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Zusatztext This book is intended to introduce graduate students to the methods and results of nonlinear diffusion equations of porous medium type, as practised today. The present text, remarkable for generality and depth, is also notable for its author's concern, throughout, to keep the important issues about varieties clearly in the foreground ... [the book] succeeds admirably, in the reviewer's opinion, in introducing its difficult subject at a level appropriate for preparing future workers in the field. Klappentext This text is concerned with the quantitative aspects of the theory of nonlinear diffusion equations; equations which can be seen as nonlinear variations of the classical heat equation. They appear as mathematical models in different branches of Physics, Chemistry, Biology, and Engineering, and are also relevant in differential geometry and relativistic physics. Much of the modern theory of such equations is based on estimates and functional analysis. Concentrating on a class of equations with nonlinearities of power type that lead to degenerate or singular parabolicity ("equations of porous medium type"), the aim of this text is to obtain sharp a priori estimates and decay rates for general classes of solutions in terms of estimates ofparticular problems. These estimates are the building blocks in understanding the qualitative theory, and the decay rates pave the way to the fine study of asymptotics. Many technically relevant questions are presented and analyzed in detail. A systematic picture of the most relevant phenomena isobtained for the equations under study, including time decay, smoothing, extinction in finite time, and delayed regularity. Zusammenfassung This text is concerned with quantitative aspects of the theory of nonlinear diffusion equations, which appear as mathematical models in different branches of Physics, Chemistry, Biology and Engineering. Inhaltsverzeichnis Preface Part I 1: Preliminaries 2: Smoothing effect and time decay. Data in L¹(R^n) or M(R^n) 3: Smoothing effect and time decay from L^p or M^p 4: Lower bounds, contractivity, error estimates and continuity Part II 5: Subcritical range of the FDE. Critical line. Extinction. Backward effect 6: Improved analysis of the critical line. Delayed regularity 7: Extinction rates and asymptotics for 0 8: Logarithmic diffusion in 2-d and intermediate 1-d range 9: Super-fast FDE 10: Summary of main results for the PME/FDE Part III 11: Evolution equations of the p-Laplacian type 12: Appendices Bibliography Index ...
