Type Ii Blow Up Solutions With Optimal Stability Properties For The Critical Focussing Nonlinear Wave Equation On Mathbb R 3 1

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

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On Stability of Type II Blow Up for the Critical Nonlinear Wave Equation in $mathbb {R}^{3+1}$

The author shows 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 Krieger, Schlag, and Tataru (2009) and Krieger and Schlag (2014) 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 + frac 2rpartial _r +V(lambda (t)r) $ for suitable monotone scaling parameters $lambda (t)$ and potentials $V(r)$ with a resonance at zero.

Infinite Time Blow-up Solutions to the Energy Critical Wave Maps Equation