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arXiv · 1410.4079

Blow-up results for a strongly perturbed semilinear heat equation: Theoretical analysis and numerical method

Van Tien Nguyen·Hatem Zaag
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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 precisely 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 in 1988. Note that our method is applicable to more general equations, in particular those with no scaling invariance.

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Van Tien Nguyen, Hatem Zaag. 2014-10-15. Blow-up results for a strongly perturbed semilinear heat equation: Theoretical analysis and numerical method. https://doi.org/10.2140/apde.2016.9.229

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