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Blow-up results for a
strongly perturbed semilinear heat equation: theoretical
analysis and numerical method
Van Tien Nguyen and Hatem Zaag |
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Vol. 9 (2016), No. 1, 229–257
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DOI: 10.2140/apde.2016.9.229
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Abstract |
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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
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Mathematical Subject Classification 2010
Primary: 35K10
Secondary: 35K58
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Milestones
Received: 25 November 2014
Revised: 20 August 2015
Accepted: 11 October 2015
Published: 10 February 2016
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Authors |
| Van Tien Nguyen | |
| Department of Mathematics New York University Abu Dhabi Saadiyat Island, PO Box 129188 Abu Dhabi United Arab Emirates |
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| Hatem Zaag | |
| LAGA, CNRS (UMR 7539) Université Paris 13, Sorbonne Paris Cité 93430 Villetaneuse France |
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