Existence and stability of a blow-up solution with a new prescribed behavior for a heat equation with a critical nonlinear gradient term

TL;DR

This paper constructs and analyzes a finite-time blow-up solution for a heat equation with a critical nonlinear gradient term, providing a detailed blow-up profile and stability analysis.

Contribution

It introduces a new blow-up solution with prescribed behavior for a heat equation with a critical gradient nonlinearity, including stability and profile description.

Findings

Constructed a finite-time blow-up solution.

Provided a sharp description of the blow-up profile.

Proved the stability of the solution with respect to initial data.

Abstract

We consider the semilinear heat equation, to which we add a nonlinear gradient term, with a critical power. We construct a solution which blows up in finite time. We also give a sharp description of its blow-up profile. The proof relies on the reduction of the problem to a finite dimensional one, and uses the index theory to conclude. Thanks to the interpretation of the parameters of the finite-dimensional problem in terms of the blow-up time and point, we also show the stability of the constructed solution with respect to initial data. This note presents the results and the main arguments. For the details, we refer to our paper \cite{TZ15}.