Book

Smoothing and Decay Estimates for Nonlinear Diffusion Equations

Juan Luis Vázquez

Published in print August 2006 | ISBN: 9780199202973
Published online September 2007 | e-ISBN: 9780191707919 | DOI: http://dx.doi.org/10.1093/acprof:oso/9780199202973.001.0001

Series: Oxford Lecture Series in Mathematics and Its Applications

Smoothing and Decay Estimates for Nonlinear Diffusion Equations

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This book 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 book is to obtain sharp a priori estimates and decay rates for general classes of solutions in terms of estimates of particular 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 is obtained for the equations under study, including time decay, smoothing, extinction in finite time, and delayed regularity.

Keywords: classical heat equation; nonlinearities of power type; singular parabolicity; decay rates; asymptotics; time decay; smoothing; extinction in finite time; delayed regularity

Book.  248 pages.  Illustrated.

Subjects: Applied Mathematics

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Table of Contents

Introduction in Smoothing and Decay Estimates for Nonlinear Diffusion Equations

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Preliminaries in Smoothing and Decay Estimates for Nonlinear Diffusion Equations

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Lower bounds, contractivity, error estimates, and continuity in Smoothing and Decay Estimates for Nonlinear Diffusion Equations

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Subcritical range of the FDE. Critical line. Extinction. Backward effect in Smoothing and Decay Estimates for Nonlinear Diffusion Equations

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Improved analysis of the critical line. Delayed regularity in Smoothing and Decay Estimates for Nonlinear Diffusion Equations

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Extinction rates and asymptotics for 0 < <i>m</i> < <i>m</i> <sub> <i>c</i> </sub> in Smoothing and Decay Estimates for Nonlinear Diffusion Equations

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Logarithmic diffusion in 2D and intermediate 1D range in Smoothing and Decay Estimates for Nonlinear Diffusion Equations

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Superfast FDE in Smoothing and Decay Estimates for Nonlinear Diffusion Equations

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