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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 II. DIRICHLET AND NEUMANN PROBLEMS

ASYMPTOTIC BEHAVIOUR II. DIRICHLET AND NEUMANN PROBLEMS

Chapter:
(p.521) 20 ASYMPTOTIC BEHAVIOUR II. DIRICHLET AND NEUMANN PROBLEMS
Source:
The Porous Medium Equation
Author(s):

Juan Luis Vázquez

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198569039.003.0020

This chapter contains a complete study of the large-time behaviour of solutions of the porous medium equation, ut = Δum with m > 1, posed in a bounded domain of the n-dimensional space with homogeneous boundary conditions. Asymptotic profiles are obtained and full proofs of the convergence results are given. Section 20.1 looks at the theory for non-negative solutions, while Section 20.2 covers the general theory without a sign restriction. This study is used to exhibit some of the most common concepts and techniques used in establishing the asymptotic behaviour as t → ∞ of solutions of nonlinear evolution equations. The main ideas involved are rescaling, the existence of special solutions, a priori estimates, ω-limits, and Lyapunov functionals.

Keywords:   PME, asymptotic behaviour, lateral propagation, Dirichlet problem, Neumann problem

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