Stable blowup profile for a semilinear Heat Equation with spatially inhomogeneous nonlinearity

TL;DR

This paper investigates the stability of self-similar blowup solutions in a semilinear heat equation with inhomogeneous nonlinearity, analyzing spectral properties and stability for various coupling constants.

Contribution

It establishes finite co-dimension stability of blowup solutions for a range of coupling constants and analyzes the spectrum of the linearized operator in this context.

Findings

Proves stability for all admissible coupling constants c in (0,c*)

Analyzes the spectrum of the linearized operator and identifies stable blowup in the cubic-quintic case

Provides an upper bound on unstable eigenvalues using a modified GGMT criterion

Abstract

We study the focusing semilinear heat equation with an additional defocusing H\'enon-type nonlinearity, the coupling of which is measured by a constant $c >0$. For $c \in (0,c^*)$, the model admits a closed-form self-similar blowup solution in every space dimension $d \geq 1$. Restricting ourselves to the three-dimensional case, we study the stability of this solution under small non-radial perturbations. By working in intersection Sobolev spaces with additional angular regularity, we prove finite co-dimension stability for all admissible values of $c$. Furthermore, we analyze the spectrum of the underlying linearized operator and we prove stable blowup for the cubic-quintic case and $c$ sufficiently close to $c^*$. Finally, we discuss the situation for small values of $c$ and use a modified version of the classical GGMT criterion to give an upper bound on the number of unstable eigenvalues.