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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The porous medium equation in one dimension


Author: Barry F. Knerr
Journal: Trans. Amer. Math. Soc. 234 (1977), 381-415
MSC: Primary 35K15
MathSciNet review: 0492856
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Abstract: We consider a second order nonlinear degenerate parabolic partial differential equation known as the porous medium equation, restricting our attention to the case of one space variable and to the Cauchy problem where the initial data are nonnegative and have compact support consisting of a bounded interval. Solutions are known to have compact support for each fixed time.

In this paper we study the lateral boundary, called the interface, of the support $ P[u]$ of the solution in $ {R^1} \times (0,T)$. It is shown that the interface consists of two monotone Lipschitz curves which satisfy a specified differential equation. We then prove results concerning the behavior of the interface curves as t approaches zero and as t approaches infinity, and prove that the interface curves are strictly monotone except possibly near $ t = 0$. We conclude by proving some facts about the behavior of the solution in $ P[u]$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1977-0492856-3
PII: S 0002-9947(1977)0492856-3
Keywords: Nonlinear degenerate parabolic second order equation, porous medium
Article copyright: © Copyright 1977 American Mathematical Society