Vol. 9, No. 1, 2016

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Blow-up results for a strongly perturbed semilinear heat equation: theoretical analysis and numerical method

Van Tien Nguyen and Hatem Zaag

Vol. 9 (2016), No. 1, 229–257
DOI: 10.2140/apde.2016.9.229
Abstract

We consider a blow-up solution for a strongly perturbed semilinear heat equation with Sobolev subcritical power nonlinearity. Working in the framework of similarity variables, we find a Lyapunov functional for the problem. Using this Lyapunov functional, we derive the blow-up rate and the blow-up limit of the solution. We also classify all asymptotic behaviors of the solution at the singularity and give precise blow-up profiles corresponding to these behaviors. Finally, we attain the blow-up profile numerically, thanks to a new mesh-refinement algorithm inspired by the rescaling method of Berger and Kohn. Note that our method is applicable to more general equations, in particular those with no scaling invariance.

Keywords
blow-up, Lyapunov functional, asymptotic behavior, blow-up profile, semilinear heat equation, lower-order term
Mathematical Subject Classification 2010
Primary: 35K10
Secondary: 35K58
Milestones
Received: 25 November 2014
Revised: 20 August 2015
Accepted: 11 October 2015
Published: 10 February 2016
Authors
Van Tien Nguyen
Department of Mathematics
New York University Abu Dhabi
Saadiyat Island, PO Box 129188
Abu Dhabi
United Arab Emirates
Hatem Zaag
LAGA, CNRS (UMR 7539)
Université Paris 13, Sorbonne Paris Cité
93430 Villetaneuse
France