Stable self-similar blow up for energy subcritical wave equations

Abstract

We consider the semilinear wave equation \[ \partial_t^2 ψ-Δψ=|ψ|^{p-1}ψ\] for $1<p\le 3$ with radial data in ${\mathbb{R}}^3$. 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.

some typos fixed