Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

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

Pages: 1761 - 1774, Volume 17, Issue 6, September 2012      doi:10.3934/dcdsb.2012.17.1761

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Xinfu Chen - Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, United States (email)
Jong-Shenq Guo - Department of Mathematics, Tamkang University, Tamsui, Taipei County 25137, Taiwan (email)
Bei Hu - Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN 46556, United States (email)

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.

Keywords:  Dead-core rate, porous medium equation, strong absorption.
Mathematics Subject Classification:  Primary: 35K20, 35K55; Secondary: 35B40.

Received: March 2011;      Revised: August 2011;      Available Online: May 2012.