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

TL;DR

This paper proves the stability of certain finite-time blow-up solutions for the energy-critical nonlinear wave equation in three spatial dimensions, showing they are stable under small perturbations near a self-similar scaling rate.

Contribution

It establishes the stability of Type II blow-up solutions for the critical wave equation along a co-dimension one Lipschitz manifold, near a self-similar scaling, extending previous analysis.

Findings

Stability of blow-up solutions under perturbations

Construction of a Lipschitz manifold of initial data

Optimality of stability results in the critical setting

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.