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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 CAUCHY PROBLEM WITH GROWING INITIAL DATA

THE CAUCHY PROBLEM WITH GROWING INITIAL DATA

Chapter:
(p.279) 12 THE CAUCHY PROBLEM WITH GROWING INITIAL DATA
Source:
The Porous Medium Equation
Author(s):

Juan Luis Vázquez

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198569039.003.0012

This chapter examines general conditions on the initial data under which the PME can be solved. The emphasis is on non-negative solutions and the data are locally integrable functions. Signed solutions are also considered. Section 12.1 introduces the problem and the main growth assumptions. Section 12.2 derives the well-known Aronson-Caffarelli estimate for non-negative solutions. Section 12.3 covers existence under optimal growth conditions. Section 12.4 deals with the uniqueness of growing solutions. Section 12.5 deals with further properties of the solutions and Section 12.6 studies the properties of some particular classes of solutions obtained under special assumptions on the data. As a complement to the results of this section, Section 12.7 considers the questions of boundedness of solutions defined locally and their approximation by smooth solutions. Section 12.8 constructs solutions in conical domains, which shows that growth rates as high as desired can be admissible if the aperture of the cone is small enough.

Keywords:   PME, non-negative solutions, signed solutions, Aronson-Caffarelli estimate, bounded solutions, periodic solutions

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