Stable blow up dynamics for energy supercritical wave equations

Abstract:

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