Stable blow up dynamics for energy supercritical wave equations

Abstract

We study the semilinear wave equation \[ \partial_t^2 ψ-Δψ=|ψ|^{p-1}ψ\] for $p > 3$ with radial data in three spatial dimensions. There exists an explicit solution which blows up at $t=T>0$ given by \[ ψ^T(t,x)=c_p (T-t)^{-\frac{2}{p-1}} \] where $c_p$ is a suitable constant. We prove that the blow up described by $ψ^T$ is stable in the sense that there exists an open set (in a topology strictly stronger than the energy) of radial initial data that lead to a solution which converges to $ψ^T$ as $t\to T-$ in the backward lightcone of the blow up point $(t,r)=(T,0)$.