The decay of global solutions of a semilinear heat equation

Pavol Quittner

doi:10.3934/dcds.2008.21.307

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2008, Volume 21, Issue 1: 307-318. Doi: 10.3934/dcds.2008.21.307

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The decay of global solutions of a semilinear heat equation

Pavol Quittner^1,

1.
Department of Applied Mathematics and Statistics, Comenius University, Mlynská dolina, 84248 Bratislava

Received: November 30, 2006

Revised: July 31, 2007

Published: December 2007

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Abstract

We are interested in the time decay estimates of global solutions of the semilinear parabolic equation $u_t= \Delta u+|u|^{p-1}u$ in $\R^N\times\R^+$, where $p>1$. We find several new sufficient and/or necessary conditions guaranteeing that the solution for $t$ large behaves like the solution of the linear heat equation or has the self-similar decay. We are particularly interested in the behaviour of threshold solutions lying on the borderline between global existence and blow-up.

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Mathematics Subject Classification: Primary: 35K55, 35K57, 35B40; Secondary: 35B45.

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