## Blow-Up in Quasilinear Parabolic EquationsThe 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. The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers interested in a thorough study of the subject. Editorial Board Lev Birbrair, Universidade Federal do Ceará, Fortaleza, Brasil Honorary Editor Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia Titles in planning include Yuri A. Bahturin, Identical Relations in Lie Algebras (2019) |

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### Contents

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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 |

535 | |