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Compactness and single-point blowup of positive solutions on bounded domains

Published online by Cambridge University Press:  14 November 2011

Stathis Filippas  and
Frank Merle
Stathis Filippas
Affiliation:
Université Cergy-Pontoise, Département de Mathematique, Avenue du Parc, 8 Le Campus, 95003 Cergy-Pontoise Cedex, France
Frank Merle
Affiliation:
Université Cergy-Pontoise, Département de Mathématique, Avenue du Pare, 8 Le Campus, 95003 Cergy-Pontoise Cedex, France and Université Paris VI, Lab. d'Analyse Numérique, 4, Place Jussieu, 75252 Cedex 05, France
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Abstract

This paper is concerned with the blowup of positive solutions of the semilinear heat equation

with zero boundary conditions. The domain Ω is supposed to be smooth, convex and bounded. We first show that, under the assumption that the initial data are uniformly monotone near the boundary, solutions that exist on the time interval (0, T form a compact family in a suitable topology. We then derive some localisation properties of these solutions. In particular, we discuss a general criterion, independent of the initial data, which in some cases ensures single-point blowup.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 127 , Issue 1 , 1997 , pp. 47 - 65
Copyright
Copyright © Royal Society of Edinburgh 1997

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