On the stability of type I blow up for the energy super critical heat equation

TL;DR

This paper investigates the stability of type I blow-up solutions in the energy supercritical heat equation, introducing spectral analysis and bifurcation techniques to establish non-radial stability for localized initial data.

Contribution

It develops a bifurcation approach and spectral gap analysis to prove non-radial stability of self-similar blow-up solutions in energy supercritical regimes.

Findings

Spectral gap in weighted spaces controls stability.

Non-radial solutions are stable under localized initial data.

Provides a framework for existence and stability of blow-up solutions.

Abstract

We consider the energy super critical semilinear heat equation $$\partial_t u=\Delta u+u^{p}, \ \ x\in \mathbb R^3, \ \ p>5.$$ We first revisit the construction of radially symmetric backward self similar solutions and propose a bifurcation type argument which allows for a sharp control of the spectrum of the corresponding linearized operator in suitable weighted spaces. We then show how the sole knowledge of this spectral gap in weighted spaces implies the finite codimensional non radial stability of these solutions for smooth well localized initial data using energy bounds. The whole scheme draws a route map for the derivation of the existence and stability of self similar blow up in non radial energy super critical settings.