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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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OPTIMAL EXISTENCE THEORY FOR NON-NEGATIVE SOLUTIONS

OPTIMAL EXISTENCE THEORY FOR NON-NEGATIVE SOLUTIONS

Chapter:
(p.309) 13 OPTIMAL EXISTENCE THEORY FOR NON-NEGATIVE SOLUTIONS
Source:
The Porous Medium Equation
Author(s):

Juan Luis Vázquez

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198569039.003.0013

This chapter studies the existence and uniqueness of solutions of the Cauchy problem for the PME posed in the whole space, which take a Radon measure as initial data. Section 13.1 constructs limit solutions for data measures with the growth condition found as optimal in the previous chapter (in the non-negative case). The theory is continued in Section 13.2 where it is proven that any non-negative solution defined in a domain QT has a unique initial trace. In Sections 13.3 and 13.4, it is proved that the initial trace determines the solution in a unique way. This is a landmark in the theory of the PME and completes the basic theory of the Cauchy problem developed in previous chapters.

Keywords:   Cauchy problem, PME, whole space, Radon measure, Pierre's uniqueness theorem

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