## 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

Chapter | 1 |

Generalized solutions of quasilinear degenerate parabolic equations | 14 |

Remarks and comments on the literature | 35 |

Asymptotic stability of the fundamental solution of the Cauchy problem | 47 |

Asymptotic stability of selfsimilar solutions of nonlinear heat equations | 53 |

Quasilinear heat equation in a bounded domain | 61 |

The fast diffusion equation Boundary value problems in a bounded | 67 |

Conditions of equivalence of different quasilinear heat equations | 74 |

Proof of localization of unbounded solutions for B 2 0 + 1 absence | 238 |

Asymptotic stability of unbounded selfsimilar solutions | 257 |

Blowup regimes effective localization for semilinear equations with | 274 |

Remarks and comments on the literature | 306 |

The operator functional comparison method for solutions of parabolic | 324 |

Conditions for absence of heat localization | 348 |

Criticality conditions and a comparison theorem for finite difference | 365 |

Chapter VI | 373 |

A heat equation with a gradient nonlinearity | 84 |

Selfsimilar solutions of the semilinear parabolic equation | 93 |

A nonlinear heat equation with a source and a sink | 99 |

The structure of attractor of the semilinear parabolic equation with | 107 |

Remarks and comments on the literature | 124 |

Heat localization inertia | 130 |

Effective heat localization | 158 |

Remarks and comments on the literature | 174 |

Asymptotic behaviour of unbounded solutions Qualitative theory | 200 |

Conditions for finite time blowup Globally existing solutions | 214 |

Approximate selfsimilar solutions in the nondegenerate case | 386 |

Approximate selfsimilar solutions in the nondegenerate case Integral | 398 |

Remarks and comments on the literature | 413 |

Boundary value problems in bounded domains | 430 |

A parabolic system of quasilinear equations with a source | 447 |

The combustion localization phenomenon in multicomponent media | 467 |

Finite difference schemes for quasilinear parabolic equations admitting | 476 |

Remarks and comments on the literature | 502 |