Sharp non-explicit blow-up profile for perturbed nonlinear heat equations with gradient terms

TL;DR

This paper analyzes blow-up solutions for perturbed nonlinear heat equations with gradient terms, establishing a single point blow-up property and detailed blow-up profiles using self-similar variables.

Contribution

It introduces a new analysis of blow-up profiles for perturbed heat equations with gradient terms, extending understanding of their blow-up behavior.

Findings

Proves single point blow-up property.

Determines the final blow-up profile.

Provides a sharper description of blow-up behavior.

Abstract

We consider a class of blow-up solutions for perturbed nonlinear heat equations involving gradient terms. We first prove the single point blow-up property for this equation and determine its final blow-up profile. We also give a sharper description of its blow-up behaviour, where we take as a profile some suitably chosen solution of the unperturbed semilinear heat equation. The proof relies on selfsimilar variables with involved arguments to control the gradient term.