Construction of a blow-up solution for a perturbed nonlinear heat equation with a gradient and a non-local term

TL;DR

This paper constructs a blow-up solution for a perturbed nonlinear heat equation with gradient and non-local terms, providing its profile and employing spectral analysis and topological methods.

Contribution

It introduces a novel approach to handle non-local and gradient perturbations in blow-up solutions of nonlinear heat equations.

Findings

Existence of a blow-up solution with a specific profile

Method for controlling spectral directions using kernel properties

Application of topological arguments for spectral control

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. We prove the existence of a blow-up solution, and give its blow-up profile. Our proof relies on the following method: we linearize the equation (in similarity variables) around the expected profile, then, we control the nonpositive directions of the spectrum thanks to the decreasing properties of the kernel. Finally, we use a topological argument to control the positive directions of the spectrum.