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Global blow-up for a semilinear heat equation on a subspace

Part of: Parabolic equations and systems

Published online by Cambridge University Press:  24 August 2015

C. J. Budd ,
J. W. Dold  and
V. A. Galaktionov
C. J. Budd
Affiliation:
Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK (mascjb@bath.ac.uk)
J. W. Dold
Affiliation:
School of Mathematics, Alan Turing Building, Upper Brook Street, University of Manchester, Manchester M13 9PL, UK (john.dold@manchester.ac.uk)
V. A. Galaktionov
Affiliation:
Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK (masvg@bath.ac.uk)
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Abstract

We study the asymptotic behaviour as t → T–, near a finite blow-up time T > 0, of decreasing-in-x solutions to the following semilinear heat equation with a non-local term:

with Neumann boundary conditions and strictly decreasing initial function u0(x) with zero mass. We prove sharp estimates for u(x, t) as t → T–, revealing a non-uniform global blow-up:

uniformly on any compact set [δ, 1], δ ∈ (0, 1).

Keywords

non-local parabolic equationintegral constraintglobal blow-upasymptotic behaviour

MSC classification

Primary: 35K55: Nonlinear parabolic equations
Secondary: 35K65: Degenerate parabolic equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 145 , Issue 5 , October 2015 , pp. 893 - 923
Copyright
Copyright © Royal Society of Edinburgh 2015 

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