Construction and stability of a blow up solution for a nonlinear heat equation with a gradient term

TL;DR

This paper constructs a blow-up solution for a nonlinear heat equation with a gradient term, analyzing its stability through a reduction to finite dimensions and index theory.

Contribution

It introduces a method to construct and analyze the stability of blow-up solutions for nonlinear heat equations with gradient terms.

Findings

Successfully constructs a blow-up solution with a prescribed profile.

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

Reduces the problem to a finite dimensional one using self-similar variables.

Abstract

We consider a nonlinear heat equation with a gradient term. We construct a blow-up solution for this equation with a prescribed blow-up profile. For that, we translate the question in selfsimilar variables and reduce the problem to a finite dimensional one. We then solve the finite dimensional problem using index theory. The interpretation of the finite dimensional parameters allows us to derive the stability of the constructed solution with respect to initial data.