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

TL;DR

This paper analyzes blow-up solutions for a strongly perturbed semilinear heat equation, deriving blow-up rates, profiles, and classifying behaviors, supported by a new numerical mesh-refinement method.

Contribution

It introduces a Lyapunov functional approach for analyzing blow-up and develops a novel mesh-refinement algorithm applicable to non-scaling-invariant equations.

Findings

Derived explicit blow-up rates and profiles.

Classified all asymptotic behaviors at singularity.

Implemented a new numerical mesh-refinement technique.

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.