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The Porous Medium EquationMathematical Theory$
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Juan Luis Vazquez

Print publication date: 2006

Print ISBN-13: 9780198569039

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198569039.001.0001

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ASYMPTOTIC BEHAVIOUR I. THE CAUCHY PROBLEM

ASYMPTOTIC BEHAVIOUR I. THE CAUCHY PROBLEM

Chapter:
(p.454) 18 ASYMPTOTIC BEHAVIOUR I. THE CAUCHY PROBLEM
Source:
The Porous Medium Equation
Author(s):

Juan Luis Vázquez

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198569039.003.0018

This chapter begins with a study of the behaviour of solutions of the PME for large times. The cornerstone of the presentation is the interplay between asymptotic behaviour and self-similarity. It is also shown that large time behaviour gives rise to the formation of patterns. Section 18.2 contains a proof of the asymptotic theorem for non-negative solutions using the so-called four step method, based on rescaling and compactness. The convergence of supports and interfaces for compactly supported data occupies Section 18.3. Section 18.4 examines the so-called continuous scaling and the associated Fokker-Planck equations. Section 18.6 introduces another functional, the entropy. Section 18.7 delves in to the peculiarities of asymptotic behaviour in one space dimension; this allows us to establish optimal convergence rates. Section 18.8 contains a proof of asymptotic convergence for signed solutions, and the extension to cover integrable forcing terms. Section 18.9 gives an introduction to the special properties of the large time behaviour of the FDE.

Keywords:   PME, Cauchy problem, non-negative solutions, supports, interfaces, Lyapunov method, entropy approach, asymptotic behaviour, FDE

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