Single point blow-up and final profile for a perturbed nonlinear heat equation with a gradient and a non-local term

TL;DR

This paper studies a perturbed nonlinear heat equation with gradient and non-local terms, proving single point blow-up and identifying the final blow-up profile, extending previous work on blow-up solutions.

Contribution

It establishes the single point blow-up property and characterizes the final blow-up profile for a perturbed nonlinear heat equation with non-local and gradient terms.

Findings

Proved the solution blows up only at a single point.

Determined the asymptotic profile of the solution near blow-up.

Extended previous results to include gradient and non-local perturbations.

Abstract

We consider in this paper a perturbation of the standard semilinear heat equation by a term involving the space derivative and a non-local term. In some earlier work (Abdelhedi-Zaag JDE 2021), we constructed a blow-up solution for that equation, and showed that it blows up (at least) at the origin. We also derived the so called "intermediate blow-up profile". In this paper, we prove the single point blow-up property and determine the final blow-up profile.