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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 III. STRONG SOLUTIONS

THE DIRICHLET PROBLEM III. STRONG SOLUTIONS

Chapter:
(p.181) 8 THE DIRICHLET PROBLEM III. STRONG SOLUTIONS
Source:
The Porous Medium Equation
Author(s):

Juan Luis Vázquez

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198569039.003.0008

This chapter addresses the question of how regular the solutions constructed in previous chapters actually are. Section 8.1 considers the question of further regularity of the time derivative ut . Both in the case u = 0 and in the signed case, it is proved that ut is a locally integrable function. This allows the introduction in Section 8.2 of the more stringent concept of solution called strong solutions, i.e., weak solutions such that both ut and ΔΦ(u) are locally integrable functions. Strong solutions have nice calculus properties. Some of those properties are examined in detail. The concepts of super- and subsolutions are also discussed.

Keywords:   PME, Dirichlet problem, strong solutions, super solutions, subsolutions

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