Stabilization towards a singular steady state with gradient blow-up for a diffusion-convection problem

Philippe Souplet; Juan-Luis Vázquez

doi:10.3934/dcds.2006.14.221

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2006, Volume 14, Issue 1: 221-234. Doi: 10.3934/dcds.2006.14.221

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Stabilization towards a singular steady state with gradient blow-up for a diffusion-convection problem

Philippe Souplet^1, and
Juan-Luis Vázquez^2,

1.
Département de Mathématiques, Université de Picardie, INSSET, 02109 St-Quentin
2.
Departamento de Matemáticas, Universidad Autónoma de Madrid, 28046 Madrid

Received: October 2004

Revised: February 2005

Published: March 2006

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Abstract

This paper is devoted to analyze a case of singularity formation in infinite time for a semilinear heat equation involving linear diffusion and superlinear convection. A feature to be noted is that blow-up happens not for the main unknown but for its derivative. The singularity builds up at the boundary. The formation of inner and outer regions is examined, as well as the matching between them. As a consequence, we obtain the precise exponential rates of blow-up in infinite time.

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Mathematics Subject Classification: 35K60, 35K65, 35B45.

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