Complete and energy blow-up in indefinite superlinear parabolic problems

Julián López-Gómez; Pavol Quittner

doi:10.3934/dcds.2006.14.169

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2006, Volume 14, Issue 1: 169-186. Doi: 10.3934/dcds.2006.14.169

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Complete and energy blow-up in indefinite superlinear parabolic problems

Julián López-Gómez^1, and
Pavol Quittner^2,

1.
Departamento de Matemática Aplicada, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040-Madrid
2.
Institute of Applied Mathematics and Statistics, Comenius University, Mlynská dolina, 84248 Bratislava

Received: April 2004

Revised: February 2005

Published: March 2006

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Abstract

We study the continuation of solutions of superlinear indefinite parabolic problems after the blow-up time. The nonlinearity is of the form $a(x)u^p$, where $p>1$ is subcritical and $a$ changes sign. Unlike the case $a>0$, the solutions will never blow up completely in the whole domain but only in a certain subdomain. In some cases we give a precise description of this subdomain. We also derive sufficient conditions for the blow-up of the associated energy.

Keywords:

Indeﬁnite superlinear parabolic problems,
a priori estimates,
complete blow-up,
single-point blow-up.

Mathematics Subject Classification: 35B45, 35J65.

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