have that 11I1II1"u'lI belong to a bounded set, for example, in W:(O, T: w\(m), In addition to (17) we shall need another identity, to obtain which we multiply each of the equations (15) by gllll' sum all the equalities in j from I to Ill, and then integrate in I, As a result we have --';';I (T 1.
435
§ 2 Boundary value problems in bounded domains
VII Some other methods of study of unbounded solutions
434
} 1.'1U1
ds= { , .10
sllch Ih(/[ (11(1).
{(IJ(II I11 (S,x)dxds,
To'
(1.1' I .->
11'\) ......... X
.Ill
From the above estimates it follows that 111 111 1"12 11111 are bounded in H\(wr) (wr = (0, n x n). However the embedding HI (wr) into l}(wr) is compact 1296, 3621, and thereforc from the sequence 11 111 we can pick a subsequcnce 11 11 , such that 111,"1"011" -)- 11I1"Ou in l}(wrl and a.e, Hence by other estimates we conclude that q(lI ll ) -)- q(lI) weakly in L 2 «(vr), Let us set A(II) = -~(IIII"II), Then it is not hard to see that AU/ are bounded in L""'(O, T; H 1 (U». From that we can conclude that Alu ll ) ......... X weak-* in L "'(0, T; H \ (U). The proof of the equality X = A(II) proceeds as in [2061. using (18), Passing now to the limit, and using standard arguments, we have that u is a global generalized solution and satisfies the inclusions (7), (8), To conclude, let us observe that the inclusion (8) also allows us to establish the weak Maximum Principle (for the method of proof 0 consult 1125, 3711>,
Proo( Let us start by establishing the first claim of the theorem. Applying estimate ( 13) to the equality (17) we derived above, we obtain 4((T + I) /., ----,-, «(T
+ 2)-
.
I
{11I1I/1,,12 I1I1 ,!'
II'".
+
dol
1.·llll
0
(22)
II /)
Hence by the condition 1\1 > I the functions (111//11,,;111 111 )' are bounded in L2 W, T; L 2 (U», while 111 111 1"11//1 are bounded in C"(O, T: He'I(U)), In a similar manner we hav!.: from ( 18) that I
d
"t 2
/
- - - I -,·II U I//(I)II I "",lll ::: (T (I .
+_
It is not hard to show that under the conditions of Theorem I the global solution
(23)
ct () stabilizes as I ......... 00 to the stationary solution U > () in U, ~U" I + UfJ = () in U, U = 0 on au (existence and uniqueness of the non-trivial function U are
II
t
:::
~-
(I
_\
AI
N
)
{
""
d,
I
(1.\/
L 11-:-:-(1
11 111(1)1
"
11//1(1»)11
-'
}
.
1.'d!1
2
established, for example, in 17, 211>. Stabilization of 11"11(1,.) to U,,+I in L d!) as I -)- 00 follows from the existence of the Lyapunov function
.I (11)(1)
= ::jI LN -
;0,1
II
~. '/ /I" I
2
( 10)
I (I) 11
Cll!)
which is non-increasing on trajectories 111(1, '), I > O}, and, in particular, from the inclusion (11 11."/2)1 E L2(R, x n) (see the estimate (17'), where c is independent of n. (This means that the norm II(u ll "/2)IIIY'iJll is "small"in a neighbourhood of I = 00, and that is in principle sufficient to prove stabilization to the unique stationary solution U ct 0, U 1 == 0,) Stabilization to the trivial solution, u'" I ......... 0 in l}(!l) as I ......... 00, is impossible; this is proved using, for example, the arguments used in the proof of Proposition X, 4 of eh. II.
*
From that it follows, in particular. that the functions 1111/ are bounded in L'" (0, T: L"! 2 (n», and that 111 m I" 1111/ are bounded in L 2 (0, 1'; l (n». Applying to the right-hand side of the above inequality the estimate (13) and the Wilder inequality, we obtain
Hi
I
d
,,12
-+~-I-III1I1/(I)III""ll!1 - -(AI'" (T
1. I I
.
I)
".n) "/1"'2111
(medsu
()11 2 \/H.11
11/1/ I
1.''''I.lli'
and thercl'ore
"Local solvability of lhe prohkm in Ibis case will be demonstrated laler,
*2 Boundary value problems in bounded domains
VII Some other methods of study of unbounded solutions
436
Passing to the limit m :::: fL --> 00 is performed as in the proof of Theorem I. Estimate (20) follows from the last inequality. Let us move on to prove the second part of the theorem. Integrating (15) for j = I, fJ :::: IT + I, over the interval (0, t), and passing to the limit m = fL --> 00, we have as a result (1I(t), WI)
= (110, WI) + (I
Using the Htilder inequality following estimate: E(I)
==
(11(1), WI)
~
(II" fl. WI)
(1I0,lJ l) '
- AI)
f' (11,,+1 (.1'), WI) ds.
~ II 11' I II;: ,'~l!J (II,
+ (1·-
(23')
.10
WI
AI)llwill,:,';ol
l'r+ I. we arrive at the
f'
.10
437
/.3. The case fJ > IT + I. This is the most interesting case: both the stable set 'Wand the unstable set 'V' are then non-empty. Below we shall assume that fJ > IT + 1 for N = 1.2 and that fJ E (IT + I, (IT + I )(N + 2)/(N - 2)) for N ~ 3. First we shall prove a theorem concerning local solvability of the problem. Theorem 3. AssLlme thai IT + I ::.:: fJ < IT + I + 2(IT + 2)/N. Theil there exists a CIIIlStllllt T. > 0 sllch that Oil the illiernl! [0, T.J prohlem (5). (6) has (/ solwion, which sati.\:ties the inclllsions (7). (X) . The proof relies on the following lemma [2R2. 283]. Lemma 2. Lei a ./illlCIioll v he sllch that Ivl"v E H!}(n). Then .!iJr £Illy fJ > I if N = 1,2, llIldlf)r 1 < fJ < (IT + I)(N + 2)/(N - 2) if N ~ 3, 11'1' h(/\'e the estimate
E,nl(S)ds.
Hence it follows that E(I) ~ F(r) a.e .. where F(r) is the solution of the problem
Here /' :::: N(fJ - I )/((fJ + IT+ 1)[2(1T+ 2)+ NIT]), C 2 = {I for N = I. 2 alld C 2 :::: [2(N - I)/(N·- 2)J2" for N ~ 3.
and therefore
+ (IT+ 2)/[2(IT+ I )1l21'
Proof of Theorem 3. Let us usc Lemma 2: as
I -->
o
T;.
I'
. /. -
()(III/1(t,.t))I.t:,::: (,lit 2
. II
Remark. It follows from (10) that the global solvability condition A I > 1 will be necessarily satisfied if mead! < I K( N) IN, i.e. for sufficiently "small" domains
n.
Let us consider separately the case AI = I, fJ :::: IT + I. Then from (22), (23) 2 it follows that the functions (1111/11,,/2 111/1)' and 1111/11"12 11,,, are bounded in L (oJ/). These estimates are not sufficient to pass to the limit. In this casc it is not hard to provc global solvability using thc standard com. Id . V ,,(x) = all',1/1,,+11' panson th eorem. n ec(,I thc f' unction (x) IS for any (Y > ()
n.
an.
a stationary solution of the problem !J.V:~II + V:~t I :::: 0 in U" :::: 0 on Thcreforc if 110 ::.:: V" in n, thcn II ::.:: U" in R+ x n. It is of intcrcst thaI though thc boundary valuc problcm has a continuum of . ' ILl . ose'IS asymptotically stationary soIutlons \ ,,:::: aWlI/llnll ' a > ()) . on Iy onc 0 f th stablc. and 11"11 --> a~I.lwl(x)in l}(n) as I --> 00, where
This follows from thc identity
(1I(t). WI)
==
(110, WI)
(see (23') for
AI
= I) 13451·
rT
!
{~II ~(1I1/1 iJ I I" 111/1. )\\2 , I
/" = ~(fJ -_1)_. 2(IT -+ 2) + NIT
}'" II 111/1 11"2 I." "ill)'
1 L Ih
I
=
ih
'.'101
f3(2 - N)
-
2(IT
+ {IT + I)(N + 2) (IT + 2) + 2) + NIT
> O. (24)
where by thc restriction fJ < (IT + I) + 2(IT + 2)/N wc have thai "I < 1. This allows us to apply Young's inequality to the right-hand side or (24), as a rcsult of which wc dcrive the estimatc
Substituting this estimate into inequality (I X). which is firsl differentiated. we obtain
tI
/f+ ')
.
I"
I( , .. 1'1 )
-/ 1I111,,(t)II""~lll} ::.:: clllll/l(t)II/::"ill} . I "
.
Hence it follows that 111/1 are boundcd in L ..... (O. T.:I.(1!2(n)) if T, > 0 is sufficiently small. Then from (17) ami (25) it follows that the functions (\11 11, 1"/2 11 1/1)' arc bounded in CW, T.: L2 (n)). 1111/11"111/1 arc bounded in L'''''lO, 1'.: H(I}(n)), and 0 this concludes the proof.
VII Some other methods or study or unbounded solutions
438
This theorem is presented to illustrate proofs of local solvability using a minimal mathematical apparatus. In the course of the proof of Theorem 3 there appears an inessential restriction from above on the size of (3. This restriction does not arise in the construction of the local solution as a limit of a monotone sequence of classical positive solutions. Uniform boundedness of the sequence of classical follows from comparing each of solutions in a domain of thc form (0. T.) x these solutions with the spatially homogeneous solution (see I. 2. eh. O. Sequences of strictly positive classical solutions lIf' which converge monotonically to II as E -+ O. can be constructed in a number of different ways. For example. as lIf we can take solutions of the original equation (5) with different conditions: lI. = E + lIo for I = 0 in !!. lI. = E on (0. T) x an. Then lI. :::: E in (D. n x !!. and therefore on each solution lI. equation (5) is uniformly parabolic. We could also do this differently: leave boundary conditions as they were. and regularize the equation by replacing in (5) the operator C.lI,,+1 by (IT + I)V', ((lI C + E C)"/2V'lI). A priori estimates, that guarantee. in particular. convergence of lI. to II as f -+ 0 are derived in practically the same way. Let us move on now 10 construct the stable setW'. First let us prove certain auxiliary statements.
n
*
Lemma 3. I.£'1 Iv(x)I"v(x) E I-lil(!!)' L£,{ 11.1' .I'£'T .I(v)
I = -£1(1
IT+ 1)
-
2
(3
I
·h(v).
d.\,
Th£'11
\I'e
= Iv
(27)
: Ivl"v E I-li)(!!): 0:::: J(Av) < d. A E 10. Ill·
The distinguishing property of this set. which follows from the method of its construction. is retleeted in
= ''W. u 10j !Iolds.
Lemma 4. EqlloliTy 'W'
lV'.
= {v
Here
: Ivl"1' E I-lil(!!)' a(v) - h(v) >
O. J(v) < dj.
Proof Let v E 'lV. v ¢ O. Then
and therefore o(v)/h(v) -~ I. whence v Then sup .I(A1')
E
'lV•. On the other hane!. let v
= J(v)
E
'W •.
< d.
Ai' 111,11
E
o
'lV.
Lemma 5. 711e quonTiTy d in (26) is finite; T!Ie set 'W' is hounded alld cOlllailled
2
L. N II -;-:-'(ll'l"v) iJ 11 i" 1
11-'
Some other important properties ofW are contained in
+ IT + I
\\'!I l' 1'1' a(v) :..-::
Let us introduce the set
so that v
··_........_· ......
439
2 Boundary value problems in bounded domains
. h(v) 1.'llII
= I. (I>(v(x)) dx. J!!
in l!Ie hall {
v ..
11'1"1 1 E .
I . 2((3 + IT + I) } H 0 (!!) • a(11) < --~ . ..~·--d . . .... (3 - ((T + I)
!I(I\'£' T!Ie illeljl/(/liTV
d
= j,
(26)
inf supJ(Av) > O. "/jill' A >0
Proot: Under thesc assumptions the boundary valuc problem
"r,.
"/cO
c.v"
2
IT+ I /31"11 --·---A h(v):
(3
+ IT +
v"
= 0 on an.
has a positive solution in !! for some I' E R, (see 1332j). (Moreover. such a solution exists for all I' > 0; its positivity follows from the formulation of the corresponding variational problem: see [339. 96. 297].) Let us set v = V~/I"tll. Then V"II E llil(n) and
Proot: It is 110t hard to see that 1 '( II J(Av) :::: -A~" 'a(!') -
+ /lv{:lllfi II = 0 in!!:
I
therefore. using the estimate (11). we obtain O(V))I/II!\,rllll]
supJ(Av) =.1 v ( - ,\-.0
_ (3 - (IT + I) 1((3+IT+ I)
[
h(v)
!~. l}J.I~.":~ ..1 [ (h(v))c!,,'11 ]
1/113
{tTl-
d <.I
==
-
III
> ",(3--::-_(_IT_+_I..,..)
1((3+IT+
I)
c' "";,",'.'::;','"
>
{
a(v)]III/3"IJTII"} [ h(v)
v _.._-
Hence by the equality (/(v) - /lh(v) == O. we have
o.
1 [1
.
£1 > .I(v)
I
= -a(v) 2 .
f3
(T + I + IT +
I
f3 _. (if + I) a(v). - 2(f3 + IT + I)
b(v) >
o Let us now prove a theorem concerning global solvability of the problem.
=
+
Theorem 4. LeI f3 > IT+ I fin' N 1,2 alld IT+ I < f3 < (iT I)(N + 2)/(N - 2) for N :::: 3. ASSIIlIle Ihal Ihe illilialfimclioll 110 ill (6) is slIch Ihlll 110 E 'W. Theil fin' allY T > 0 Ihere exisls a gelleralized SOllllioll of Ihe problelll (5), (6), which salisfies Ihe illcillSiolls (7), (8) alld helollgs 10 '1
qj"'W
ill the .I'l'l
Iv 11111"11
in n, then by the Maximum Principle II :s U" in R 1 x n. Theref~e u cannot stabilize to U as I -'> 00, even though the difference 110 - U in C(il) as a -'> I can be arbitrarily small. Similarly, choosing a > I we can prove instability of a stationary solution from above. It is not hard to show, using Lemma 4, that if the condition 110 E 'W holds, the function lI(I. x) stabilizes as I -> 00 to the trivial solution U == O. Indeed, if //0 E1f, then by (28) for III -'> 00 we have .1(11(1» :s .1(110) < d. However. if U =1=0, then .I(U) :::: £1, and stabilization of II to U as I -'> 00 is impossible. Let us show now that in problem (5), (6) for f3 > IT + I all sufficiently large initial functions 110 belong to the unstable set 11' (the analysis here employs methods different from those of ~ 6, eh. V).
110
Take v E 'W, v =1= O. Then by Lemma 4 a(v) - h(v) > 0 and therefore
E H!)(!!)l).
:s U"
Theorem 5. LeI IT
+
I < f3 < IT
.I(lIo) Proof As in proofs of previous theorems, the solution of the problem is constructcd
by Galerkin's method. Then the equivalent of (17) can be written in the form 4((T + I) --'-;-'0' (IT
if II
+ ~), . 0
) '1\2 111I",(s)I"I'II",(sll
1.'llll
ds::;.~ .1(110111) - .I (lIlIl (I)).
(28)
Hence it follows immediately that lI", E 'W for all I > 0 and sufficiently large Ill, sincc 110 E 'W, .1(110) < £1 and IlIo",IITIIO'" -'> II;~+I in H(\(!!) as III -'> 00. Then by Lemma 4 a(II",(I» :: b(u",(I»), and therefore 4(iT . !1T12 1I 111 (.1)} . '11 - - - -+ - -I) j ' /./ II I 11111, (.1) (IT
+ 2)'
. 0
+ f::.J.IT + I) ~(f3 + IT + I)
2
.. £1.1
l'llll
{t. 11·/L(III",(1)!"I1",(I»)11 /1
d.t,
Theil Ihe problem (5). (6) does (0, T.
+ I + 2(iT + 2)/N
alld
(29)
< O. IIIIOIII."·'(l!1 > O.
1101
IUII'e a global SOllilioll, alld \I'e c(/Ilfilld To E
J, where
==
T,
f3 + IT + I (measn)1 /3 (f3 - 1)(f3 - (iT + 1)(
I"
I
/3
)/(lrl-llilIolIl.''''1l11 < 00.
slIch Ihul
Proof Recall that the restriction f3 < IT+ I +2(IT+2)/N ensures local solvability of the problem. Let us show that II cannot be defined for all I > D. By passing to the limit in (28) we obtain .1(11(1)) :s .1(110) a.e., and therefore
+
2 1'(1)1
441
2 Boundary value problcms in bounded domains
§
VII Somc other mcthods of study of unboundcd solutions
440
}:s .1(110",)
<
c.
2 Therefore the functions (111,"1"/ 2 1111,1' are bounded in L (wr) and !//",I"II", are bounded in C"(O. T; H!I(n». From that point on the theorem is proved as the o preceding ones.
Thus for f3 E (iT + I, (IT + I)(N + 2)/(N - 2) +) the problem has a global solution. if the initial function is sufliciently small. Let us note that the asymptotic behaviour of global solutions II E 'W here is completely different from the case f3 E (I, IT + II. The point is that every stationary solution U > 0 in n of the problem tiU"+ I + U13 == 0 in n, U == 0 on an, is unstable (concerning existence and stability of Usee 1331, 334, 96, 200, 296]). This is easy to demonstrate, when, for example, n is star-shaped with respect to .r = O. Indeed, in that case the function U" = aU(a'" X), III = 1f3 - (IT + I) 1/2 > 0, for any a> 0 is a stationary solution in the domain n" == IIJ!'lI x E nl. If a E (0. I), then clearly Ii c n" and sUPll" U" < sUPll U. Therefore if in the original problem
a(II(t)
Then for
111
==
OQ
•
we derive from (I H) the following inequality:
Ilu(I)II~"j~~lll)
!llIoll~':?'IlI) +
::::
+(IT+2)f3-(~+ f3
2(IT + I) {3+IT+1
< - - - - - h(lI(I» a.e. in R t
+ IT +
1)(l11easn)II/3 1/1"'21 I
r lIu(s)II~J""~I'l;lds, .10
(30) I>
O.
In deriving this inequality, we used the estimate /3·\lr·1 I
l\
Ilu(l)lI u l""'illl:::: (measH)
II
jJl/I" 1 ~)
II
11#+""
I
11(1) I""I!!)'
Taking into account the fact that IIlloll';}'\lli > 0 and using (2()), we deduce the 0 required result from (3D). Let us note that condition (29) of unhoundedness of a solution of the problem is in a certain sense the opposite of the inclusion 110 E 'Hi' (see (27)).
V II Some other methods or study of unbounded solutions
2 Equations of general form In this subsection we consider the problem (I )-(3) with coefflcients
I COllditiolls .f{Jr Mlohal sol\'abilitv jill' arhitrarv i//itial.filllctio//s tV" Dcrivatio// or Mlohal i//tcMI'llI I'stimatl's.f{}/" the solwio//.
= 0)
Here, as in the previous subsection, we apply the Galerkin method. Tbe main restriction on the coefficients cp(u), QUI) consists of the following: there exist positive constants M I, M2' such that the incquality
443
The above theorem is proved by Galerkin's method, practically in the same way as Theorem I. It is shown that the approximate solution U/II (I) satisfies for all T > 0 the inclusions (33), (34). Condition (32) then allows us to prove that (
2 A//alysis usi//M a .fill//ily or statio//ar\' solutio//s
holds; moreover,
*
For simplicity we shall also assume that c//(II)/\I -I- C/J 2(1I)1
:::: M
Here we present an application of the method of stationary states, which is diffe~rnt from the one of I. The method is used here to determine conditions of global solvability of the problem. Here we shall not be needing any preliminary results, as all the necessary material is contained in I. Below we shall assume for simplicity that a local solution of the problem (I )-(3) exists and belongs to C~/ wherever it is positive and has for all t > 0 a continuous derivative 'V,c/J(u(£, x)). Then the solution obeys the Maximum Principle and depends in a continuous monotone fashion on the initial function; in particular, the comparison theorem is valid for solutions of equation (I). The proof of these assertions proceeds by constructing the generalized solution as a limit of a sequence of classical strictly positive solutions of equation ( I). In the following we assume that cP(uo) E C l (n) . Let {U) be a family of radially symmetric stationary solutions of equation (I) (see (1.6), (1.7)). We shall only need the estimate (1.9):
3 = const ;;. 0 for u ::::
o.
(32)
For convenience, let us introduce the function
Theorem 6. Let collditiol1S (31), (32) be satisfied. Theil fiJr (/II.\' T ;;. () there exists a gelll'/"ali:,ed solutioll o(problem (1)-(3). such that 11(£. x):::: 0 a.e. ill H.filr ani' .fixed t :::: 0 alld the jiJl/owillg i//clusions hold:
(34)
*
..' .. .. U(I,U o ) ,> U(I,U o ) = c/)
I [
c/J(l}o)(1 -
I
.2
..:
110)+
]
.
__ [
,10 -
4)( U 0) ] 2N-.--
II'
Q(l}o)
-
(36)
Remark 1. Condition (31), which plays a crucial part, will be necessarily satisfied
Here Uo > 0 is the parameter of the family, r = Ixl. Let us note that in certain particular cases, for example for N = 1 or Q(II) = }LC/J(II) , exact equalities, which
if
were derived in
Q(s)/cp(s)
->
0, .\
-> 00.
(35)
Remark 2. If in (I) we set c/J(s) == Q(s), then (31) holds for M 2 = I (M 1 = 0). Therefore nl') holds if AI ;;. I. Thus in this case existence of a global solution depends only on the domain H (this situation occurs for equation (5) for (3 = cr+ 1).
* I, can he used.
Theorem 7. Undl'/" the ahol'e l/SSlllllpticlIIS the stable set or the prohlem (I )-(3) ('ontaills the set
IV
= (uo:::: 013U o ;;' 0: H
~ suppU(r;U o ), uo(x):::: U(r;U o ) in HI·
(37)
§
VII Some othcr methods of study of unbounded solutions
444
Proof The proof is based on the Maximum Principle. Indeed, if Uo E 'W then by construction of 'W we shall have for all t ::: 0 the inequality u(t, x) ::: U (r; Uo), x E n, since U(r; Uo) > 0 on on. The latter follows from the condition n ~ suppU(r; Uo). 0
Let us note that (37) allows us to describe the structure of initial functions in '11". In particular, the choice of the size of U0 in (37) determines the maximal amplitude of the global solution, sUP,dlu(t, x) ::: Uo, I > O. The magnitude of Uo depends to a large extent on the geometrical dimensions of the initial perturbation: the "wider" it is, the larger must the value of the parameter Uo be, because otherwise the function uo(x) will not be majorized from above in by the stationary solution U(r; Uo). Inequality (36) allows us to obtain a reasonably good estimate of this dependence.
3 Analysis of lin bounded soilltions /Ising the eigenfimctions method This method is similar to the one used in subsection 2, § 6, eh. V. As well as the method based on conditions of III-criticality, it is useful in the analysis of unbounded solutions of boundary value problems in bounded domains. Let us impose some restrictions on the functions that enter equation (I). We shall take the functions qJ, Q to be convex in R i ' We shall assume that the function P(s) = Q(¢-I(s)) is also convex, and that s/P(s) is non-increasing in R+. These conditions can be written down in the form
n
q/'(I) 2: 0, Q"(s) ::: 0,
(41 )
Q"(s)cf/(s) - Q'(s)cp"(s) 2: 0,
(42)
(43)
Q'(.I)c!J(S) - Q(s)c//(s) 2: D. s > 0. ~ll
Corollary 1. Let
lim lc!J(s)/Q(s)! =
(3H)
00.
,\--·x
Then the problem
445
2 Boundary valuc problems in bounded domains
(I )--(3)
n
has Mlobcl! solutiO/lS for all initiaijilllccio/ls.
Indeed, if (38) holds, by (36) we have U(r; Un) -> 00 along some subsequence Uo = Uti -> 00 in R N . Therefore for any domain n and functions l/O(.\') there exists U~, such that IIn(x) ::: U(r; U~) in fl, which by the Maximum Principle ensures boundedness of the solution uniformly in I. For equation (5) this case obtains if f3 < (T + I. Corollary 2. LeI
lim Ic!J(s)/Q(s)] \""".. 00
= JJ. = consl
> O.
the inequalities are satisfied. for example, by the coefficients of equation lor f3 ::: (r + I. As shown above. it is exactly for these values of the parameters that unbounded solutions can be expected. As usual. we shall denote by WI (x) the positive in eigenfunction of problem (9), which corresponds [0 the minimal eigenvalue AI > O. We shall choose the function 11' I, such that IIII'I III.' ill) = I. The main result is contained in the following theorem. where we have introduced the notation Eo = I' 1/11 (x)l/o(x) c!x. (5)
(39)
.Ill
Theorem H. Let condilions (41 )--(43) hold and leI the jilllction thaI
110
in (2) be stich (44)
n
Then if is ('()ntained in a ball of radius (2N JJ.) 1/2, Ihe problem (I )-( 3) has a global solwion for aI/ initial jilllclions.
7'l,en prohlem where
(I ),-( 3)
T.
This claim follows immediately from (36). Furthermore, from (36) we obtain
does not hal'e a glohal solution anc! there exists To ::: T ,.
=
Q(Eo) _.__ Q(E o ) - }q¢(E o )
(Xc _dYJ_
.1/,,,
<
00.
Q(YJ)
such thaI
Corollary 3. Let l{)(s)/Q(s) -, 0, s
· 4 00.
(40)
lim sup 1/(1, 1--··'1"11
Then, it supp U(r; Uo) -"> 101 as Un -> 00, Ihe set 'W, dejined by (37) is bOllnded in C(H). If: on the other hand, supp U(r; Uo) = RN .fi)r all sllfficiently large Uo > 0', then 1~r contains jimclions 1I0(.\') of arbitrarily large norm in C(i!).
'This possibility occurs, for examplc, if condition (1.12') holds.
x)
= 00.
(45)
Pro(J( Let us set I:'(t) = (1/(1, x),
Let us note that it is precisely in the case (40) that we should expect the appearance of unbounded solutions for sufficiently large initial functions 1I0(.r), which do not belong to the stable set (37).
\(~U
Then
L:;((l)
= Eo.
11'1
(x))
== I' I/(t,
.II!
X)WI(X) dx.
As in subsection 2, § 6. eh. V, we obtain for I:'U) the equality (46)
VII Some other methods of study of unbounded solutions
446
which we shall analyze below. By Jensen's inequality for convex functions, from (42) we have the estimate (WI, Q(II» == (10" P«(p(u») 2: P[(w" (/J(u»!. using which we obtain from (46)
However, the function .1'/ P(s) is non-increasing in R, (see (43), and therefore in view of the convexity of (/J, and thus also Jensen's inequality (w" 4J(II») 2: (/J[(w"II)1 = (p(E). we have (11'" (!J(II))/P[(U'" 4J(II»)] '5: c!J(E)/Q(E). Then from (47) we get dE ::: P[(WI,
-
[(P(E) 1 I ~ A,-) f:.' . 0(:)
(48)
From (48), (43). (44) we conclude that L'(t) ::: 0, that is. that E(n ::: Eo. Then from (48) and Jensen's inequality it follows that E'(I) 2: Q(E(t»)[ I AI4J(Eo)/Q(E oli. Hence we have that the function E(I) cannot be bounded for all I E (0,1';1 and there exists 1'1 ::: 1';, such that E(I) ~ 00 as I ~ 1',. Since E(t) '5: SUP,II(t, x), (45) follows immediately, which concludes the proof. 0
Remurk. The inequality cjJ(E o ) L" ::: J-toQ(E) , 0 < I < 1': J-tll :::: I - AI--.- > 0, Q(E II )
(49)
obtianed in the course of the proof of Theorem 8, allows us in a fairly general case to derive an upper bound of 11(1,') in the norm of L"-(R N ). Indeed. from Theorem 3 in I it is easy to obtain a condition on the functions 4J and Q, under which E(T) = 00, if l' is the blOW-Up time. Then integrating (49) over (I, T) we have
*
("
./HII
dry Q(ry) -
.
-'-- > J-tIl(7 - I).
(50)
Finally, if in the computation of [:'(1) :::: (11(1, x),II',(x» in (50) we use the estimate (1.26) of the solution 11(1, x) in terms of 11(1.0) and the structure of the function U given by (1.9), then (50) leads to an upper bound for the function 11(1.0) :::: sup,. 11(1, r). In a number of cases in terms of dependence on the nonlinear coefficients
447
3 A parabolic systcm of quasi linear equations with a sourcc
Recall that if A, :::: A, (n) > L then the boundary value problem is always globally solvable (see Theorem 6). To conclude. let us note that practically all the results of I having to do with S- and LS- blOW-Up regimes can be used to describe asymptotic behaviour of unbounded solutions of the boundary value problem (I )-(3). [n particular. for the problem with power type nonlinearities (5), (6). Theorems 5. 6 of I. as well as the results of 6, Ch. [V. remain valid. All of these allow us to provide a sufficiently exact description of the singularity of the limiting profile lIt To, x).
*
*
*
§ 3 A parabolic system of' quasilinear equations with a source This section is concerned with the study of the properties of solutions of a parabolic system of two quasi linear equations of nonlinear heat conduction with sources: II, V,
= Llul" 1 + vi'. t :::: LlV/.1. ' + 11'1.
(I) ~2)
Here J-t > 0, v > 0, P 2: I. q :::: I are constants. This system describes processes of heat diffusion and combustion in two-component continua with nonlinear heat conductance and volumetric heat release. The functions II, v can be interpreted as temperatures of interacting components of some combustible mixture. We shall be especially interested in the conditions for existence of unbounded solutions as well as conditions for their localization in the Cauchy problem. Let us observe that there are four parameters in (I), (2), and therefore, even though the nonlinearities are of power type. for arbitrary J-t, 1', [I, q the system docs not admit self-similar solutions, which, as we know. provide us with a detailed description of asymptotic behaviour of unbounded solutions and, in particular, of the localization property of finite time blOW-Up regimes. We shall consider initially the first boundary value problem for ( I), (2): U(O, x) :::: 1I0(.r) :::
O. v((J. x)
11(1, x):::: v(l, x)::::
O.
= vo(x) r >
0,
2: 0, x
x E
an.
E
n,
(3) (4)
where U is a bounded domain in R N with a smooth boundary an: 110, Vn are bounded continuous functions, 11 1\/1' E HMn). vil"1 E H(\(U). First we shall determine the conditions of its global insolvability. Then, using an approach based on the analysis of a family of stationary solutions of the system. we shall derive restrictions on the parameters of the problem for which it is always globally solvable for arbitrary initial functions 110. lin. and establish a condition for the absence of localization in finite time blOW-Up in the Cauchy problem for (I), (2).
S 3 A parabolic system of quasi linear equations with a source
VII Some other methods of study of unbounded solutions
448
It turns out that much depends on the sign of only one parameter m = pq - ( I + fL)( I + IJ). If m < 0 then the boundary value problem has only globally bounded solutions (there is no blow-up) and unbounded solutions in the Cauchy problem are not localized. This result appears to be optimal, that is, the inequality m < 0 is a necessary and sufficient condition for both these properties of solutions. In this section we study only the most general properties of solutions of the system (I), (2). A more detailed numerical investigation of systems of this kind is undertaken in § 4, where we also consider self-similar solutions, which exist under some restrictions on thc parameters of the problem.
Let us choose the function
Below, under the indicated restrictions on the parameters, we single out an unstable set 'V in the space of initial functions, such that the inclusion {1I0. Vol E 'Y implies global insolvability of the problem (I )-(4). This means that there exists a blOW-Up time To < 00 and -1'- (II II "+1 (t,') 11 2I.'tlll+ II v Wi I (I,,) 11 2UtIli ) ,_~IP -00,
(5)
so that IIWIII{,I(111
=
(8)
I.
Taking the scalar product in L 2 (n) with W I of both parts of the equations (I), (2) and integrating the resulting expressions in I, we obtain the system of equalities
f' (ttl I (.1'), WI) ds + f' (vl'(s). WI) ds. .10 .10
(u(t). WI) - (110. WI) = -AI
(v(tl. WI) - (vo. WI)
Conditions for absence of global solutions of the boundary value problem for p ?: 1 + p., q ?: 1 + 'J
WI
449
= -AI f' (V Il + 1 (.1'), WI) ds + f' (11'/(.1'). WI) ds, .10 .10
(9)
(10)
Let us introduce the notation llO
= (110, WI)
::: 0, lin
= (vo, WI)
:::
O.
( I I)
and let us set
From the Hmder inequality and (8) it follows that
II
Let us note that formally in (5) it can happen that only one of the functions II or v grows without bound as I -->- T u' However, because of the "entangled" nature of the source terms in (I), (2) the functions II, v have to become unbounded at the same ti me. In the proof of the propositions formulated below, we shall assume that the local solution of the boundary value problem satisfies the natural inclusions {1I 11 "/2\" {Vi t' 1l 12], EL 2 (0,
T; L2 (!!)1. (6)
11'·+1, jilL + I E
L00(0, T; Hil(!!)' T < To.
Under the assumption of boundedness of II, 11 these can be easily derived by Galerkin approximations. It is also not hard to establish a weak Maximum Principle, so that if II, V arc bounded in (0, T) x n, then II ::: 0, v > 0 a.e. in n. 0<1 < T.
I Derivalion or a syslems or ordinary dWerenlial ineqlllllilies
°
Let us denote by WI (x) > in !! and AI > 0, respectively, the first eigenfunction and the corresponding (smallest) eigenvalue of the problem ~W
+ Aw = 0,
x E
n,
Wlall
= O.
(7)
= P(I), ::: (Villi (I), WI )I/IIlIII = R(t).
(1I(t), WI)::: (II"; 1(1), WI)I/II'+II (v(l), WI)
(\ 3)
Furthermore, taking into account the fact that p ::: 1 + fL, q ::: I + 1', we have (vl'(t), WI) ::: RI'(t), (11 '1 (1), WI) ::: P'I(t).
( 14)
Using the notation of (J I), (\ 2) and the estimates (13), (\ 4), we conclude that the solution of the problem (I )-( 4) satisfies for all admissible I > 0 the system of inequalities P(l) _
{/n :::
-AI
('
.10
p"I·1 (.1') dx
+ f'
.10 R(t)-lio:::-A 1 (' RIlII(S)tlX+ f' .In .In
R"(S) ds.
(15)
P'I(s)d.l.
(16)
In conjunction with (15), (16) let us consider the following system of ordinary differential equations:
§ :1 A parabolic system of quasi linear equations with a source
VII Some other methods of study of unbounded solutions
450
Let the functions
P, R satisfy p(O)
the conditions
= ao
;:: 0, R(O)
= 170
:::
(19)
0,
451
such that E(I) -c; 00 as I -c; To. Therefore the solution of the boundary value problem is unbounded in the sense of (5). b) Let fL #- I' and for definiteness fL > 1'. Then, using Young's inequality
A direct comparison of (15), (16) with the problem (17)-(19) shows that for all admissible I we have the inequalities (20)
Pit) ::: ji(l), R(I) ::: R(I),
Therefore the system of equations (17), (18) allows us, in view of (20), to define the conditions under which the functions P(I), {?(I) cannot both be bounded for all I > 0, that is lim max{P(I), R(I)} = 00, I -c; To <00. In view of the inequalities ,
" I I
I /I,·! I1
1111
H() <
II VIlillll!(/lilill 111/llltll 1.'lll) 1lJ I I.'illl '
I
_
> 0 is an arbitrary constant and
fL-/J A II = - fL+ I
(I- -
+IJ)IHII!IIl""
I +fL
>0,
we obtain from (22)
LeI 1\
E
I!I" til II WI I1"llll '
I (I) :::
1I/.'illi
where
E :::
I. Then, using (23), we derive from (24) the following estimate:
this ensures that (5) holds. Let us set
2 The ('(/se fJ
Theorem I. LeI fJ = I doma;1I n he silch l!Jal
+ fL.
if
= I + I' alld
fJ = I
if
E --
/10
glolwl SOlllliollS, alld cOlldilioll (5) holds
Let us sel E(I) = ?(I) + R(f). Adding up equations (17) and (18) for il = I + 11, we have thaI F satisfies the equation
= (I
+ RIIIlI,
- A ill i',I+,'
where we also have by assumption that E(O) We have to consider two separate cases. a) Let fL = 1'. Then, using the inequality I
+ {;I I"
:::
/A 0 1111
"1/111"1 .
1
T -~-"("" , _. Eol I -, AI) .Ifill I 77 l i ,'
._~/77 - - - - "
-
fii
2"A o
<
-->
00.
l "j!(/l'1'1
Therefore in the case under consideration the unstable set 'V is all the space (110, Vo
I Uo + VII ¥=
0).
0
(26) (2))
which is equivalent to the system (17), (18). in the case A I < I. J..L > /'. The . . , . I -I, 1/111"1 dashed Ime shows the Isocline 01 zero P II : P = [AI {? JIl\ ' the dashed and
and (21), we get from (22) that
dE/dl ::-. (I - AI )2
1/.() ,. 11 " ()
It is not hard to obtain the same result in a different way. In Figure 80 we have drawn in a schematic way the integral curves of the first order equation
> O.
+ {;) II", t; ::: 0,
2 "( I
f'
(22)
I> 0,
= (uo + vo, III,)
'(I ,I') mm -
Then for all E ::: E(O) the right-hand side of (25) is positive and therefore E(t) 00 as I -c; To' where To .S T.,
./ill'
+ fL'
dE/dl
Ell --
Moreover, leI Ihe (21 )
Theil Ihe prohlem (I )-·(4) has sOllie To < 00. pro(~r
= I + I' 1111 + VII ¥= 0 ill n.
= I + fL,
"IP + RI IJ."
Hence it follows lhat there exists a lime
I
I
'= (I - A, )TI' E +,',
= To
::: 1'" where
I
> O.
dOlled line shows the isocline of infinity, P.",: P = IAIR1+I'II!II""I. The thick curve denotes a special trajectory, the separatrix 1\: P = I's(i?). For large R
'),'
T, =. 1'(
/:' I - AI)
"(0) < 00.
(27)
453
§ 3 A parabolic system of quasilincar equations with a sourcc
VII Somc othcr mcthods of study of unboundcd solutions
452
tP I
"
~ ~\ ""'.
if Fig. 80. Thc phasc planc in the casc fJ == I
+ fL,
q == I + I'. A, < I
Fig. 81. The phase plane in the case fJ == I
As time advances (the direction of evolution of the trajectories is indicated by arrows), all the integral curves bunch up into an ever narrowing neighbourhood of the separatrix P.\', which by (27) ensures that the functions P(I), R{t) grow to infinity in finite time. For comparison, we have drawn in Figure 81 the integral curves of the same equation (26), but in the case A, > I. Here the functions P. R are bounded for .tall I > D, and, morcover P(I). R(I) -> 0 as t -> 00. In some sense this indicates global solvability of the problem (the proof of this fact is found in subsection 3). 3 The C(/se p > I
Let as before
E(t) ::::
+ fL,
q > I
p(n + R(I). From (17), (18) it follows that
tiE/tit
= -Adp 1fl' + RI'I/ll + R" + pll.
I
> O.
AI >
(29)
If fL :::: I', using (23), we derive from (29) the inequality tlE/dl ::: 2 "E"I' - Co.
_ II
.
>
-
- II I"
R
1',
I >
(30)
O.
then, selling for definiteness fL >
- Do. Do
(1'111/(1' fL - I' I' + I = --) fL+ I ( fL+ 1
II,
and using the
1'1
we obtain from (29)
Let us estimate [J", R" using Young's inequality:
where Ao and Bo are constants:
+ v.
tlE/tll:::[JI"'+i,?II/l-CO.I>O;Co=Ao+Bo.
R . /l (28)
q =-~ I
Then we obtain from (28) that
If, on the other hand, fL j estimate
+ II
+ fL.
tiE/til::: 2"1:;,'1 I - (Co + Do).
I
(31 )
> O.
It is easy to see that the right-hand side of this inequality admits passing to the that is, for fL = I' it is the same as (30). Therefore we have proved limit fL ->
,.,t.
Theorem 2. LeI fJ > I + fL, fJ > I .limelio/ls l/o. Vn he sl/ch Ihal
*:I A paraholic system of quasi linear equations with a source
VII Some other methods of study of unhounded solutions
455
/
/
Fig. 82. The phase plane in the case Theil Ihe /ll'Ohll'lI/ ( I )-(4) dol'S where
1101
h(/\'C
0
fi >
I+
fL.
if >
I + 1/
Fig. 83. The phase plane in the case Pif
colldilioll (5) holds.
Of course. the functions I/o, Vo satisfying condition (12) do not exhaust all the unstable set Y. Even though Theorem 2 gives an upper bound for To. it docs not use all the information contained in the system (17). (18). A more detailed and explicit description of the set '11' is provided by the analysis of the integral curves of the equation RI'-}'1Plt,·
dR
P'I - AI RI
<
(I + fL)( I + 1/)
glohol SOltlliolls, £Iml for some To ::: I' ,.
terized hy P. subsection 2.
dP
1J
I J.L •
P> n. R > n.
(33)
which is equivalent to that system. These arc given schematically in Figure 82. where we have distinguished the . 11/11+"1 -I t. 1/,( isoclllles of zero Po: I' = [AI RI'I . of infinity. I'ex: I' = lAIR 1'1 . The thick curves show the separatrices A - Band C - D. The set'1/' contains all the points IP. Rl = (tlo. WI). (VII.1JI1)) lying above the separatrix A-B. Trajectories through points in this region converge as P. R -> 00 to the separatrix C - D, which ensures unbounded growth of the functions Prl), I?(I) in finite time. Let us note that the behaviour of the integral curves lying under the separatrix A - B indicates the existence in the problem of a stable set 'W'. Here it is charac-
R -~ 0 as
I --;. 00.
A rigorous construction of the set 'W' is done in
Remark. [n the derivation of the system (17). (18) we used the restnctlons p ::: I + J.L, (I ::: I + v (see (14)). Nonetheless. the phase plane of the system ( 17). ( 18) correctly rel1ects typical behaviour of solutions of the original problem (I )-(4) for arbitrary values of the parameters. It is not hard to sec that the phase plane picture depends on the sign of the one parameter 11/ ::::; fiq - (I + J.L)( I + II), which agrees well with results obtained below. a) If m ::::; D. that is. pq = (I + J.L)(I + v), then global insolvability and global solvability of the problem for arbitrary till. Vo hold for AI < I and AI > I. respectively: phase portraits in these two cases arc the same as those in Figures 80. 81 (sec subsection 3). b) If m > O. that is. pq > (I + J.L)( 1+ 1'). then it follows from the analysis of the system (17). (8) that there arc non-empty stable and unstable sets (see Figure 82 and subsection 2). c) [n the case 11/ ::::; fitt - ( I + J.L) ( I +1') < () the phase plane has the appearance as in Figure 83. Here there are no trajectories to which there correspond unbounded solutions. so that the stable set can be the Whole space of initial functions. We should note that the Galerkin method. which was used in 2 to study parabolic equations. cannot provide us with all the above results. Roughly speak-
*
* 3 A parabolic system of quasilinear equations with a source
VII Some other methods of study of unbounded solutions
456
ing, in the process of using this technique to derive a priori bounds for solutions, the "control" parameter m :::: pq - (I + j.L)( I + p) does not arise. For example, the global solvability condition for the system for arbitrary 110. Uo is given by the two inequalities 17 < I + j.L, q < I + P. Clearly, this domain in parameter space is much smaller than the (optimal) set pq < (I + j.L)(I + p).
2 The stable set for pq > (1
Clearly, using the comparison theorem, the stable set can be defined in terms of the solution of the stationary problem (39), (40). However, such a construction would not be optimal. We start with a negative result. Proposition 1. LeI Ihe domain n be star-shaped with respect to the point x Assllme Ihat N > 2(1 + I/a) al/d
f3
+ JL)(1 + II)
Below we shall assume that the functions tI. v are in C~;" wherever they are positive and have in n continuous derivatives "111 1+", "1v l +1" (these assumptions are quite natural: see 3, Ch. I). Then the solution of the problem satisfies the Maximum Principle and depends monotonically on initial functions. The stable set can be defined in two ways. In the course of applying one of the methods, some (later shown to be insignificant) restrictions on the parameters of the problem have to be imposed: this analysis allows us to consider quite interesting properties of the stationary solutions of the problem (I), (2), (4).
*
>
IN/a + 2(1 + l/a)J1N - 2( 1+ l/aJl
Thel/ the problem (39), (40) has
Let pq >- (I + j.L)( I + p). Let us consider the functions U. V. which solve the stationary system of equations (\). (2), which we write for convenience in the form (34) fl(IUI"U) + IVI" I V = 0, fl(IVII"V)
+ lUI"
IU
= O.
x
E
n
(35)
I/O
(4\ )
I
l/ol/-lrivial,lOl/-l/egatil'e solllliol/s.
0= -(fl(lflUI 1/" IflU).lJI)
!
+ (UI!,
11')
a
= -(lflUI I /
It
I flU. flw)-
"UII/".IA"",U)(.I /. - ---IIUII/I"'ll' N fJil 11'-;-(1,-,
f3 +
cJl/
I
.
I
= 0,
V(x)
= O.
x E
an.
(42)
I
where iJ/an stands for derivative in the direction of the outward normal to a11. [t is not hard to see that
a
w(x) ::: 0, -;-) (lM!1
\'I"
I
M!}(x) ::: 0, x E
(1/
an.
This follows from the fact that n is star-shaped with respect to X :::: 0, the boundary conditions (40), as well as the assumptions U ::: 0, flU ::: 0 in n. Therefore
(obviously, for positive U, V this system coincides with the original one.) The functions U. V satisfy the conditions U(x)
= o.
Proof. Following 1332, 333], let us take the scalar product in L"(11) of both sides of equation (39) with the function w(x) = L~ I x,ll \, Then, taking into account the boundary conditions, using Green's formula, we obtain
. illl
/ The s((lliol/w}' solutiol/
457
(43)
(36)
and then from (42), using the easily verified identity Let us make the change of variables IUI"U functions U, V we obtain the problem
-+ U,
IVI,LV
-+
V. Then for the new -(IM!II/"
I flU
,
flw)
== (
?)
N ._ I+I/a -
IlflU11
1 t 1/" I"'''IH)'
(37) U
= V = O.
x
E
we obtain the inequality
an.
Here we have introduced the notation a = p/(j.L + I), f3 = q/(ll + \) (let us note that af3 >- I). Solving the first equation for V, V = -lflUI I /" IflU and substituting into the second one, we obtain the following problem for the function U: (39) - fl(lflUI I /" I flU) + lUll; I U = 0, x E n, U
= flU = O.
x
E
an.
(40)
N ( I+I/~
N I!I I ~ - 2) IIflU· II l111/" t"'''!lll -- f3 + IIiUIlU"llll :::: O.
(44)
On the other hand, by taking the scalar product with U and integrating by parts. we have from (39) that
*:\ A parabolic system of quasilillear equations with a source
VII Some other methods of study of unbounded solutions
and therefore (44) means that we must have the inequality I ( 1+Nl/lY -2- f3+N)
111/"
II C1U II/I·''',ll) ::cO.
(45)
However, if (41) holds the left-hand side of (45) is non-negative, and therefore 0 the only admissible solution is the function U == o. Let us notc that the Strong Maximum Principle in this case gives us a strict inequality in (43), and therefore in (45). so that this proposition is valid if we have equality in (41).
459
Remark. The conditions p > I, q > I are more or less related to the requirement that the solution be unique. There are reasons to expect that uniqueness holds even if just the one condition pq 2: I is satisfied (at least if pq 2: I the spatially homogeneous problem II' ::;:: vI'. v' == 11'1. I > 0; 11(0) == v(D) ::;:: 0 has only the trivial solution; if. on the other hand, pq < I. then non-trivial solutions exist). .f. A ditrerl'1I1 ml'lhod or COIISlrtlclillg Ihe stahle SI'I
Let us show now that a non-empty stable set 'IV exists under the condition pq > (I + /-1-)( I + p) even if by Proposition I there is no stationary solution. Let us consider first an example that gives a partial explanation of the reason why absence of a positive stationary solution of the problem is not that important.
2 A .fll/llil." or Slciliollary SOlllliolls
Let U be a positive in n solution of the problem (39). (40). Returning to the original notation, we have that the functions
are a solution of the problcm (34 )--(36). Lct the domain U be star-shaped with respcct to the point x ::;:: 0 (that is. from the condition .lEn it follows that ax E U for all {/ E (0. 1)), It is not hard to see that the family of the functions
V,,(x)
= a"'1/Il'! II/!,",
(46) (I'!
111/'
I
I ilV I (UX).
where a E (0. I) is an arbitrary constant (the parameter of the family), satisfies equations (34), (35) and is strictly positive in domains n" with boundaries au" = Ix lux E i/Ul, and that U" = V" = 0 on au". Let us note that U c n" for any a E (D. I), so that U" > O. V" > 0 on an. Using the family (46) we can now determine the stable set 111' of the problem (1)-(4).
3
The .I'lahle .I'l'l Jill' pq > (I
+ /-1-)( I -+ II)
Theorem 3. IA'lp > I, if > I. alld a.l'.I'lIIJ/e Ihm Ihere e.l'i.l'l.I' a//mHril'ial SlClliollarr .I'ollllio// U I. V I ot' (34)-(36). 711l'1I Ihere I'xislS a //o//-elllpl." .I'lOhll' .I'el '111' = {( lI o. vo) 11111 2: 0.1111 ::: 0; 3 a E W, I): 1I0::C. U", Vo ::: V" i// HI. .I'llch Ihal it' (110, lIo)
E'lIf, Ihe pro!JIeIJ/ (I )--(4) has a glohal (boll//cll'cI) .I'o/lIlioll.
Proof: The theorem follows from the Maximum Principle. Under the above the assu;nptions concerning smoothness of the solution everywhere in R, x inequalities II ::: U". v ::: \I" hold. Here the valuc of the parameter a E (0. I) is chosen so that 110 ::c U", 110 ::: V" in n. 0
n.
Example. Let N > 4. P == I + /-1-. q ::;:: (I + I')(N + 4)/(N - 4). that is. condition (41) is satisfied. For these values of parameters a :::: I, f3 == (N + 4)/(N - 4) > I. and therefore the boundary value problem (39), (40) does not necessarily have a solution. However. in this case equation (39).
has a strictly positive solution
.j ,IN·.j)/H. b' . whcre CN(a) = !a N(N - 4)(N- - 4)J ' a > D IS an ar ilrary constant. It is clear that the family of positive stationary solutions of the original system
(34), (35). U,,(x)
.
V,,(.x)
_
=
A
.
\-uU(.x)]
= \C,y(cl)ll/IIII')/(a 2 + Ixl"),N.lJ/lw !I·II. l +.I tI 2 )1/II+fL l 12C (a)(N - 4)j /ll+fL)(Na"/7_ --------'--'1 I' N/I'II.j II (0- + x -) -I'
1/11'/1) _ N
-
(47)
constructed using this positive solution, can be used to construct a set '111', since U" > 0, V" > 0 in U. It will have almost the same form: 'lV = (110, vo) 1 Uo 2: O. vo 2: 0; 3a > 0: 1I0::C U". vo::c \I" in nl. Let us notc that the family (47) of positive in R N functions dennes in a similar way the stable set for the Cauchy problem for the system (I), (2). It is not hard to show that for all values of f3 that satisfy condition (41) (lY = 17/(/-1- + I). f3 = If/(l! + I )), the system (34). (35) has a family of solutions that are strictly positive in R"'. For example, from Proposition I (subsection 2.1) it follows that in this case there exists a radially symmetric solution whieh is everywhere positive (see subsection 4.1 of 3, eh. IV). Using such a family of stationary solutions in R N , the stable set is easily determined. However, we shall proceed in a different manner.
*
VII Somc othcr mcthods of study of unboundcd solutions
460
Indeed, to determine W, it is sufficient to establish local solvability of the system (34), (35), for example, for sufficiently small Ixi. The following simple assertion, which will be frequently used in the sequel, is true: Lemma 1. For arbitrary p, q, JJ-,
Il
111 WI II'C
==
= I.
V(O)
U'(O)
==
V'(O)
== o.
3 Conditions of global solvability of the boundary value problem for pq ~ (1
+ J!)(1 + v)
As before. the main tool is the analysis of the family of stationary solutions. I The case pq < (I
+ JJ-)( I + I)~
(48)
Theorem 5. Let pq < (I + JJ-)( I + Il), P :> I. q :> I. Then the bOllndary vaillc problem (I )-(4) has a global (bolillded IIl1ijfmllly ill t) soilltioll {f)r arbitrary initial fllllctiolls 110, Vo,
(49)
Pro(!{ We shall be needing only one property of the functions U", V" of the family (46) (U I • VI were defined in Lemma I): U" ->x. V" -> 00 as a -> 0+ in R N This means that the stable set (50) (or (51» covers the space of all initial
h[/\'e the estimates
Local solvability and the indicated properties of the functions U I, V I follow from applying the Banach contraction mapping theorem to the integral equation U,,·tl(r)
461
0 the system (34), (35) has in the ball
:>
WI == {lxl < J2N} a strictly positive, radially symmetric, mOllOtOlle decreasing sollllion U I (r), V I (r), r = Ixl. which satisjies the cOllditiolls
UfO)
I
§ :I A parabolic systcm of quasilincar equations with a source
=
I_r
dp
t' (N
Id(
.10 pN - I .Ill
[I _t. .10
l" tNIU'I(t)d(]l'iIJl;11
d7)
7)N
I .10
II
which is equivalent to the problem. By (46) the functions U I (r), V I (r) define a one-parameter family of stationary solutions of the system (I), (2). Let us set w" == Ixlax E (od. Then for any a :> 0 U", V" are defined and strictly positive in W". Now we can determine the stable set without any additional restrictions on the parameters of the problem. Theorem 4. Let pq
:>
(I + JJ-)( 1+ Il). Theil the prohlem (I )-(4) has a nOli-empty
stable set
n
n.
2 The case pq
n <;
0
:>
W,,; 110:::: U".vo :::: V" ill
n}.
= (I + JJ-){ I + t / )
Here the situation is completely different: existence of the global solution of the problem (I )-(4) depends on solvability of the system of stationary equations (34). (35) with boundary conditions (36), Below we prove the following simple (but not optimal with respect to the admissible domains n) result. Theorem 6. Let pq
''W == ((110, vo) I 110::: 0, vo::: 0; 3a
n
functions. In other words. in the case of any bounded domain for arbitrary Llo. Vo E C(D) we can always find a :> O. so that. first of all. C (U" == Ix I ax E WI} and. secondly. 110(.1:) :::: U,,(x). lIo(.r) :::: V,,(x) in n. Then, since U" :> O. VI/ :> 0 on an (that is. we always have 1I0(ol) < U,,(x). vo(ol) < V,,(ol) on an), using the Maximum Principle we conclude that II :::: U". v :::: V" in R j x and therefore 0 the solution is bounded from above uniformly in t.
the domaill
n satis!.'·
= (I + JJ-)( I + JI),
P
:>
I. q
:>
I, alld let the diameter D n
(~l
the cOllditioll
(50)
Dl! < J2N.
Using the estimates (49). we can also distinguish another stable sel. which is smaller than (50). but illustrative:
'W' = {(1I0.VO) I Llo::: O. vo::: 0;
3a
1I0(X) :::: 1I 21 1't 1ltili lll
:>
0 :
n
<; Ilxl < J2N/1I};
(
Theil the prohlem (I )-(4) has a glohal solutioll Fir lilly illitiaijilllctiolls uo, Vo· Proof For P(I == (I + JJ-){ I -+- l') the functions (46) are not defined, In this case there is a family of stationary solutions
")lilHIi
1-·
(52)
ai~'-
(51 )
where II :> 0 is a parameter and {} I. V I are defined in Lemma I. Henee we immediately have that III
VII Some other methods of study of unbounded solutions
462
§ 3A
Condition (52) means that the domain n can he placed in a ball of radius J2N. Without loss of generality, we shall assume that C WI. But then by (53') for any /10, Vo E ('(n) there exists a sufficiently large 11 > 0, such that /I ::: Va' V ::: Va in n for I = 0, and, since C lVI (that is, V"' V" > 0 on an) these inequalities hold for all I > O. 0
n
n
Remark. It appears that a necessary and sufficient condition of solvahility of the problem (37), (38) for af3 = I is the inequality /lJ > I. As shown by the proof of Theorem 6, existence of a positive solution of the prohlem (37), (38) implies glohal solvability of the boundary value problem (I )-(4) for fJll = (I + J-L)( I + 1') for practically all 1I0, Vo. Let us note that the same condition A I > I was obtained non-rigorously in subsection I.
paraholic system of quasi linear equations with a source
(2) self-similar solutions are possible for /)( p + I) = J-LUI + I). Furthermore, the questions of existence, and even more so, of stability of unbounded self-similar solutions f(Jr systems of equations remain largely open. I The I/Iaill resull (colldiliolls .fllr Ihe appe{lrallce or Ihe
lI(O, x)
= 1I0Cr)
1I0, Vo E (,( I"
,N
:::
0, lJ(O, x)
), /1 0I'll E
= vo(.r)
N HIR o( ),
::: 0, x ERN;
HS
blOW-lip regime)
By analyzing a family of stationary solutions we can ohtain a sufficient condition for absence of localization in the Cauchy problem. It appears also to be necessary. Theorem 7. LeI PlI < (I
+ J-L)( 1+ ").
P > I. q > I. Theil all 1l/ll}(Ill/lded mllilions
or Ihe Cauchy fJrohJt'1II (I). (2). (54) (Ire nol /ocali;:ed.
Remark. By comparing with spatially homogeneous solutions of the system (I), (2), which satisfy the equations (55)
lI'=V". iJ'=lI'I, 1>0.
4 On localization of unbounded solutions of the Cauchy problem It turns out that the spatial structure of the family of stationary solutions contains information ahout quite a subtle property of unbounded solutions of the Cauchy prohlem for the system (I), (2), namely, localization. An unbounded solution of the Cauchy problem for ( I ), (2) with initial functions with compact support, (54)
1 N V11+ E fflo( R ), o
463
we sec that a necessary condition for existence of unbounded solutions of the if Cauchy problem is the inequality fJq > I. Without loss of generality, we shall assume that lim max{u{t, 0), 1'(1,0)) = 00, I - , To < 00. Let us consider the family of stationary solutions (46), where VI, VI are detined in Lemma I. By the condition pq < (I +J-L)(I +/') the functions V"' Vee -> 00 in R N as a -> oj. Therefore we can find {I(I E (0, I), such that for all (/ E (0, {III I we have lhat supp (UII + VII) C lV ee • U(l ::: lJ", I'll ::: Vee Proof
111 lV".
will be called loc{I!i;:ul if for all I E 10, To), where To < 00 is the time for which the solution exists, the functions /1(1, x), v(l, xl arc non-zero inside some ball 11.1'1 < L < 001 (L does not depend on I), and are identically equal to zero for 1.1'1 ::: L. If. on the other hand, perturbations penetrate arbitrarily far from the point x = 0 as I ~, To (that is, as the solution grows to infinity), then we say that Ihere is /w lo('{/Ii;:lllioll in the Cauchy prohlem. Here we shall not he establishing the conditions for a solution of the Cauchy problem to be unbounded, since our main aim is di fferenl. That eould be done in a relatively simple way, for example, hy constructing unhounded subsolutions as in 3, Ch. IV. Let us note that all the results of subsection I (Theorems I. 2) extend also to the case of the Cauchy prohlem, since every unbounded solution of the boundary value problem in an arbitrary domain n E R N is a subsolution of the Cauchy problem. The localization phenomenon in the Cauchy problem for systems of equations is conveniently studied by constructing self-similar solutions (see 4). However, such solutions exist only for some values of parameters. Thus, for the system ( I),
*
*
Let us fix an arhitrary (/ E (0. allj. Then it follows from the Maximum Principle that the solution u(l, .1'),1'(1. x) cannot be larger than the function U ee , V" in (Vel as long as 1I ::: V"' I' ::: V, on iJw eI (and thus as long as supp (li + v) C lV ee ). Therefore by unboundedness of the solution u, v, for any a E (0, (I(d there exists tee < T(I, such that supp 11I(t ee' x) + 1'(1". x) I rt (V". Hence, by passing to the Ii mit a .....,. 0+, we obtain the required result. 0 In a similar fashion we can establish the following assertion: Theorem 8. LeI I'li < (I /10.
+ J-L)( I + 1'),
P >
I, 1I > I. lIlld leI the illilial jil/lCliolls ill I' Ixl. Th(n" ir the sol/ltioll
v[) he /'lIdia!l, ' s."l/1l/Ielric and Iloll-illcreasillg
or Ihe C{luchy prohlelll ( I), (2), (54) is 1Il1holinded, jill' {III." jixed .r E R N
\Tnl
J·.I"
max (lI{t, x), v(t, .\) I
i.e" 01 le{lsl II/Ie or Ihe jil/lctiollS slll/ce.
/I
CXJ,
or v hecol/les ill/illile liS I -> Til' Oil Ihe ,v/lOle
*3 A parabolic systcm of quasilincar cquations with a sourcc
VII Somc othcr mcthods of study of unboundcd solutions
464
°
Therefore for 1/1 = pq ..- (I + fL)( I + Il) < unbounded solutions express features of combustion in the HS blow-up regime. We are able to provide specific bounds for the amplitude and the size of the support of the generalized solution of the problem for m < 0. Briefly, their derivation is as follows. Let us write down a simple homothermic solution, which satislles equations (55). For pq > I it blows up in finite time: 11 (1) 111
= CI(T o -
1)11"111/11"/
= C 2 (T o -
II. v",(I)
0<1 < 7 0 <
I)
1'1+
11 / 11"1 II
(56)
00,
where
CI
=I(q+I)I'(p+I)/(pq-I)I'-+III/II'"
II
C2
= I(q + I )(p + 1)'1 /(PlJ -
II
I ),/111 1/(I"{
Let us use the fact that the system of equations (I), (2), which obeys the Maximum Principle, has to have an "intersection" property at least for one component of unbounded solutions having the same blOW-Up time (see ~ 3. 4, 5. Ch. IV). On that basis. let us compare the solution of the Cauchy problem and the strictly positive unbounded solution (56). Lemma 2. Let To < 00 he the hlow-Ilp time Fir all IIllholinded sollitiOIl or the problem (I). (2), (54). l17enFir OilY I E 10, To) either
max 1I(t,
x) ::.:
C I (To
(57)
_.1). IIHII/II"{ I)
465
2 S- and LS-n'Mimes In the case III ;? 0 the condition U", V" -+ 00 in R N as {/ -+ 0+ is not satisfied and therefore Theorems 7. 8 are invalid. We can expect unbounded solutions to be localized for III ::.: o. Here for m > 0, when solutions of the family (46) become infinite as t/ -+ 00 only at the one point x = () (which is characteristic of the LS-regime), the functions U", V" allow us to bound from below the singularity of the unbounded solution as I -+ To (see ~ I). On the other hand, if III = 0, then the family of stationary solutions (53) becomes infinite as {/ -+ 00 at least for all x E lUI (S-regime). In this case we have, for example, the following statement. which estimates the fundamental localization domain of the S-regime. It is proved as Theorem 7. Theorem 10. Let pli = (I + fL){ I + 1'), P > I, 11 > I. LeI Ihe illitial fimctiOlls lin, lin be radial/v svmmetric, lIoll-increasillg ill r = lxi, alld assllme jilrthermore that Sllpp (un + vol C WI. IJ the solutioll or the Callchy problem (I), (2). (54) is ll/ZliOlIIllled, Ihell lim maxlu(l. x), v(1, xli = 00 t-.. T(I
everywhere iii
WI·
In other words for PlJ :::: (I + fL){ I + 1') an unbounded solution of the Cauchy problem cannot be localized in a ball with a radius less than .../TN.
\
or max v(l. x) ::.: C 2 (T o --
1) l'Itll/II"/
3 Comparisoll wilh sel(similar reslIlls
(58)
I)
It is of interest to compare the results with qualitative derivations. obtained by
\
Comparing now II. v at each moment of time with the family (46) of stationary solutions (U I, V I are taken from Lemma I). as was done in the proof of Theorem 7. we arrive at an estimate of the support of the unbounded solution.
using unbounded self-similar solutions of the Cauchy problem for (I J, (2). Let
Theorem 9. Let lUI < (I + fL)( I + 1') olld let To he Ihe hlow-Ilp time or WI unbounded solution o(the Couchy prohlem (I), (2). (54), where the inilialfilllction.1 have compacl supporl ami ore mdit/lly symmetric, Uo :::: uo(r), Vo = vo(r), r = Ixl· Then jill' eoch t slllficielllly c/ose 10 To eilher
the latter condition heing equivalent to the inequality 11 > 1'/ fL if (61) is satisfled. Then, as can be easily seen, equations (I), (2) admit unbounded self-similar solutions of the form
111
meas sUPPru (t, I' ) >
1/( P
I)
= fLUI + I),
IIH(~),
(62)
II{III/(I,,{II/((;).
(63)
(51)) v.\(1. x)
or Obviously, for III :::: pq - (I + fL)( I + 1') < 0, PlJ > I, estimates (51)). (60) guarantee, in accordance with Theorem 7, the appearance of the HS blOW-Up regime as I -+ Til: meas sUPPr(1I + li) becomes inflnite as I -, To·
~=
= (T n -- t)
x (Tn -- I)'"
iI::::
lJfL-l'(fL+ I) I 2(lJfL _ 1')- ==:
() < I < Tn <
(61J
PlJ > I:
IIs(t, x) :::: (Tn - I) (/,111/11"1
/tI([J+ll
'IIr:::: 2L N--C. ' I 211'-+/1+11 (1'11 .- t) 'I- I' 1 /11 II II"{ II
VII Somc othcr mcthods of study of unboundcd solutions
°
The functions (J ::: 0, I ::: satisfy the following elliptic system of equations obtained by substituting the expressions (62), (63) into (I), (2) and taking (61) into account:
b.~O"11 - I1V~(). I; -
4 Thc combustion localization phcnomcnon in multi-component mcdia
467
where II is the exponent in the expression (64) for the similarity coordinate ~, which determines the law of motion of the fronts of self-similar thermal waves formed in each component in the HS-regime.
_t:O + I" = 0, qfl - II (65)
§ 4 The combustion localization phenomenon in
multi-component media Let this system have a non-trivial solution in R N , satisfying the conditions 0(';) -+ D,n,;) -+ 0 as \1;\ -+ 00. Then it follows from the form of the similarity coordinate'; in (64) that localization of the solution lis. t's. or its absence. depends on the sign of the parameter I = qfl - V(fl + I). The critical value I = 0 (that is. q = /l(fl + 1)1 fl: S-regime, ,; = x amI (62), (63) is the solution in separated variables) divides the space of parameters of the problem into two regions. For I ::: 0 the self-similar solution is localized ( I > O. that is, q > ll(fl + 1)1 fl. LS-regime), while for I < 0 there is no localization, and 11.<;(1, x), v.\(I. x) -+ 00 in R N as I -+ I'll (HS-regime). Let us compare the self-similar critical value (66) with the critical value (InO.
= (I + fl)( I + II),
(67)
derived by the method of stationary states. By (66) we have from (61) that fl. = fl(£I, + I )/I' - I == fl(/l + I )/,'. and therefore fJ,q. = (I + fl)( I + 1'). which is exactly the same as (67). Therefore in the "self-similar region" of parameter values, the condition of Theorems 7. X concerning absence of localization for pq < (I + fl)( I + II) is not only sufficient, but also necessary. It appears that this conclusion is valid also without the self-similar condition (61). Let us observe that the rule for growth of the amplitude of the unbounded selfsimilar solution (62), (63) is the same as in the right-hand sides of the estimates (57), (5X) as far as the form of dependence on I is concerned. Those estimates were obtained by comparison with the homothermic solution (56). It is important to note (and this fact again underlines a certain optimality of the results that follow from the method of stationary states), that sharper estimates of the size of the support of the solution (59), (60) in the HS-regime coincide with those of the self-similar solution. It can be easily checked that if condition (61) is satisfied. then we have the equalities lIl(p+l) 2(p
+ fl +
I )(pq - I)
This section can be consiuered as the continuation of the previous one. It deals with a qualitative and numerical analysis of combustion with finite time blOW-Up in multi-component media. Most properties of unbounded solutions of quasi linear parabolic systems of equations presented here have not as yet been rigorously justified. Below we consider two different systems of equations. We shall concentrate on properties of solutions, for which there are no analogues in the theory of finite time blow-up developed earlier for a single quasilinear parabolic equation. We shall also discuss questions related to the eflicacy of similarity methous. It turns out that unboundeu sclf-similar solutions that can be constructed for systems with power law nonlinearities, are not always "responsiblc" for the asymptotic stage of the blow-up process. It can happen that the asymptotic stage of the combustion process is described by self-similar solutions of completely different equations, that is. a.s.s. appear here.
t A system of equations with a source
Here we consider a parabolic system of quasi linear equations. which is a generalization of the system studied in 3. It is conveniently written down in the following form:
*
1/,
= "1(1/"'1/,), +-qII//J,v)':,
l',
= ":(v"
v,),
+ £/:1'/;:1/)",
(I) I>
0. .1' E R.
Here (T, > 0, 13, ::: I. y, ?: I (i =: 1.2) are fixed dimensionless parameters. The number of dimensional positive constants ",. q, can be reduced by rescaling 1-+ '0'. x -+ .\0.1'.1/ ---. Uoll. v--> Vov. If 8 = (T21f3l -- (YI +- 1)\ - (Tdf32 (Y2 + 1)1 j: O. then this method can be used to get rid of all dimensional constants of the system. by setting
lIl(q+ I)
c-----:------ =
2(q
+ II +
I )(pq - I)
11.
10
= £II
11}1 0
13,\/)':
- o ' ·\0 -
,,1i2 IJ ",12 I
0
'0.
~,\,
\
'
~ 4 The combustion localization phenomenon in multi-component media
VII Some other methods or study of unbounded solutions
468
Q2)"2/ h (k2)IY'+I-lhl/,j
Un::::: ( - ' ql kl
_
Vo -
-
kl
ql
Therefore instead of (I) we shall be considering the equivalent system
v,
(11'" 11., ).,
+ //3
= k(v"'v.,), + lP'U Y '.
1
(2)
vY"
I>
O. x
E
(3)
R.
where k ::::: I if 8 =I D, and k > 0 is arbitrary if 8 ::::: O. For (2), (3) we formulate the Cauchy problem 11(0, x) ::::: uo(x) :::
O.
v(O. x)
= vo(x)
:::
o.
x E
R.
PI + (X1')I2/(X2
(PI -
If, on the other hand, 8 ::::: D then any three constants can be taken to be equal to one; only one dimensionless parameter remains in the system, for example,
I/ t :::::
> I. This inequality is equivalent to the condition -')11')12 + I )(P2 - I) < D. which is satisfied in this case (since Pi < I + ')Ii). Similarly, it can be checked that if (XI > 0, (X2 > 0, the second component v also blows up in finite time. From the identity (7) it follows that the blOW-Up times of 11(1) and v(l) are the same. An interesting situation arises if D'I and a2 have different signs. for example, if a I > D. a2 < D. Here. as can be seen from (7), Co > D, and since P2 > I + 1'2 > J. v blows up in finite time: v(1) ---> 00 as I ---> Til < 00. The function 11(1) in this case remains bounded: 11(1) ---> «(XICO)I/I>I. I ---> Til' The nature of homothermic combustion is still more varied if (XI < D, (X2 < D. The constant Co in (7) can be of either sign. For Co ::::: 0 equations (8), (9) lead to finite time blOW-Up in both the components. If Co < O. then 1/(1) blows up, while v(l) remains bounded: if Cn > 0 it is the other way around. Therefore if (XI < 0 (or (X2 < 0). evolution of the components II, v can differ: one can blow up in finite time, while the other remains bounded.
have
(Q2)"'/<5 (k2)IYlfl'-fJll/<5
(4)
2
Ill, :::::
/ A/lalysis ot'spalially homogc/lcolIS SOlllliolls First of all we have to work out under what conditions unbounded solutions are possible. This can be done by considering spatially homogeneous solutions of the problem, which do not depend on x and satisfy the equations
==
II li , (I)v Y2 (1), v'(1)
11(0)
Let
(x,
= ')Ii -I-
I --
P,
==
=I 0
==
v 1i '(I)lI Y1 (I). I>
110 > 0, v(O) ;;:: vo >
(i
,\;e/f~silllilar
so/wiollS
Let us introduce the notation
where 110. Vn arc bounded functions with compact support.
1/'(1)
469
O.
o.
a,/ p.
== 1,2). Then the system has the first integral
== (PI _. I )(P2
--
I) --
')11')12-
If the conditions ( 10)
(5) (6)
p
1111 <
O.
1112 <
(II)
0,
hold, the system (2). (3) admits unbounded self-similar solutions of the following form: 11.1'(1. x) == (To - I)IIIIO(~), vs(l. x) == (To - I)III'f(~).
( 12)
~
where the functions equations
using which it reduccs to thc lwo (uncouplcd) autonomous equations
= .1'/('1'0 -
(J :::
D,
I :::
(0'" 0')' - 110' ~ k
1>
O.
(9)
From this we derive conditions for occurrence of linite time blow-up in each componcnt. If, for example, (XI > O. (X2 > O. then for II to blow up. it suffices to
I)",
/l
==
(IIII(TI
+ 1112(T2 + 2)/4.
0 satisfy the systcm of ordinary differcntial
+ II/I () + Oli, p' == 0,
(f"'r)' - nr~ + 11121' +
f1i,{JY'
== O. ~
(13) E
R.
and the usual conditions: (i(~), I(~) ---> D as I~I -:> 00. As can bc seen from (12), many properties of self-similar solutions. which are expressed as I ---> Til < 00. depend on the sign of the parameter 11If /l < 0 then both components evolve in the HS blow-up regime. which is not localized, liS. Vs - , 00 in R. I ---> Til'
~ 4 The combustion localization phenomenon in mulli-component media
VII Some other methods of study of unbounded solutions
471
IJ,IJ
/,2
10
0,8
ero U(IJ,:E) /
/ IJI/1:)
/
?
Fig. 84. Similarity functions of the S-regime fh =' 1.6, YI ~., 1.05, Y, n.R, k = I
(II
1.5,
IF~
1.9,
On the other hand. in the case 1/ c::. 0 unbounded solutions arc localized: if 1/.\, Us grow without bound as I -> 1'(') only at the point x = O. For /I = 0 the S hlow-up regime develops: the functions 1/.\, Vs arc a solution in separated variables. and therefore grow without bound as 1 -> TIl on respective fundamental lengths L o = meas supp fJ. L I = meas supp f. Localization domains of each of the components are. in general. different. As an example. we show in Figure 84 the spatial profiles of the functions O(~), /(0 in the case of the S-regime. Here L o < L/. Numerical studies show that for 11'1 > 0, a~ > 0 the self-similar solution of the S-regime is unique and stable (in the norm of the special similarity transformation. see ~ 2, 5. eh. IV), For sufficiently general initial conditions. the system evolves to a stable dissipative structure on a bounded domain, with each component effectively localized on its respective fundamental length L o or L 1 · An example of such evolution to a self-similar S blOW-LIp regime can be seen in Figure 85. The initial perturbations /lo(.r). vo(.r) are not symmetric. Therefore initially two thermal waves in /I, l' (t = II, 1 = 1,) appear. These collide at time 1 = I, and generate a thermal structure (t = 14)' which evolves in a self-similar manner in the S-regime. For (fl < 0 (or < 0), when the nature of homothermic combustion of the two components can be significantly different. self-similar solutions do not appear. As a rule, in numerical computations one of the components blows up in finite time as 1 _.)- 1'11' while the other remains bounded as 1 -> To' Therefore selfsimilar solutions with coordinated combustion of the components comprise a sort of /I
o
UU".E)
\;..;!,..:::~:.. U(l;,.1:) 50
20
Fig. 85. Formation of the self·similar S-rcgime structure after interaction of concenlration waves in tbe medium: f31 =' f3, =' I, YI =' Y~ =' I, IFI =' IF~ =' 2, k =' 1: 'I ,= n.27~, ':2 =' 0,672, h, =' 0.7X12, '.j O.7XD
> 0 O_S-regimel. then
(f,
unstable boundary, which separates sets of solutions with different (uncoordinated) patterns of evolution of /I and v. 3 The gel/eral
1'(/.1'1'
A natural question arises: what happens in the general case when condition (10) for the existence of self-similar solutions is not satisfied" First, numerical computations show that for al > O. a~ > 0 (that is, when fJl < 1+ 1'1, fJ:2 < 1+1'2) both components always blow up in finite time. Secondly, if the similarity condition (10) is not satisfied, the evolution of the two components proceeds, in generaL in an uncoordinated fashion, and can be markedly different. For example, it is possible for the lirst component to blow up in tinite time in the LS-regime (unbounded growth on a set of measure zero; localization). while the second component evolves in the HS-regime, and its blOW-Up set covers the whole space as I -> TI~I' A numerical computation of such behaviour can be seen in Figure 86. Other situations are also possible. For example. II can evolve in an S-regime, while v evolves in an HS-regime. Alternatively, both components can evolve in the HS-regime, but with different speeds of propagation of thermal waves. To conclude, let us write down for the general case sufficient (and apparently necessary) conditions for the absence of localization of unbounded solutions of the problem. that is, for the occurrence of the HS-regime at least for one component.
§4
VII Some other methods of study of unbounded solutions
472
The combustion localization phcnomcnon in multi-component media v,:::: -POl!"U f
u,u
I> D, x E R.
It describes the process of combustion in a nonlinear medium. with volumetric energy production, which takes into account decrease in the density of the substance due to its depletion as a result of burning (v = v(l. x) ;:: D is the density of the combustible substance). Here f3 > I. /1 > O. f > 0 are dimensionless parameters: ko. C/O. po are positive dimensional constants. Diffusion of the second component is not taken into account: in fact this has no bearing on the tinal results. It is easy to get rid of the constants k o. C/O. po by rescaling the dependent and independent variables r. x. II. v. Therefore in the following we shall consider the dimensionless system
20
/2
urll,x)
III
o
,
r!O,08
20
vt
Fig. 86. Mixed combustion regime with finite time blow-up (LS in the first componerll. HS in the sccond): f31 "" /h "" 1.5, 1'1 n =, 1.5, ITI I. 1T2 = 3. k := I: II := 0.0745. 12 co 0.0771
= (U"II,), + v",/',
= -V"II'.
r > O,x
( 14)
Here a > 0 is an arbitrary constant. U I. V I is some stationary solution. for example, one similar to that constructed in Lemma I. 3:
*
= 1'11'2 --lf31 f,
- (ITr
= I + Yi + IT,
+ 1)J1f32 -
- ( IT 2
+ 1)1.
= vo(x)
;::
(16)
O. x E R.
Observe that for /1 :::: 0 (v is the order of the chemical reaction of combustion) the two equations of (15) are uncoupled and the temperalUre distribution 11(1, x) satisfies an equation considered in preceding chapters. We have to find out how depletion inl1uences the asymptotic stage of evolution of finite time blOW-Up regimes in a nonlinear medium. and under which conditions localization is possible. It is of intercst to note that in this case. in general. unbounded self-similar solutions do not describe correctly the asymptotic stage of the blOW-Up process. We shall discuss reasons for that later.
f3,. i:::: I. 2.
I Unbounded selrsimila(' soilitions
It has to he expected that it is precisely the signs of these parameters that determine much of the asymptotic behaviour of unbounded solutions. In particular, an elementary analysis (which uses the method of stationary states) of the family (14) leads for small a > 0 to the following result:
Proposition I. LeI f3, > I, Y, > I and film < O. f2/m < O.
(15)
E R.
Let the combustion process be initiated by giving an initial temperature prolile and some initial concentration of the combustible substance: U(O. x) = uo(x) ;:: O. v(O, x)
[t is not hard to check that equations (2). (3) admit the following one-parameter family of stationary solutions:
m
473
A great advantage of the system (15) is that essentially for all valucs of the
parameters it admits self-similar solutions: us(l,x)=(T o -1)
'l1,en even
vsu, x)
= (To -
r)113 (Hlil/rtf(~),
(l'
1/"0(0.
= fl + (I l
- jJ)(f3 -- I)
(17)
#- n,
(IS)
IInhollnded solUlion Illlhe problem (2). (3), (4) is nol IO('£lli::ed.
where
~
is thc similarity coordinate, Irj!t2rr)
2 A system of equations with depletion
Below we consider the Cauchy problem for the system
(19)
If diffusion of the sccond componcnt is taken into account, then anothcr dimensiOllal constant appears in the system, and self-similar solutions will exist under an additional restriction on the parameters.
VII Some other methods of study of unbounded solutions
Similarity representations of the temperature O({;) :': 0 and of the density f( {;l :': () arc determined from the system of ordinary differential equations
~O'{; - ~O +rrjlJ = O.
(OrTO')' -
2a
(E
.r w = 0,
+ Il( _
a
La
(27)
(20)
{; E R.
= 0, f(CXl)
-
= II"
IT(IT
rr
+ 2)( I
+ (I
- /I) . - IJ)(rr 2)
+
Therefore O(~), f(~) are functions with compact support: the size of the support determines the fundamental length of the S-regime: < x.
(22)
which have a simple physical interpretation. From (17) we obtain the conditions for existence of linite time blow-up as I -) To < 00:
= EIJ + (I
b"
(21 l
and the boundary conditions O(CXl)
475
and is easily integrated. The function O(f) is found from the quadrature
I¥
_a: (T.r ~ + /3 -
£.l'
4 The combustion localization phenomenon in multi-component media
In (18) the concentration Vs cannot increase with time. Therefore we need another restriction on the parameters, (24)
Naturally, for Il = 0, E = 0 this equality detlnes the fundamental length in a medium without depletion: L s = 27T(IT + 1) 112 I(T (sec § I, eh. IV).
As usual. we classify self-similar solutions according to how the domain (halfwidth) of intensive combustion depends on time. It follows from (19) that its size depends on time according to
2 Asymplotic behaviour (!l bloll'-up regimes
(25) Therefore three cases are possible: a) if IX < (T, then x"r(t) ~ 00 as I ~ TIl' and by the blOW-Up time the combustion wave covers the whole space (HS blowup regime); b) if IX = IT, then x"r(t) == conSt > 0, and the intensive combustion domain is constant in time (S-regime); c) if £.l' > IT, then .1,-[(1) ~ 0, t ~ To, the intensive combustion domain shrinks, and unbounded growth of the temperature is observed only at the one point x = 0 (LS-regime). We shall not consider in any detail the analysis of the system of ordinary differential equations (20), (21); there is no need. Let us only note that in the S-regime (a = IT) it simplifies drastically; for I) < I (21) becomes the equality
fW
= [ IT( 1- 1/)] 1/11, (T -
"j
[0({;1I(/11
/'1
(26)
E
(T
where
Maxima of the initial distributions correspond to the self-similar solution (26), (27), which blows up at Ti~ = I. However, the spatial profiles of lIo(.t) and vo(x) arc not the self-similar ones. As a result, since vo(x) is too large (as compared with the self-similar one), only a part of the energy of the initial distribution vo(.r) is needed to cause tlnite time blow-up in temperature II. This can be clearly seen in Figure 87,/J. Therefore the process of substance depletion stabilizes at the asymptotic stage of the finite time blOW-Up process in II, and as t ~ (the real value of the blow-up time is To = 0.525 < I), the density v(t, x) does not change much in the intensive combustion domain. Therefore as t ~ T(), the equation for the concentration falls away. and asymptotics of temperature blOW-Up is described by the single equation
To
while the lirst equation of (20) takes the form
((}trrJ')' - ~o + a"OItT+ll'l/II-"1
The above self-similar solutions arc structurally unstable, that is, their (coordinated in both components) spatio-temporal structure is not observed in numerical computations. Figure 87 presents the results of numerical simulation of equations (\ 5) for values of parameters, which formally correspond to the S-regime (a = IT). The initial functions are non-zero on an interval of length 2L s , where (see (28»
= O.
(26')
(29)
*5 Finite differencc schcmcs
VII Some other methods of study of unbounded solutions
476
where Qo is a constant equal to (D(T(;·, x) )", the average value of the limiting density in the intensive combustion domain. But for these parameter values f3 > (1' + I, which ensures development of the usual LS-regime, which can be clearly seen in Figure 87, a. And in general, an S-regime perturbed in temperature and density from above, degenerates as f - ; T(l into a self-similar (with respect to (29)) LS-regime, since from the inequality (T > f, which always holds for the S-regime (a = (T), it follows that f3 > (T + I in (29). If, on the other hand, in the S-regime (a = (1') the initial profiles are below those of the self-similar solution, the result most often is complete depletion of the combustible substance in the whole space and energy is no longer produced by the medium. The consequence is that the temperature does not blow up in tinite time, but satisfies instead an equation without a source, II,
477
u(t,x) ()
.9·
(30)
= (lI"lI,)\.
Thus the self-similar solutions (17), (18) comprise an unstable boundary between two large classes of "degenerate" equations (29) and (30). In the LS-regime (with respect to (15), a > (T) initial functions lying above the self-similar ones lead to the development of the LS-regime, which corresponds to f3 > (T + I in (29). Figure 88 shows the result of numerical computation of the LS-regime with initial data of the S-regime, as in Figure 87. As t -; T[l = 0.291 6 > the density stabilizes, while the temperature grows in the LS-regime (f3 (1' + I 3 in (29)). For the HS-regime (a < (T), these initial data are lower than the self-similar ones. It can be seen from Figure 89 that the constant density profile vo(x) == I is also too low. Therefore initially there is fast depletion of combustible substance in areas where the initial temperature is non-zero, and then two thermal waves propagate into the surrounding space which has high density of combustible material. Due to the higher rate of energy production at high temperatures, these waves blow up in finite time (To 1.72).
L~_
,r
Iii
()
=
=
U(I"r)
b
0.8
=
§ 5 Finite difference schemes for quasilinear parabolic equations admitting finite time blOW-Up 0.1
An important place in this study of blOW-Up regimes is occupied by results of numerical computations on the non-stationary problems being considered. In this section we analyse properties of difference schemes for a quasi linear parabolic equation with power type nonlinearity in one space variable: (I )
Here, as usual,
(T
> 0 and f3 > I arc constants.
•
()
Fi~.
•
"
m
1.-...L__----.
87. Simulation of equations (15) in thc S-rcgimc with a: the lirst componcnt. 11: thc sccond componcnt
VII Some other methods of study of unhounded solutions
47H
u(t,X)
479
5 Finite difference schemes
aft,x')
a
8
9
i1
'I 8
6
7
5
t-lo ·""0,/9(
6
"
5
J 2
3 2
(6
0
u(l,x)
:J'
u(t"z')
b
1,0
b
',0
:c
16
0
0,8
0.8 0,6
0,6 0,4
11,4
0,2
0,2
o
o
li
Fig. 811. Simulation of equations (15) in the LS-regime with (a > IT): a: the first component. b: the sccond componelll
16
.r
IT
= 1.
f3 == 6.
E
== I. /' == D.5
Fig. 89. Simulation of equations ( 15) in the HS-regime with (a < IT): a: lhe first eomponcnt. b: the second component
IT
= 1. f3 = 3. E ==
I.
I'
== D,S
480
We consider for (I) the boundary valuc problem in the domain (0, ') l. , = const > 0, with the conditions 11(0, x)
= lIo(X)
:":
O. 0 < x < I;
11;;+ I E
H/j(O, I).
{I >
0,
(2)
The results we obtain here give an indication of difllculties that arise in numerical computation of blOW-Up regimes. The main emphasis is on the study of implicit (nonlinear) difference schemes. which. in view of their many advantages [3461 as compared with explicit difference schemes. were used in all the numerical computations. Below we consider such classical questions of the theory of difference schemes as conditions of solvability of the discretized problem on a time level. conditions for global solvability of the discretized problem. and conditions of convergence of the finite difference solution to the solution of the original differential problem. We treat in detail the question of conditions under which there is no global solution, that is. to the appearance of finite difference blOW-Up regimes. Related to that are two comparatively unusual properties of the implicit scheme: if the solution grows at a fast enough rate, it can happen that at a certain time level either the solution is non-unique. or it does not exist at all. that is, the scheme is no longer solvable, All these properties (unboundedness. non-uniqueness. and non-existence of solutions) arc possible in the case f3 :": (T + I. when the difference operator corresponding to the right-hand side of (I) is not coercive. For I < f3 < (T + I and sufficiently small steps in time. a global solution always exists; moreover, it is unique. Many of the obtained results also hold for difference schemes for equations of the type (I) with sufficiently general nonlinearities. At the end of the section we brietly consider explicit (linear) schemes: the weak Maximum Principle is analyzed. and conditions for unboundedness of the difference solution arc established. Let us introduce a uniform grid in space with step size h = I/(M + I). where M > 0 is an integer: a system of time intervals {T)}. TJ+I ::: T) and the corresponding grid in time. W r . Everywhere. apart from subsection 1.4. we shall take the grid W T to be finite and uniform: Tj == T = T/(N + 1),0::: j ::: N, N > 0 is an integer. T > 0 is a constant (in subsection 1.4 7 1 -> 0 as .i -.. 00, and W T is a non-uniform grid). Let us denote by H" the set of grid functions
w".
v"
= I v, I Vo = VM I I = 0, Vi
Implicit (nonlinear) difference scheme
X E
Following [3461. we write down the implicit difference scheme corresponding to the problem (1)-(3):
(3)
11(1.0)=11(1,1)=0,1:,,:0.
:":
0, i
=
I, 2, ... , M}.
481
§ 5 Finite difference schemcs
VII Somc other mcthods of study of unhoundcd solutions
, /T (11--11) j
-{> ( .) = ('''+1) II ,,+11,1,.\
(4)
E W,XW",
(5)
lit
I. II = IIi is the desired grid where we have introduced the usual notation u = 2 function defined on the discrete set WT.Ii = W r X (VI" v" = (Vk I - 2 Vk + Vk+ 1) / h denotes the second difference operator. and 110" is the projection of lIo(X) onlO w". In the fonnulation of the problem (4), (5) and all the subsequent results. we assume that the difference solution 1/ is non-negative (otherwise the operation of taking an arbitrary power is not defined). This is one of the properties of implicit schemes for parabolic equations. It is easily verified that the scheme
([/-II)/T j
= (!i1I"u)" + (max{O, ii})/i.
(6)
(I, x) E WT.'"
which is identical to (4) for iI :": O. cannot admit for any T" h negative values of iI if II :": 0 in (furthermore, ii > () in WI, if II ¢ 0). This follows from an analysis of (6) at a point of negative minimum in x of the function it (sec 7. eh. V). An analogous weak Maximum Principle holds for the differential problem (see 2). Let us introduce the necessary finite difference functional spaces. The space of grid functions Vii = {v,1 i = 0, I, _.. , M + I. Vo = VM II = O} is equipped with the scalar product and the norm
w"
*
*
(7)
Norms in the grid analogues of the spaces 1:1(0, /). q > l. and H /)(0, /) have. respectively. the form M
Iv"I/I,'1
= ( 11 L
)
1/11
IV,!'I
,--I
Ilv"lih2
= (11 ~
2 \ V'II,: V11 ) 1/2
We denote by II -11;'.2 the norm dual to II· 1Ih.2 with respect to the scalar product (7):
VII Some other methods of study of unbounded solutions
482
Lemma l. Any jilllction v"
We have the equality
In the grid analogue of the space C(O, /), the norm takes the form
H" sati,\:fies the estimates
E
2(
(8) i'J +
IV" 1".i'J"
A II ()
<
-
v",'r+ 1111f!+'" ".2
III
A" I
2
II/I,r+ II '
A _
" AI 1j1,,(X)
=
4. 2 7Th = 1'SIll -, h21
tan(7Th/(2/) h
.
(9)
7TX
Slll-.O < X < I. I
Existence of a root of the equation P,,(u) Let initially I < {3 < IT + I. Then
1/1/,
in (10) is chosen so that
11/,,,1,,.1
= l.
( 13)
RAJ:
-4
= 0 means that the scheme (4) is solvable.
, '
+ IIft'T!' ! 11-".2 J
"
a I -1[d",'\"I. "./3+" 1 I'
(15)
Using the inequality (13) as well as the easily checked estimatc (16) we obtain from ( p " ('II),
(15) ,,,+!
II
)"
> -
IT
I I I 1"+2 +'2:; 11",,,+2 -
I' Ii'J+
"./3+
A 1'1 21 '7+ II
I II ".{:I+"·I·I'
Let us estimate the second term using Young's inequality. As a result we have '1111'<711 - A 11 '1 21 ,,+!l ". i'J+
(II)
in
.
(10)
respectively, the first (smallest) eigenvalue and the first eigenfunction of the difference problem The function
Illl/H,,+ 11/12«T+ I Ii
() -
( 14)
7
In the same way for grid functions vT .", defined on the nodes of the grid tv T .". we introduce the extension operator qT' defined by qTP"VT." = p"vj,+I, qTq"V T." = q" viII I for all j7 < t < (j + 1)7. j := n, I .... ,N (the grid tv, is assumed here to be uniform). Let us denote [3461 by
(12)
v""+111 ".2'
Let us consider the continuous operator p" : RAJ Let us introduce the extension operators PI" q" by assuming that p" v" is a continuous function, which is linear on each interval (ih, (i + I )h), such that p"v,,(ih) = v, (i = D, I, ... , M + I); q"v" is a pieceswise constant extension of the grid function v" E V", which is equal to Vi for all ih < x < (i + I )h. It is clear that p"v" E H!,(O, /), q"v" E V'(D, /), and that
483
5 Finite difference schemes
1II
We observe thatljJ,,(x) > 0
AJ:=
w".
We start by considering the question of solvability of the scheme on a time level, that is, thc qucstion of cxistence and properties of thc transition operator 13 46 1from onc time level to thc next. Below we dcnote hy All. AI, ... various constants independcnt of 7. h.
-
IT+ 2(1J
1-{3
+
I)
+ A 2,
[(3+IT+ 1]1/3+
and then the final estimate has the form '('
,,,,·1 ),,/'. AII'I"I,,!II 2 ""./11-<711--
f (·,,11).11
(\
/2+
I 1 11 "+-2) IT +'2:7 " 1.<7+2
.
(17)
'
I SI!/Jicient conditions for solvability of'the difference schellle at a ./ixed tillle !lTd
I. Wc shall show first that for {3 < IT + I and also in the case {3 := IJ + I, A'I' > I (this imposes an upper hound on the length of the interval 10 , II: see (9» the scheme (4) is solvable with respect to the grid function Ij for any magnitude of 7. For this purpose we shall need the following assertion, which is the finite difference analogue of Lemma I or 2 (see [296, 3461l.
*
From this. hy thc Brouwer fixed point theorem for continuous operators in a finite-dimensional space (see. for example 12961l. we conclude that the equation P,,([I) := 0 has at least one solution in the hall ( 18)
~
VII Some other methods of study of unbounded solutions
484
Outside this ball there are no solutions. since there. as follows from (17), (P,,(U), il'T+1 )" :> O. Let us move on now to analyse the case f3 = if + I. Then from (15), (12) we have that ( PI,(ti). ti"
Therefore for
A~' :> I
11.)1, :> (A'I' ..--
21 I )1'1 '" II II " .'1<'" _ , II
-
+I -II II 1"+-2 "."I" -
~LT (T .
the equation P,,(ii)
= 0 has at least one solution ti. such that
'121
I I )(IT
II
1
<
(A'I' _
"12
+
I or f3
= IT + I, A'i
:>
I. Theil filr allY
T :>
exists at leasI 0111' soilitioll U E H" of Ihe schcme (4). \l'hich belollgs to scls (18) or (19), \l'hile there are /10 solutions olilside these sets.
j
2 COllditiolls jill' lloll-lIlliqllelless I!f Ihe differellce soilitioll
{; /T
I!f the
or f3 = IT + I. A'i ::: I the parabolic operator of the scheme (4) is no longer coercive. so that the Brouwer theorem, which uses coercivity of the operator, is not applicable and Theorem I is invalid. Therefore we shall seek for sufficiently small T a solution ft close to II. Let us set/i-II :::: ~ and let us introduce the continuous operator F" : R M ..... R M : IT
G,,(~)
= (T(~' + T -T
I/IIJ II)IJ+TI(~'+T 1/lfJ 11)"tll,,_ •
1/(fJ-I)+II-~•.
Let us consider the expression
= ((~ +
T l/ifJ II)fJ - T'IJ/
+ Tn (f + T 1/t!J I) )"11
= {III:> O.
L" ~li,
T <: --------.
- (Iule
C II
:>
0
(20)
+ CII)fJ + 2h- 2 (lule + CO)"f 1 .
7111'11
the schel/ll' (4) has
(II, ~)" -
1£17,.2
= /1
1/2.
= {lo , 1-1/2 T 1/(fJ·11
+ /2 + /1 -
1z.1~. 2
and therefore. by setting (23)
we obtain h:::-1~1i.21(~+T
1/t!J I))',T.,III'I. 'T :> _TIIJ ~ -
l, ::: -1~lll.2lulh2
Therefore by the Schauder fixed point theorem 110 II we have Theorem 2. LCI cOllditioll (20) hold. such that. l/loreol'l'1'. 1ft - IIII' ::: CII'
+
+
Obviously I~[e ::: {loh 1]0
CIII into itself (here Co
k=1,2 .... ,MI·
(4).
on the sphere 1~11,.2
= 1z.11~lc :::
(22)
Existence of a root of the equation (h(~) = 0 implies solvability of the scheme
(Gitli),~li,
Existence of a Ilxed point of the operator F" implies solvability of the scheme (4). We have the obvious estimate
= (;13
This equation is the same as the original one if we ne~lect the term (ti"+I),, and set.1I == O. The second solution has the property that IUIe -> 00 as T ..... O. M M Let us set ~ = ti- T I /IIJ '. I) and define the continuous operator G" : R -'> R :
+I
Hence it follows that F" maps the set Xe" is an arbitrary constant), if
(21)
Below we shall show that as far as the dependence of T",f on [lIle is concemed, this estimate is optimal. It is important for large \1I1e. when the solution exhibits linite time hlow-up, In the class of uniformly bounded IIIIe we obtain the usual for parabolic equations estimate T",I = 0(h 2 ) for h « I (see [346\).
T:>
:>
l +7"12[11\".11. ) I. ~ (7'"- fJ lII [fJ (, (
\o! -
0 Ihere
0111'
As the estimates obtained in subsection 1.5 show, under the conditions of Theorem I, the difference scheme (4) has a unique solution for sufficiently small O. 2. For f3
_.
Let us show that for f3 ::: IT + I and sufficiently small T the implicit scheme (4) has in addition to the small solution constructed in Theorem 2 another. large solution. which is close to the root (; = T' l/ifJ II of the difference equation
Thus we have proved Theorem 1. Let f3 < (T
Remark. Setting Co = IlIle in (20), we obtain the following estimate of the practically maximal possible time step Tsol. for which the scheme is solvable at a fixed time level: 2
VII Some other methods of study of unhounded solutions
486
which holds for all (3 > I, 11]1 ::: C., where C. Then, choosing 7 so small that
= C.({3)
3 Condilio/ls IiI' !lOll-Cxisle/lce of Ihe differe/lcc solutio/l
> 0 is some constant.
(25) taking (24) into consideration, we obtain
487
A second manifestation of non-coercivity of the operator of the scheme (4) for {3 ? IT + I is the fact that it can actually be locally insolvable (on a given timelevel). To establish the conditions for non-existence of the solution, let us use the estimate ~ .,,+1) _ 2: - ,-I I·,,+1 , .\. E IV". (U \\ hin view of which we deduce from (4) the inequality
Thus we have the inequality
(28)
(G,,(f).£)11 2:
2: {3
~
I_uo {oo - {3
~
I [111111,2 +
~i71f!
(",211/1/jll( 1+ C.({3))"'1/1/2]},
which the function II must satisfy everywhere in (1)11, It is clear that it is sufficient to verify this inequality at the point at which max II is achieved, that is, to determine the conditiolls under which the inequality
Hence (G,,(f). flJ, ::: 0 for all
(29) has no solutions in R t · Let us consider first the case {3
= IT + I,
Then (29) assumes the form
It remains to check that for small 7 conditions (25), (26) are compatible, SubstilUting into (25) the value of Uo from (26), we have ~
1] o
4
~ /,,1(211 7 1/1/j·ll+ = 1f=1' II 11,2 f3 __
I
/ s12[1 I
+(' .. ({3)1",1/112 7 1f3 1,,+lil/lf311
and, as is easily seen, has no solution if
•
(27) whence 1]0 -, 0 as 7 --> 0 if f3 > IT + I, so that condition (25) docs not contradict (26) for small T. Thus we have proved
(30) Now let
f3
>
IT
+ l.
Theorem 3. LeI {3 > IT + I, The/l jill' sir/licie/llly .I'mul/ 7 Ihe dijf'ere/lcc schcme (4) hos ill uddiliOlI {(J Ihc solwio/l cO/lsll'IIClcd ill Thcorem 2. u/lolher sollllio/l, II' {3 = IT + I rlll'/l Ihis cO/lclusio/l is slil/ \'lIlid. pl'lll'ided 1/11/1 (sec (27))
f't f
Fortunately the second (large) solution, which has no physical interpretation, IS unstable, and a correct solution algorithm for the implicit scheme [3461 converges only to the required solution. In this context, we might observe that in any neighbourhood of the solution 0 = 7- 1/1 /;- I) the operator
is not contracting, and therefore the solution (; cannot be obtained hy the method of successive approximations. This testifies to its instability,
Using Young's inequality h2
I :::
4~fJ
+ f,
~ E Rt
,
= {3 '-J~~.12 [~~%t~T(J+ll/lfJ
I"t III
we sec that (29) has no solutions if everywhere in R·t •
l; <:
-
1111('
27 ._-€
h2
7 jJ + -E, , 2'
Henee we obtain conditions for insolvability of the scheme for {3 >
lule
>
2€ f3-1(2)1/1(l11 ,7 + ---{37 h{3
IT
+
I: (31)
~
VII Some other methods of study of unbounded solutions
488
Theorem 4. Let (3 ::::;
IT
+
I.
insolvaiJle, In the case
f3
>
IT
Then, it' condition (30) holds, the scheme (4) is + I solutiolls do 1I0t exist if (31) is sati.l:tied.
Inequalities (30). (31) provide us with quite sharp estimates of the size of the time step. for whieh no iterative process of solving the implicit scheme (4) will converge (for the simple reason that the difference solution does not exist). These estimates can be utilized in real numerical computations. Therefore let us consider (31) more carefully. Let us set all::::;
21f3_-
~~~l! [:i~~i~rJ+ll/liJI"+111
h
o
f!._~_~ (~Y/lf3
::.::
(34)
where 1jJ" is the first eigenfunction (10) of the problem (II). Taking thc scalar product of the system of equations (4) with 1/1". we obtain the sequence of equalities
(35)
By normalization of If;" > 0 we have the H(ilder inequality
lull' ::: ao7(h2ri3/11l -("1111
+ hOT
1/1f311
I ),,_> ('''II .1/"" I )il/ '1' ' ,,::::; (('rr+I)I3I IIT -1 II ,1/"
( U'f3 • .1.)
and is satisfied. for example, in the case
= j3aod o(h")
1/1f3--("1
III,
do
h _jf3IW--11
o _ = [ao(j3 -
= fo (h" )I f3
= 12i3(j3oodoll
E-
(32)
£ _
-- ~ 7 1
I)
At the same time condition (21) for the solvability of the scheme with lull' taken from (32). has the form 7 ",I
U
' II
.
('IT+I ,,)i3/Ifrflljl U
,Vi""
-
,"('IT,I "I U
I )Ifrfl ·i ,1/""
31 /I rT ,llj.
Thence. again applying the Hblder inequality (/t" \ I, r/I" lh ::: (ii. 1/1" )/;'
I.
we deduce
the inequality
(36)
I III f3 - I" , III .
+2"'"(j3a od o )"j
(33') I,
Let t· o be such that
and has the same dcpendence on thc size of the spatial grid as (33). Hence we conclude that condition (21) for the solvability of the schcme for j3 > IT + I is optimal for largc lull' (for cxamplc. whcn the difference solution is unbounded).
_ I _ ,,,/.__ IT+
/-4) -
,,\
Then from (36) we conclude that
t::
>
~o
\
i3
()
E in 111 r for sufficiently small
7 j.
j =
O. I ..... and therefore (38)
Let us move on now to detcrminc the conditions for global insolvability of the difference problcm (4). (5) for j3 ::: IT -+ 14 . Recall that the time grid here is not uniform: 7 j -7 0 as j -> 00 and 7 1 ::::; To <. 00. where To is thc time of existence of the solution. The proof of unboundedncss of the solution will utilize the method used earlier in 2. as well as in 6, eh. V.
2=J"0
*
41t will be shown in subsection 1.5 that. just as in the differential casc no unbounded solutions if f3 <. (J + I.
(37)
>.
4 Ullhounded ditTel'ell('e .\'(}lutiolls
*
llT
U
in view of which we obtain from (35)
(33)
fo
I. Let us set
deriving which we have taken into account the fact that
II
Then (31) has the form
lull'
489
5 Finite difference schemes
(~
2). there arc
Since
liile = max iik ::: I
in order to determine conditions for (5). we have to lind a system (7 II and that from (38) it would follow difference solution will blow up in
i.:,
t E l11 r ,
(39)
unboundcdness of the solution of problem (4). of time intervals. such that L 7} = To <. 00 that £1)1 -> 00 as j -> 00. Then by (39) the finite time, that is lu 1 lc' ----. CXJ as j ---. 00.
VII Somc othcr mcthods of study of unbounded solutions
490
Let Ti
whcrc A > 0,
(t'
= Ap . It), j = 0, I .... ,
(40)
> 0, P > I are constants. Then To
= Ap"/(plt
- I) <
00.
(41 )
Let us dctermine thc conditions for A, cr, p, for which EI)1 :::
Eop),
.i
= D.
I....
491
The first one of these is optimal, since for A~' > I the solution always exists (see Theorcm 1). Lct us prcsent an interesting corollary, which has a differcntial analogue (see
*3,Corollar)'. Ch. IV). LeI f3
E lIT + I, IT + 3) alld leI liS he gil'ell a .fimction lIo(.t) ::: 0, x E R" Lei 11.1' .fix 1111 arhilrarv 17 > O. Theil .fiJI' s!~tjicienlly large M Ihere is a collection 01' lillie sleps {T) l. Sllli.I:f)·illg To = L T/ < 00, sllch Ihlll Ihe solutioll of Ihe .fillile difference pmhlel11 (4), (5) .fiJI' I = (M + l)h llnd uo" t 0 is ullhollnded. Il'luOl,ih.1 > 2 .fiJr sOl11e M > I. Ihen Ihe .I'll/III' conclllsioll is I'alid also ill Ihe Cllse
f3 = IT + 3.
For that, it is cnough that
Proot: For large I f.lo in (37) satisfies the estimatc (42)
Substituting into (42) Af.lo E {;
I pil!) It-II:::
T)
p, j
from (40) and simplifying, we obtain the condition I = 0, I ..... This condition will hold if
+
(43) Therefore we have proved Theorem 5. LeI f3 ::: IT + I and leI Ihe illilial.fimclion uo" ill (5) he sllch Ihal (37) holds, LeI Ihetinile d!l1erence pmhlel11 (4), (5) he solvllhle Oil a sequellce 01' lil11e sleps (40), where Ihe cOllstallls A, a. p slllisl.\- (43). Then 1171' sollllion exisIS for lime (41), and
Therefore undcr thc above assumptions f.lo > D for sufficiently large I, which by Theorem 5 cnsures global insolvability of the problem (4), (5). D 2. To conclude, let us give an example of an unbounded solution of the finite diffcrcnce problcm (4), (5), which can be written down explicitly. This cxample shows, in particular, that thc requirement (40) of Theorcm 5 conccrning solvability on a sequcnce of time steps, is not especially burdensome, Let f3 = IT + I. As in the continuous case (see I, Ch. [V), we shall seek the differencc solution of the problem in separated variables:
*
(44)
Substituting
*
Remark. [n 2 it is shown that in the differential (continuous) case thc problem for f3 = fT+ I has an unbounded solution if A? = (1T / /)2 < I. If. on thc other hand, All> I, then it is globally solvable. From condition (37), which for f3 = IT + I takcs the form A'II < I, and the easily verified inequality A'I' < A? (see (9)), we thcn easily conclude that it is possible for the !initc difference problem to have an unbounded solution, while the differential problem is globally solvable. This will happen if the length of the interval I is such that A? > I but Alii < I. Inequality (38), which was derived in the course of thc proof of Thcorem 5, can be used to analyl.e thc problcm of insolvability of thc schcmc on a givcn time-Ievcl. For cxample, from (38) it is casy to dcrivc thc following cstimatcs for insolvability of the schemc at the .i-th time stcp in the case f3 = IT + I. Thcy arc sharper than thc cstimates of (30):
II!
into
(4)
we obtain the following problems for the grid functions
SJ,lh
.S' T
S I
;:-2
~,S'''II
(45)
IT
(46)
Let there be given a systcm of time intervals (40), where p > I, a ;;:-; a solution of the problem (45) is the function SI
= pi, .i = 0,
I .... , A
= ITp·(rrfll(p -
I).
IT.
Then (47)
Let us construct a solution of the problem (46) in the particular case IT = 2. Let us tix an arbitrary M > 0 and sct h = 2sin(37T/[2(M + I)]l. [n this case the length of the interval is 37Th ( arcsin 'h) Iii = 2 2
0< 17 S 2.
(48)
(T + I. n,en for sujjiciel/tlv .I'll/all T the difference problem has a unique global solwion. If {3 = IT + I ,then the .I'll/Ill' a.n·ertion is true IInder the additional requirement
Theorem 6. Let {3 <
Then it can be easily seen that the solution of the problem (46) has the form
fh
= { 2 [ 3 ( 1-
4. 0 h 2 S\l1-
a"h)]~'1 }1/2 s\l1(a"kh), . T
k
= O.
I. .... M
+ I,
(49)
"
AI
a"
where = 7TII". By (44), the functions (47), (49) dehne an unbounded difference solution of problem (4). (5) for (T = 2. {3 = 3, I = I", which grows without bound at all points of 10", conserving its spatial structure. As h ~ 0 the difference solution (49) converges to a solution of the corresponding ordinary differential equation (see § I, eh. IV): 0(.1')
= (3/4)1 / 2 sin(xI3).
0 < x < 10
= 37T.
(50)
The length of the support of this solution. In, defines the fundamental length of the nonlinear medium. The difference fundamental length (48) is close to 10 = 37T for h « I. For large' h the difference can be significant: for example, I" = 9 for h = I, '" = 6 for h = 2. Let us note that the grid function (49) is not the projection onto of the solution (50), the differential analogue of problem (46) for IT = 2, thougb it has a similar structure. The dissimilarity is even more substantial in the case (T = I. when equation (46) also has a simple solution:
w"
(h
= A" sin 2 (h"kh) + Ih. k = O.
I, .... M
. h = --I arcsln·-
2h
2'
1
B"
= 2K/Y
\ _ ~ I 1'(' I 11/2 . 0 < h ::: 2; 1,,·~ ~K" - )
K"
In
- [2(2K" -
1)1 '-I.
K"
=I -
2 h2
[
I
(
4
Recall that below the grid
Wr
is assumed to be uniform.
IT
21
(51 )
> 1.
In both cast's {3 < rr+ I and {3 = (T+ I. AI> I, the difference sollllion cO/ll'erges as h ~ 0 to the generalized solution of the dill'crential problell/ ( I )-(3) cOlIstl'lleted in Theorell/ 2 of .Ii 2. We shall need two lemmas, the first of which is verihed directly. The second is proved in [296. 3461·
Lemma 2. For all
where C 1
=C
I ((T.
r
TJ
E
R , the FIIIO\\'ing ineql/ality holds:
(3) > 0 is a constant.
Lemma 3. For an." gridtimetion v"
E
1iJ,
"'IIII/I,,tli A,. ".2 ' , Iv" I(' :: I \11 v,,'
(
')112]
=/1/1211r+ 1t l.
(53)
L_
<;
)L"tl > _1_(L"iC_TJ"t-C).
TJs
-rr+2s
L
s·
TJER j,
we obtain
which, however. does not satisfy the boundary conditions and is strictly positive. However, in the limit h ~ 0, when K" ~. 3/4, B" ~ 0, the function Ok is the solution of the di fferential problem for I ::: 47T. Let us move on now to questions of global solvability of the difference problem and of convergence of the difference solution as h ~ 0 to the generalized solution of the differential problem (I )-(3). Two separate cases have to be considered. .5 G/ohal so/l'ahility and passage to the limit Fir {3 :::
h
T.
Ir 1-
4 . , 7Th = -S\l1- 2
Proof' of Theorell/ n. Let LIS fix an arbitrary T > O. I) Lel us first consider the case {3 < IT + I. By Theorem I the difference schcme (4) with (3 < IT + I is solvable for any T, that is, thc function ft is defined everywhere in IOr.h' We shall need estimates of the finite difference solution. Taking the scalar product of both sides of (4) by ft" t 1 and using the obvious inequality
+ I.
;where h"
493
§ 5 Finite difference schemes
VII Some other methods of study of unbounded solutions
492
+
I
I '1"'2
(T+2;([II"."'2 -
11"12 11-"t'IIC ~ IIIf.lrt2) + II h,2:::::
I-I#t"tl
II ".Ii-t-"I-I·
(54)
We estimate the right-hand side of this inequality using (13) and Young's inequality, laking into account the fact that {3 < IT + I. As a result we obtain the inequality -I/Jlonl < 3:'11-"tI 2 A (55) I1 h.'2+ 1Uh.#t"ll u 4·
VII Somc othcr mcthods or study of unboundcd solutions
494
For f3 < (T + \ that can always be done. Then, summing the inequalities (59) in j from 0 to N. and applying Young's inequality. we obtain
and then from (54) wc have I
;T
1 -
495
1 _"'1
"1-1
+ 2 ~ 1II 1lur,1 +"2 11 II
1
\ 14+
Ik2 :::
I I I 1"+1 ~ -;; + 2 II h.,,\1·
(56)
1_ IHr j2
,
N
C')2
"'...
-
r ll
I
Hence we obtain the estimates _
I hr
_11__ . -
~
IT
12\2
II
+ ~11(uN+' )"111171 ')
T
1. .
<
--
11.2-
+ I IU N-J IlfJ·",,+1 III ,,+111 2 I I" I ,H'.'I + '_)' UIIII 11 .•' -
- f3 + IT +
<.. - - - - -
(57)
I.
(T+ I I IfJ+"'1-1 ~. 111(/N+I)"+'11 2 +1\ - f3 + (;+I Ullh h./3hf>l I :::: 4: I h.1 (,
and, by (53), the inequality ')
I.nax
III}
lI'r N
'I
\
Ie:: AJ ( 21\4 + .-~ ---1\)
) liI21". I II
(5X)
TIT+2
7/ E
R I' where C 1
= C 1 (IT) 1
c,2 \1/11"12_111',,'12\- -.. T---.-~-
(4)
\vith (!I'"
1 _II" + I)
IT (62)
Thu~ restriction (60) on the size of the time step T ensures global boundedness of the difference solution of problem (4). (5) for f3 < (T + L Let us note that by (53). from (62) follows the estimate
is a constant. As a result, we have
> 0
(-,,112 1111 III' _.
I
:: - -2T
1._
max IlI i +' Ie ::: A,/.
1111
,,111) II,1._1
IJ~ /'••
+
IT
+
I
I
_ Ii t "I
I I/H" \ I
, i
I -
II h./H"+')
+
+ CITlmaxlllle.llllcll
fJ
_
the solution is unique. Let III. 1/2 be two solutions of the problem. Then from (4) we have that III - 1/1 == T(!I'(I 1 - lI~t I h, + T(t/~3 - a;\ Taking the scalar product of this equality with 11'(' I -- lI~t I. we obtain
U' I" j2 \-
T
I
(64)
(59)
1
11' /I, j2
(63)
N
Let us show now that in the case
h,2
+ f3 + (T + 1 ~ (I II Ih./3 I"
Hence we have the (6\ )
To derive other estimates, we take the scalar produet of and use the inequality (52) as well as the inequality
for all ?;.
(to derive the last inequality we used the estimate (13)). estimates
U
\
In the derivation of (59) we also used the inequality (( 1/" I , h
\. //"11
-
II" \ ')" ::: -
(i[l/'r' I II ~
1 -
1111" + I II ~.1) I 2
(e -
(since ?;(?; - 7/) ::': 7/")/ 2 for any?;. 7/ E R I )· Let us choose N so large (that is, T = TI(N + I) so small), that CIT[max{llllc.llllellfJ I ::: C 1 /2 for all 0 ::: j ::: N. It follows from (58) that for this it is sufficient that T
( 21\4
I )lfJ + :') ---1\) T (T + 2
11/111" II II
C 2 I IJ < -_.-1\ - 2C, 3
(60)
Hence by (63), (64) we conclude that (!It- 1/10 11'(11 - II~+ I)h == O. that is. tl, == 111. To prove the validity of passing to the limit. we shall need another estimate. - Ul"\Vf(~~,".~L~ - - - -f").. From (4) we have li(fl
-<
1I)/TIl~1
AJ1():""''':': " ~.,
,...
:.'
,~
<
\.
•
~
VII Some other methods of study of unbounded solutions
496
497
5 Finite difference schemes
from which the estimates (6\)-(63) follow. Subsequent analysis is carried out as in the case {3 < iT + I. 0
Therefore, taking into account (8), and using (62), we obtain (65)
Thus in the cases {3 < (7 + I and {3 = (7 + I, A~' > I, no matter how flne the grid in time (which, obviously, can be taken to be non-uniform), there is no blOW-Up of finite difference solutions. which agrees with the conclusions of 2 concerning the differential problem, This agreement is even more clearly seen for
*
For convenience, let us introduce the notation
{3>(7+1.
From (57), (61), (62), (65), using the results of [3391, we obtain the following estimates: the functions (jTP,,(u i )tr+l are uniformly bounded in LOO(O, T; HA(O, I)) and in C"(O, T; ly't!I/I"tIJ(O, /)); ({,({h(Ui),,!1 are uniformly bounded in LX(O, T; Otrt!I/ltr+II(O, I)): ({T({,,(U J )lttr12 in L x(o. T; L!(O, /)); V T({T({,,(U J )lltr 12 in L!(O, T; L!({)./)); ({TP"II! in L"(O, T; Ltrl fro. /)); VT({TP"U J in C'';(O, T;HI(O,/)): [({TP"(ui),,tl]-,-, in LX(O, T:H1(O./)): ({TilhlIlJ)(3 in L""(O, T:L1(O./)): ((,,(u Ntl ) in c't!(O,/). This information is sufficient to be able to use a compactness theorem [861 to pass to the limit as T, h -+ 0 (details can be found in [296, 339]). As a result we have another proof of existence of a global generalized solution of the problem (I )-(3) for {3 < iT + I, which satisfles all the inclusions of Theorem I, 2. Observe that we have in addition that II, E L!(O. T:H-1(0./)). 2) Let us consider the case {3 = (7 + I. Applying the inequality (12) to estimate the right-hand side of (54) for {3 = (T + I, we have
*
I
I _ ,nO
------ --jul (T
+2T
-
7
//.<'1.
+
(
(66)
I
Hence (see (51)) we have the estim<Jte (58) with A~ I
I
A"
= 0 and
II U-IT t I II!h,! <_ -+ '~TI\I ) -- ," _I I IUOI, I"I/.<,I!I,! (T
<'
:::
the inequalities
A II T
6 Fillite differellce stable sets alit! passillK to the Iimit.f{JI" {3 >
so sm<J1l that (see
T
*
v"
Jh(V,,)
=
I
(7
-a,,(v,,) -
2
+
I
--h,,(v,,), {3+(7+ I
1 IT f I lJ smg " T L>effimd. I . . were a" ( v,,) = II VIIITI III!",!' IJ" ( v,,) -- IV" 1# 11,# lIT f I' It ,IS not Ilard h to prove (sec Lemma :'\ in 2)
*
Lemma 4. Let {3 > (II I
(7
+
I, Theil we hm'e the ille({uality
'j" 1(' )-,. = ,/ocll",ul,,¥=OA.O In sup. "I\V" ~
{3-«(7+1)/'-I#+)I<"III/111 «"llI>().
2({3+(T+ I)
Now we can define the finite difference stable set ''W" (which is non-empty by Lemma 4):
''W h
= 11'11 I Vh
E
H", 0 ::: J,,(AVh) < ("', A E [0,
(60))
Lemma 5. We hill'e the CqllillitYW"1I
Then it is not h<Jrd to see that the following inequality is s<Jtisfied: ii ttr/21 -2 "'T I-----.-----l
tr12 · - III
C2 N + i....J ! ,,0
T
2
h,!
+ -2I
( 1-I ) h
I 1!ltrlll) ::: '2I (II 110111111! ".! - 110" h,2Itr+ II • " <"
A
I
~ II(u tV i- I ). Irl I Il,,::: /,-
+I
Here we shall show that for {3 > (7 + I we can construct a difference stable set 'W". which has a structure quite similar to the one for the differential problem constructed in 2. The latter will be denoted below by ''W o, to emphasize that it is lV" for h = ot, Let us deflne for all E H" the functional
From the construction of 'lV'" we obtain (see Lemma 4, Let us take
(7
= 'W~, U (0).
(67)
Ill·
*2)
where
Let us state the main result of this subsection. Theorem 7. Lct {3 > (T + I illld ilSSIII//(' tlwt 1I0h E ')11'". Thellfiir slIflicielltly smull T the .tillitc dif!i'relli'r' prohlem (4). (5) hilS iI glohal solutioll which helollgs to'lV'" fiJI" all t E w T, Morcol'l'r, the solutioll is ullique. ~/: .titrthcrmore 110 E 'Woo. the .tillite ditfi:rellcc solutioll COil verges os T, h -+ 0 to the gellt'roliz.ed solutiOll or the origillill difli'l'cllliol proiJlem, which was cOllstrllctcd ill Theorem 4, 2.
Remark. In the conditions of Theorem 7. no refinement of the time step 7 (which by itself has to be sufficiently small) can lead to the finite difference solution being unbounded. Therefore, as follows from Theorem 7, inequality (37) cannot hold if I/0" E 'W'", that is. J
'''( I tr+ I· fi ::::' () - 1\1 1/0".11"),, I/O/I
Thus, u l E 'W". Then from Lemma 5 we have that (I,,(u l ) > h,,(I/I). Using this inequality to estimate the second term in the right-hand side of (69). we have
,
C2!(I/I)I+tr 12 _(1/0)IHT I2\'
____. 2
,.,.
E YV",
(3--((J+I)
*
and therefore I
= ~(//I(I/O") .!..
(J + I {3 - (IT + I) ,- -(3"-+-(J--+-"-I'J/J(I/O/I) ::': -2([3 - · ' - - - ( / 1 (1/0/). + IT + I) I I
(53)
("
, 2:
T
1/ _..•._ ,_..,_. II T
Uniqueness for T < [3 proved as in Theorem 6.
min{J.C'21(2C I )} 7:::: (I + A 12 )f3 + 2h- 2 ( 1+ A 12)';';1
(68)
ensures solvability of scheme (4) at each time level. Let us make the first step in time. From Theorem 2 (see condition (20) for Co = I) it follows that under the restriction (68) the scheme is solvable. Since by (68) lulle :::: Ii/lie + I :::: A I 2 + I. I I Ie. III°Ie I IfJl :::: C 21(2C I ), 7lmaxI!u
2
1
J_
11.2
j--':'{)
that is. 'W'" is bounded in C(lu,,). Let us show that the condition
0
J,,(LI ),
I + -((J+I) llii tr • I IIi' 2([3+1T+1)
I [(r/21
\--I+tr'2
N
2
that
II~,:::: .,
and therefore II/lie:::: AI}. This justifies making the next step in time under restriction (68) on the magnitude of T and so on. Thus. if (68) holds. the difference scheme has a global solution ii E '1,V" , lii!e:::: A I 2 for all () < .i :::: N. and furthermore ~
Therefore it follows from
Itrtl'
·--·---11(1/) 2([3 + (J + J)
Theorem 7. This largely follows the lines of proof of the "differential" Theorem 4. 2. If UO/ , E'U(h. then by Lemma 5 we have (1,,(1/0") - h,,(IIO/, ) > 0
.h(uo,,)
J._
11.2
Hence
(l(
1 {3.-((T+Il.1 trt.l} I + -----·-11(1/) II,,:::: -Jlril/\ T 2( [3 + (J + I) T
-
T
This inequality is an additonal characteristic of the difference stable setlV". Pro/I(
499
§ 5 Finite difference schemes
VII Some other methods of study of unbounded solutions
498
I
A:~fJ of a uniformly bounded solution 1/
E
'W" is 0
The tinite difference solution constructed above satisfies estimates which allow us to pass to the limit as T. h -> O. Here it is only necessary to observe that in the case I/o E 'lV o the inclusion I/o" E'Ui" holds for all sufficiently small h > n. As a result we have the existence of a global generalized solution of the problem ( I )-(3). which is in 0W'~; for all r > D. To conclude. let us note that by the estimate (21) for global solvability of the 2 tinite difference problem in the set 'OW". it is sufficient that 7 = G(h ) for h « I. The restriction (6~) on the size of the time step T is essential. since too large steps can sooner or later "throw" the solution out of ·I,V'". and it will become unbounded. The necessary shortening of the time steps T j will be automatically performed by the iterative algorithm we are using, when Theorem 4 (on the non-existence of solution) comes to the fore.
and. consequently. we obtain from (59) the inequality
2 Explicit (linear) difference scheme (69)
1
Let us prove that u l E 'W". Indeed. let III if- W·". Then since III -> 1/ as O 7 -.,. 0 (see proof of Theorem 2), and U E 'lV". there exists a 7 = T •• such that 1/1 E iJ'W'". By (67) this means that 1,,(1/1) = £I". Hence we have a contradietion with (69), since by assumption 1,,(1/0) < £I".
Below we shall consider briefly unbounded solutions of the explicit difference scheme for the problem (I )-(3): (70) II
Il
=: I/Il" ::':
0, .\
E
ILl,,: liE H", r E
IrJ r ·
(71 )
500
The main difference between this scheme and the implicit one lies in the fact that the "elliptic" operator in the right-hand side of (70) is computed from the values of the grid function u not on the next time level as in (4), but on the current time level. Therefore, obviously, the question of solvability of the scheme (70). (71) does not arise. We shall not discuss in detail the comparative advantages and disadvantages of the two schemes (this was already mentioned earlier). Such a detailed analysis can be found in [3461. Let us consider one basic question: when is the solution of the explicit scheme (70). (71) non-negative for any jnitial function uo(.r) ~ O. x E Wh. that is. when does the solution satisfy the weak Maximum Principle? If it does not. (70) does not necessarily make sense (it will not be possible in (70) to perform the operation of raising to an arbitrary power). It turns out that in some cases (f3 < (T + I) the weak Maximum Principle is not satisfied at all, while for f3 ~ fT + I it requires very severe restrictions on 7). h.
T)
==
2 Unbounded d!lIerence solutio/ls (~t the explicit scheme
Here we shall consider the appearance of finite difference finite time blow-up in the problem (70), (71) for f3 ~ (I' + I. To that end we could ~p.rly the eigen~unction method. which was used in subsection 1.4; for the explicit scheme It IS much simpler. . Below we employ a different method. In the following we take the gnd W T to be non-uniform, 7) - , 0 as .i --> 00, To := 2::;:0 7 ) < 00. . Let the Maximum of an initial function I/Oh "1= O. the value \uoille. be attall1ed at some point .\ E Who It is easily seen that at that point the value of the grid function j /1)+1 is estimated in terms of u as follows: (74)
u
O
:=
IUOh Ie > O.
Here we are assuming that 1/ ~ 0 in (UT x (UII (see Theorem 8). From (74) it is not hard to derive conditions for the solution to be unbounded. Let us write (74) in the following form:
I The weak Maximum Principle
The time grid here is taken to be uniform,
5 Finitc diffcrcncc schcmcs
~
VII Somc othcr mcthods of study of unhoundcd solutions
501
7.
Theorem 8. For f3 < IT + I the weak Maximum Principle does nor hold. It f3 = (I' + I, a neces.l'lIrr a/ld sutyicie/lt co//(!irioll .fill' it to hold is rhe inequality h 2 ~ 2. In the case f3 > (I' + I such 1I conditio/l is the .!iJl/owing:
u"
I :,:
Let the initial function
UIIII
(72) {Proof We start from the fact that a necessary and sufficient condition of nonnegativity of ii for any u ~ 0 in WII is the requirement that the function
+ 7)(U I )fJ
u)
[I - 1~2
(Uj)'H I
fJ
J. t E
CUT'
be such that J.L0:=
2 h-
If i
I - ,Iullille
Then it immediately follows from (75) that therefore we have from (75) that
1·/3
(76)
> O.
> u l for all
lIlll
.i = 0, I, ..., and (77)
(73)
be non-negative cverywhere in R+. This follows from thc form of (70) resolved with respect to It: ii, := P,dllkl + A( utili + l. if we set Ilk I := U1+ I := 0 for some 0 < k < M (such a situation could obtain. for example. at the initial moment of time). An elementary analysis of the function (73) leads to the conclusions of the theorem. 0
It remains to choose suitable sizes for the time steps 7;. For example. let us set
utt'i
Let us note that in the dependence of Ton h, inequality (72) resembles condition (33') of solvability of the implicit scheme on a time level for f3 > (1'+ I. However. there is a crucial difference between the two: while (33') is needed only at a fully developed stage of blow-up (jllie large), without (72), in general. the computation simply cannot be started. Naturally. for certain initial functions l/Oll condition (72) may not be required.
VII Somc othcr mcthods 01" study 01" unboundcd solutions
502
Let us show that this is indeed finite difference finite time blow-up. By (78). (79) we have the following estimate for the time of existence of the difference solution:
Let us note that to get finite time hlow-up in the explicit scheme. it is necessary to choose small time steps. For example. in the framework of the above approach. we derive the following estimate for the magnitude of To (the first step): if (76) holds. we have the inequality
luo"le
.~ (2/lI c)I/I13
("1111
and therefore from (7R) we obtain Til
== ,!Llllo"I;.li Mil
< 3-211'131/1#1"IIII(hC)I/3 11/1/3 1,,'111 Mo
= OI(hc)l/i
II/I/.! 1"'1 11
1. h«
= (XI)
I.
Naturally. in view of (7X). (70) suhsequent steps will he even shorter. This which estimate has the same dependence of TO on h as the optimal inequality ensures solvability of the implicit scheme. Thus to have blow-up in the explicit scheme. its apparent simplicity notwithstanding. we still have to compute with very small time steps. The reason for this is clear: from (70) it immediately follows that marching in time with large time steps never leads to finite time blow-up. This is true. for example. for a uniform grid lOr. Let us also note that a restriction of the form of (X I) is needed 10 have the weak Maximum Principle (see Theorem 8).
crn.
Remarks and comments on the literature § I. The presentation of results of ~ I largely follows 1123. 127. 13\. 169. 170!
q 1311
contains a brief account of the results pertaining to the one-dimensional
503
equation). Certain results of ~ I were obtained earlier in [1201. Theorem 6 for IT = O. P > «(3 - I)N /2. p ::: I. and more general initial functions was proved by a different method in 13791 (see also theorems 3.2 and 3.3 in [26J). Let us note that 1379. 261 make significant use of the semi linearity of equation (4) for (T = O. i.e. of the ability to invert the linear operator iJ/ill - U and reduce the problem to an equivalent integral equation. Therefore the melhod of [379\ is not applicable in the quasilinear case when I./J' (1I) F consl. In the case (T = O. N = I and arbitrary lIoLt) Theorem 6 was proved in 11701: an exact form of the envelope of the family (U(I': Uoll in Theorem 5 was also established there: lI(T!). x) ::: G(r) = Co!xl- 2/1/31). where Co > C. is a constant. Some classes of equations of the form (30). (31) were studied in [127.150,3471 (see ~ 7. eh. IV). In the most general form degenerate a.s.s. of similar quasi linear equations were considered in [160]. Many other examples of the use of the method of stationary states are contained in II64J: for other possibilities see 1137. 174. 175. 180. 181. 1891· Another approach to construction of lower bounds for unbounded solutions of quasi linear equations of the form (I) has been developed by [3061 (see also 12231. where boundary value problems in a bounded domain are considered). In its final results. this approach is similar to the method of stationary states. For example. in [3061 it is shown that for I < (3 < (T + I the solution grows without bound on the whole space. while for (3 = (T + I a "lower bound" for the localization domain is obtained. The main results of this section are based on intersection comparison with the given set ~p of particular stationary solutions to the quasi linear heat equation (I) with one space varable I' = Ix!. A similar comparison can be performed with respect to an arbitrary set :1!, of other solutions if it is sufl1ciently large ("complete" in the sense of existence and uniqueness of tangent solutions in spatial variable). In this case we arrive at the notion of generalized B-convexity/concavity properties of the solutions with respect to the given functional set ~'1!,. Under certain assumptions these properties are proved to be to be preserved in time or to appear eventually in time. see general results in 11941. Observe that the "criterion" of complete blow-up for a general one-dimensional quasi linear heat equation with source [193] is a straightforward consequence 1'0 such intersection comparison with the set of travelling wave solutions depending on the variable ~ = x - At. and looks like the property of eventual B-convexity. § 2. Theorems of subsection \ are proved in 1121. \251. A similar problem was considered. using a different approach. prior to that in [372 I: the results obtained there are not quite optimal. The main claims of subsection 2. I are contained in [120. 1251 (results of similar generality were established in 12941J. The greater parl of conclusions of subsection 2.2 can be found in 1125]. A generalization of the concavity mcthod to study unbounded solutions of parabolic equations and systems of equations with a given type of nonlinearity was undertaken in [1241· Later conditions for appearance of unboundcd solutions of quasi linear parabolic equations were established in 1307. 2941. 13071 employs the method of eigenfunctions. which
504
Open problems
VII Some other methods of study of unbounded solutions
is not unlike that of 1120. 125] (see Theorem 8 of § 2); 1294] uses the same approach as [1241. A similar analysis of the boundary value problem for the quasilinear equation II, = I/I(II)U" + (Mil) in the one-dimensional case appeared earlier in [2251; in particular, a version of the method of eigenfunctions was used there. A brief survey of the literature on unbounded solutions can be found in 1157] and [290]. § 3. The majority of results of § 3 is contained in 11611 (see also \157]). Other examples of the use of the method of stationary states to derive lower bounds for unbounded solutions can be found in 1159. 164. 174]. The problem of computing upper bounds and thus proof of localization for systems of equations is almost completely open. The single result of 11051, obtained by the method of [1081. deals with the semilinear system (1), (2) for j1. = /1 = 0 and p = q. Let us note the paper 193] (see also 1290j). which considers the Cauchy problem for a semi linear system and shows that for (y + I )/(pq - I) ~ N/2, y = maxlp, q}, every non-trivial non-negative solution blows up in tinite time. thus determining the critical (in the sense of Fujita [ I 12 J) exponent of the source. § 4. Numerical 'and qualitative results of subsection I are taken from 1273.271)1· The analysis of subsection 2 comes from [ 1421· § 5. All the main assertions of § 5 are proved in I 182, 183]. Studies of unbounded solutions of explicit finite difference schemes for the semilinear (IT = 0) equation using different methods were conducted in 1311, 3121. Interesting results concerning localization in the context of an explicit-implicit scheme for the equation with IT = O. when the source term II/I is taken not from the next, but from the current time level. are obtained in [641. In particular. it is shown there that for f3 = 2 the difference solution becomes infinite at three central points. while for Ii > 2 it happens at a single point. The results of § 3 and the proposition of § 4 are based on the derivations of 1161, 1591. Let us note that the practically optimal result concerning global solvability of the boundary value problem for pq < (I + j1.)( I + /1) (Theorem 5 of § 3). established by using the method of stationary states. is hard to obtain by using the usual techniques of a priori estimates. for example, those employed in Galerkin's method. This is indicated by the analysis of 11571. as well as. for example. the results of 1299], where restrictions of the form I' < I + j1.. II < I + II are obtained. These restrictions are the natural ones for a single equation (see § 2). They characterize the easily explainable relation between intensities of processes of heat diffusion and combustion. which is necessary for the occurrence of thermal perturbations of finite amplitude. As shown in § 3. these conditions are far from being optimal for systems of equations.
505
Open problems I. (§ I) Describe the whole class of coefficients (k, Ql for which the absence of localization condition (16) (for N = I) is not only sufficient. but also necessary (for the case k(lI) = u". Q(II) = uiJ • f3 ~ IT + I, IT > n. this has been done in
[ 129j}. 2. (§ I) Derive an upper bound for II(T o ' x) of the form (29') in Theorem 5 for arbitrary initial functions lIo(.r) (for a particular class of 110 this has been done in \131,172.173]). 3. (§ I) Prove localization of unbounded solutions of equation (I) with general coefficients (30) for If ~ 2 (in the case k(11) == L Q(II) ::= (I + II) ln#(I + II). f3 ~ 2. this has been done in 11891 (f3 = 2) and 11771 (f3 > 2); see also [192] and § 7. Ch. IV). 4. (§ 2) Determine conditions for which the behaviour of unbounded solutions of the boundary value problem (5). (6) as I --> To (110) < x. f3 E (IT + I. (IT + I)(N + 2)/(N - 2), ) is described by the self-similar solutions constructed in § I. Ch. IV. Analyze the asymptotic behaviour of unbounded solutions of the problem for f3 ~ (IT + I )(N + 2)/(N - 2) t (let us note that in this case there is an unusual class of global solutions; see 1314]). 5. (§ 3) Is it possible to construct a family of explicit solutions of equation (39) in R N for the lTitical value of the parameter f3 = IN/a + 2(1 + l/a)I/[N2( I + 1/ a) I. a f I. similar to the one given in the example in subsection 2.4 for the semi-linear case It::= I, f3 = (N + 4)/(N - 4)') 6. (§ 3) Determine conditions for localization of compactly supported unbounded solutions of the Cauchy problem for the system (I). (2). Is the condition m = I'll - (I + j1.)( I + 1/) ~ 0 sufficient for that'! 7. (§ 3) What a.s.s. describes asymptotic behaviour of unbounded solutions of the Cauchy problem for the system (I). (2) in the cases wben it does not have self-similar solutions') 8. (§ 3) Determine conditions for solvability of the elliptic system (65). What is the structure of the set of its solutions for various values of parameters (some numerical results are contained in [273.279\). 9. (§ 4) Find conditions for localization of unbounded solutions of the problem (15). (16).
10. (~5) Are the finite difference solutions of the explicit scheme (70), (71) localized for f3 ~ IT + I in the case of an initial function IIO/i ~ n with "compact support" (that is. can it happen that Ix E W/i Iller o . x) ::= oo} f IO/i)'? The implicit scheme does not have this asymptotic property.
113] S, Angenent. The t.ero set of a solution of a parabolic equation. J. Reine Angew. Math .. 390 (1'188).7'1-86.
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it
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Index ......
_-----_.----------_._--------_. -_._._--------
localization domain 13~. 176. 2H4. 415 depth I ~3. 2X4. ~~6 LS blow-up regimc 60 LS-rcgime I ~4. 177
approximatl: Sl:lr-similar solution 49 aVl:raging mcthod 20 I blow-up rl:giml: I ~O timl: 130 Caul:hy problem 2 classkal solution ., l:ombustion l:igl:nfunctions 190 critical linitc difTcrcnl:c solution critical solution ~ IX criticality conditions ~ 19
Maximum Principle ~ mcthod of stationary statcs 414
clTcctivc localization 134. 177. 2X4. depth I ~4 domain I ~4 l:xplicit difference scheme 4<)9 fast diffusion equation 67 first boundary value problem 2 front of a wave I ~9 front point I ~6 fundamcntal length 1XO generalizcd solution 23 global insolvability 6 global solvability 6 half-width I ~9 heat IOl:alization 147. ~~5. X5 IIilbcrt spal:e" 1 (R) 50 liS blow-up rcgimc 60 IIS-regimc 134 implicit diflcrenl:c sl:hcmc 4X I instantancous point sourl:C 149 intcrscl:tion l:omparison mcthod 239 intcrval 240 point 240 Jensen's inequality II
Nagumo Icmmas xvii nonlinear hcat equation with sink I with sourcc I
~6X
~47
opcrator comparison theorcm ~~ I
~2X
radially symmctric function 20 resonancc length 19'.> Ifl-critical solution
~~2. ~5.1
S blow-up rcgime 59 S-regimc 134. 177 second boundary valuc problem 5 similarity representation 42 stable sct 4~O standing thcrmal wave 59 stricl localizatitlll I ~~ solution 176 suhsolution 4 super-slow diflusion cquation H~ supcrsolution 4 tangency intervtll 240 point 240 thcrmal crystal 157 thcrmal wave I ~9 total extinction 67. 6X uniformly parabolic cquation 7 unstable sct 4~O wcak Maximum Principlc
