Dead-core rates for the porous mediumequation with a strong absorption

Xinfu Chen; Jong-Shenq Guo; Bei Hu

doi:10.3934/dcdsb.2012.17.1761

Article Contents

2012, Volume 17, Issue 6: 1761-1774. Doi: 10.3934/dcdsb.2012.17.1761

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Dead-core rates for the porous medium equation with a strong absorption

1.
Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260
2.
Department of Mathematics, Tamkang University, Tamsui, Taipei County 25137
3.
Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN 46556

Received: February 28, 2011

Revised: July 31, 2011

Published: August 2012

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Abstract

We study the dead-core rate for the solution of the porous medium equation with a strong absorption. It is known that solutions with certain class of initial data develop a dead-core in finite time. We prove that, unlike the cases of semilinear heat equation and fast diffusion equation, there are solutions with the self-similar dead-core rate. This result is based on the construction of a Lyapunov functional, some a priori estimates, and a delicate analysis of the associated re-scaled ordinary differential equation.

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Mathematics Subject Classification: Primary: 35K20, 35K55; Secondary: 35B40.

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References

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