## 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

#### 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.

#### Article information

Source
Anal. PDE, Volume 9, Number 1 (2016), 229-257.

Dates
Received: 25 November 2014
Revised: 20 August 2015
Accepted: 11 October 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1510843208

Digital Object Identifier
doi:10.2140/apde.2016.9.229

Mathematical Reviews number (MathSciNet)
MR3461306

Zentralblatt MATH identifier
1334.35148

Subjects
Primary: 35K10: Second-order parabolic equations
Secondary: 35K58: Semilinear parabolic equations

#### Citation

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

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