Smoothing and Decay Estimates for Nonlinear Diffusion Equations

Equations of Porous Medium Type

Juan Luis Vazquez - Collection Oxford Lecture Series in Mathematics and Its Applications

248 pages, parution le 15/08/2006

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Résumé

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 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.

Readership: Graduates and researchers in mathematics, the physical sciences and engineering

L'auteur - Juan Luis Vazquez

Juan Luis Vazquez, Universidad Autónoma de Madrid

Sommaire

Part I
- Preliminaries
- Smoothing effect and time decay. Data in L¹(Rⁿ) or M(Rⁿ)
- Smoothing effect and time decay from L^p or M^p
- Lower bounds, contractivity, error estimates and continuity
Part II
- Subcritical range of the FDE. Critical line. Extinction. Backward effect
- Improved analysis of the critical line. Delayed regularity
- Extinction rates and asymptotics for 0<M<M_C
- Logarithmic diffusion in 2-d and intermediate 1-d range
- Super-fast FDE
- Summary of main results for the PME/FDE
Part III
- Evolution equations of the p-Laplacian type
- Appendices

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	PAPIER
Éditeur(s)	Oxford University Press
Auteur(s)	Juan Luis Vazquez
Collection	Oxford Lecture Series in Mathematics and Its Applications
Parution	15/08/2006
Nb. de pages	248
Format	16 x 24
Couverture	Relié
Poids	500g
Intérieur	Noir et Blanc
EAN13	9780199202973
ISBN13	978-0-19-920297-3