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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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THE DIRICHLET PROBLEM I. WEAK SOLUTIONS

THE DIRICHLET PROBLEM I. WEAK SOLUTIONS

Chapter:
(p.81) 5 THE DIRICHLET PROBLEM I. WEAK SOLUTIONS
Source:
The Porous Medium Equation
Author(s):

Juan Luis Vázquez

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198569039.003.0005

This chapter begins with a systematic study of the questions of existence, uniqueness, and main properties of the solutions of the PME by concentrating on the first boundary-value problem posed in a spatial domain Ω, which is a bounded subdomain of ℝ d , d ≥ 1. It focuses on homogeneous Dirichlet boundary conditions, u = 0 on ∂Ω, in order to obtain a simple problem for which a fairly complete theory can be easily developed as a first stage in understanding the theory of the PME. This is called the homogeneous Cauchy-Dirichlet problem, or more simply, the homogeneous Dirichlet problem.

Keywords:   PME, weak solution, Dirichlet problem, Cauchy-Dirichlet problem, fast diffusion

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