Blow-Up in Quasilinear Parabolic Equations

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The aim of the Expositions is to present new and important developments in pure and applied mathematics. Well established in the community over more than two decades, the series offers a large library of mathematical works, including several important classics.

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Editorial Board

Lev Birbrair, Universidade Federal do Ceará, Fortaleza, Brasil
Walter D. Neumann, Columbia University, New York, USA
Markus J. Pflaum, University of Colorado, Boulder, USA
Dierk Schleicher, Jacobs University, Bremen, Germany
Katrin Wendland, University of Freiburg, Germany

Honorary Editor

Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia

Titles in planning include

Yuri A. Bahturin, Identical Relations in Lie Algebras (2019)
Yakov G. Berkovich, Lev G. Kazarin, and Emmanuel M. Zhmud', Characters of Finite Groups, Volume 2 (2019)
Jorge Herbert Soares de Lira, Variational Problems for Hypersurfaces in Riemannian Manifolds (2019)
Volker Mayer, Mariusz Urbański, and Anna Zdunik, Random and Conformal Dynamical Systems (2021)
Ioannis Diamantis, Boštjan Gabrovšek, Sofia Lambropoulou, and Maciej Mroczkowski, Knot Theory of Lens Spaces (2021)

 

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Contents

Chapter I Preliminary facts of the theory of second order quasilinear parabolic equations
1
2 Existence uniqueness and boundedness of the classical solution
6
3 Generalized solutions of quasilinear degenerate parabolic equations
14
Remarks and comments on the literature
35
Chapter II Some quasilinear parabolic equations Selfsimilar solutions and their asymptotic stability
38
1 A boundary value problem in a halfspace for the heat equation The concept of asymptotic stability of selfsimilar solutions
39
2 Asymptotic stability of the fundamental solution of the Cauchy problem
47
3 Asymptotic stability of selfsimilar solutions of nonlinear heat equations
53
5 Asymptotic stability of unbounded selfsimilar solutions
257
6 Asymptotics of unbounded solutions of LSregime in a neighbourhood of the singular point
268
7 Blowup regimes effective localization for semilinear equations with a source
274
Remarks and comments on the literature
306
Open problems
314
Chapter V Methods of generalized comparison of solutions of different nonlinear parabolic equations and their applications
316
2 The operator functional comparison method for solutions of parabolic equations
324
3 ψcriticality conditions
331

4 Quasilinear heat equation in a bounded domain
61
5 The fast diffusion equation Boundary value problems in a bounded domain
67
6 The Cauchy problem for the fast diffusion equation
68
7 Conditions of equivalence of different quasilinear heat equations
74
8 A heat equation with a gradient nonlinearity
84
9 The KolmogorovPetrovskiiPiskunov problem
87
10 Selfsimilar solutions of the semilinear parabolic equation ut Δu + u Inu
93
11 A nonlinear heat equation with a source and a sink
99
12 Localization and total extinction phenomena in media with a sink
101
13 The structure of attractor of the semilinear parabolic equation with absorption in RN
107
Remarks and comments on the literature
124
Chapter III Heat localization inertia
130
2 Blowingup selfsimilar solutions
135
3 Heat inertia in media with nonlinear thermal conductivity
142
4 Effective heat localization
158
Remarks and comments on the literature
174
Chapter IV Nonlinear equation with a source Blowup regimes Localization Asymptotic behaviour of solutions
176
1 Three types of selfsimilar blowup regimes in combustion
178
2 Asymptotic behaviour of unbounded solutions Qualitative theory of nonstationary averaging
200
3 Conditions for finite time blowup Globally existing solutions for β σ + 1 + 2N
214
4 Proof of localization of unbounded solutions for β σ + 1 absence of localization in the case 1 β σ + 1
238
4 Heat localization in problems for arbitrary parabolic nonlinear heat equations
335
5 Conditions for absence of heat localization
348
6 Some approaches to the determination of conditions for unboundedness of solutions of quasilinear parabolic equations
353
7 Criticality conditions and a comparison theorem for finite difference solutions of nonlinear heat equations
365
Remarks and comments on the literature
371
Chapter VI Approximate selfsimilar solutions of nonlinear heat equations and their applications in the study of the localization effect
373
2 Approximate selfsimilar solutions in the degenerate case
375
3 Approximate selfsimilar solutions in the nondegenerate case Pointwise estimates of the rate of convergence
386
4 Approximate selfsimilar solutions in the nondegenerate case Integral estimates of the rate of convergence
398
Remarks and comments on the literature
413
Chapter VII Some other methods of study of unbounded solutions
414
2 Boundary value problems in bounded domains
430
3 A parabolic system of quasilinear equations with a source
447
4 The combustion localization phenomenon in multicomponent media
467
5 Finite difference schemes for quasilinear parabolic equations admitting finite time blowup
476
Remarks and comments on the literature
502
Open problems
505
Bibliography
506
Index
535
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