Type II blow up solutions with optimal stability properties for the critical focussing nonlinear wave equation on R^{3+1}

Abstract

We show that the finite time type II blow up solutions for the energy critical nonlinear wave equation \[ \Box u = -u^5 \] on $\mathbb R^{3+1}$ constructed by Krieger-Schlag-Tataru are stable along a co-dimension one Lipschitz manifold of data perturbations in a suitable topology, provided the scaling parameter $\lambda(t) = t^{-1-\nu}$ is sufficiently close to the self-similar rate, i. e. $\nu>0$ is sufficiently small. This result is qualitatively optimal in light of a result by Krieger-Nakanishi-Schlag. The paper builds on the analysis in an earlier paper by the second author.