Analysis & PDE
- Anal. PDE
- Volume 9, Number 1 (2016), 229-257.
Blow-up results for a strongly perturbed semilinear heat equation: theoretical analysis and numerical method
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.
Anal. PDE, Volume 9, Number 1 (2016), 229-257.
Received: 25 November 2014
Revised: 20 August 2015
Accepted: 11 October 2015
First available in Project Euclid: 16 November 2017
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Nguyen, Van Tien; Zaag, Hatem. Blow-up results for a strongly perturbed semilinear heat equation: theoretical analysis and numerical method. Anal. PDE 9 (2016), no. 1, 229--257. doi:10.2140/apde.2016.9.229. https://projecteuclid.org/euclid.apde/1510843208