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Admissible Lp norms for local existence and for continuation in semilinear parabolic systems are not the same

Published online by Cambridge University Press:  12 July 2007

Pavol Quittner  and
Philippe Souplet
Pavol Quittner
Affiliation:
Institute of Applied Mathematics, Comenius University, Mlynská dolina, 84248 Bratislava, Slovakia ([email protected])
Philippe Souplet
Affiliation:
Département de Mathéematiques, INSSET, Université de Picardie, 02109 St-Quentin, France and Laboratoire de Mathématiques Appliquées, UMR CNRS 7641, Université de Versailles, 45 avenue des Etats-Unis, 78035 Versailles, France ([email protected])
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Abstract

We say that a Banach space E is a continuation space for a given parabolic problem if the E-norm of any non-global solution has to become unbounded. We will prove that for large classes of parabolic systems of two equations, the space E = Lr1 × Lr2 can be a continuation space even though the problem is not locally well posed in E. This stands in contrast with classical results for analogous scalar equations.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 131 , Issue 6 , December 2001 , pp. 1435 - 1456
Copyright
Copyright © Royal Society of Edinburgh 2001

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