Stable self-similar blow up for energy subcritical wave equations

TL;DR

This paper proves the stability of a self-similar blow-up solution for a class of energy subcritical wave equations, demonstrating robustness of blow-up behavior under small perturbations in the energy space.

Contribution

It establishes the stability of explicit self-similar blow-up solutions for energy subcritical wave equations, extending previous results and providing a robust method applicable to other blow-up problems.

Findings

Blow-up solution   stable under small energy perturbations.

Method applicable to other self-similar blow-up problems.

Complemented previous stability results by Merle and Zaag.

Abstract

We consider the semilinear wave equation \[ \partial_t^2 \psi-\Delta \psi=|\psi|^{p-1}\psi \] for $1<p\leq 3$ with radial data in $\R^{3}$. This equation admits an explicit spatially homogeneous blow up solution $\psi^T$ given by $$ \psi^T(t,x)=\kappa_p (T-t)^{-\frac{2}{p-1}} $$ where $T>0$ and $\kappa_p$ is a $p$-dependent constant. We prove that the blow up described by $\psi^T$ is stable against small perturbations in the energy topology. This complements previous results by Merle and Zaag. The method of proof is quite robust and can be applied to other self-similar blow up problems as well, even in the energy supercritical case.