A continuum of extinction rates for the fast diffusion equation

Marek Fila; Juan-Luis Vázquez; Michael Winkler

doi:10.3934/cpaa.2011.10.1129

Article Contents

2011, Volume 10, Issue 4: 1129-1147. Doi: 10.3934/cpaa.2011.10.1129

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A continuum of extinction rates for the fast diffusion equation

1.
Department of Applied Mathematics and Statistics, Comenius University, 842 48 Bratislava
2.
Departamento de Matemáticas, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid
3.
Fakultät für Mathematik, Universität Duisburg-Essen, 45117 Essen

Received: August 31, 2010

Revised: October 31, 2011

Published: June 2011

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Abstract

We find a continuum of extinction rates for solutions $u(y,\tau)\ge 0$ of the fast diffusion equation $u_\tau=\Delta u^m$ in a subrange of exponents $m\in (0,1)$. The equation is posed in $R^n$ for times up to the extinction time $T>0$. The rates take the form $\|u(\cdot,\tau)\|_\infty$ ~ $(T-\tau)^\theta$ for a whole interval of $\theta>0$. These extinction rates depend explicitly on the spatial decay rates of initial data.

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Mathematics Subject Classification: Primary: 35K65; Secondary: 35B40.

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