Dirichlet boundary conditions can prevent blow-up in reaction-diffusion equations and systems

Marek Fila; Hirokazu Ninomiya; Juan-Luis Vázquez

doi:10.3934/dcds.2006.14.63

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2006, Volume 14, Issue 1: 63-74. Doi: 10.3934/dcds.2006.14.63

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Dirichlet boundary conditions can prevent blow-up in reaction-diffusion equations and systems

1.
Department of Applied Mathematics and Statistics, Comenius University, 842 48 Bratislava
2.
Department of Applied Mathematics and Informatics, Ryukoku University, Seta, Otsu 520-2194
3.
Departamento de Matemáticas, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid

Received: September 30, 2004

Revised: January 31, 2005

Published: February 2006

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Abstract

This paper examines the following question: Suppose that we have a reaction-diffusion equation or system such that some solutions which are homogeneous in space blow up in finite time. Is it possible to inhibit the occurrence of blow-up as a consequence of imposing Dirichlet boundary conditions, or other effects where diffusion plays a role? We give examples of equations and systems where the answer is affirmative.

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Mathematics Subject Classification: Primary: 35K57, 35K50; Secondary: 35B35.

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