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Regional blow-up for a higher-order semilinear parabolic equation

Published online by Cambridge University Press:  28 November 2001

MANUELA CHAVES  and
VICTOR A. GALAKTIONOV
MANUELA CHAVES
Affiliation:
Department of Mathematics, Autonoma University of Madrid, 28049 Madrid, Spain email: [email protected]
VICTOR A. GALAKTIONOV
Affiliation:
Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK email: [email protected] Keldysh Institute of Applied Mathematics, Miusskaya Sq. 4, 125047 Moscow, Russia
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Abstract

We study the blow-up behaviour of solutions of a 2mth order semilinear parabolic equation

[formula here]

with a superlinear function q(u) for |u| Gt; 1. We prove some estimates on the asymptotic blow-up behaviour. Such estimates apply to general integral evolution equations. We answer the following question: find a continuous function q(u) with a superlinear growth as u → ∞ such that the parabolic equation exhibits regional blow-up in a domain of finite non-zero measure. We show that such a regional blow-up can occur for q(u) = u|ln|u‖2m. We present a formal asymptotic theory explaining that the stable (generic) blow-up behaviour as t → T− is described by the self-similar solution

[formula here]

of the complex Hamilton–Jacobi equation

[formula here].

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 12 , Issue 5 , October 2001 , pp. 601 - 623
Copyright
2001 Cambridge University Press

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