On stability of type II blow up for the critical NLW 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 in earlier work by Krieger-Schlag-Tataru are stable along a co-dimension three manifold of radial 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. Our method is based on Fourier techniques adapted to time dependent wave operators of the form \[ -\partial_t^2 + \partial_r^2 + \frac2r\partial_r +V(λ(t)r) \] for suitable monotone scaling parameters $\lambda (t)$ and potentials $V(r)$ with a resonance at zero.

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