Smoothing and Decay Estimates for Nonlinear Diffusion Equations: Equations of Porous Medium Type

Hardcover | March 8, 2006

byJuan Luis Vazquez

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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.

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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...

Juan Luis Vazquez is at Universidad Autonoma de Madrid.

other books by Juan Luis Vazquez

Format:HardcoverDimensions:248 pages, 9.21 × 6.14 × 0.7 inPublished:March 8, 2006Publisher:Oxford University PressLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:0199202974

ISBN - 13:9780199202973

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Table of Contents

PrefacePart I1. Preliminaries2. Smoothing effect and time decay. Data in L1(Rn) or M(Rn)3. Smoothing effect and time decay from Lp or Mp4. Lower bounds, contractivity, error estimates and continuityPart II5. Subcritical range of the FDE. Critical line. Extinction. Backward effect6. Improved analysis of the critical line. Delayed regularity7. Extinction rates and asymptotics for 0mm_c8. Logarithmic diffusion in 2-d and intermediate 1-d range9. Super-fast FDE10. Summary of main results for the PME/FDEPart III11. Evolution equations of the p-Laplacian type12. AppendicesBibliographyIndex

Editorial Reviews

`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 importantissues 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.'Vicentiu Radulescu, Mathematical Reviews Issue 2007k