Refined blow-up asymptotics for a perturbed nonlinear heat equation with a gradient and a non-local term

TL;DR

This paper refines the asymptotic analysis of a perturbed nonlinear heat equation, providing sharper estimates on the gradient at blow-up by improving previous construction methods.

Contribution

It introduces a refined construction method that yields sharper gradient estimates at blow-up for a perturbed nonlinear heat equation with non-local and gradient terms.

Findings

Sharper estimate on the gradient at blow-up

Improved construction method for blow-up solutions

Enhanced understanding of blow-up profile behavior

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 works [1, 2], we constructed a solution $u$ for that equation such that $u$ and $\nabla u$ both blow up at the origin and only there. We also gave the final blow-up profile. In this paper, we refine our construction method in order to get a sharper estimate on the gradient at blow-up.