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$On Stability of Type II Blow Up for the Critical Nonlinear Wave Equation in $\mathbb {R}^{3+1}$, Paperback / softback Book$

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

by Joachim K Krieger

Part of the Memoirs of the American Mathematical Society series

Paperback / softback

Available to Order - This title is available to order, with delivery expected within 2 weeks

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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 2r\partial _r +V(\lambda (t)r) $ for suitable monotone scaling parameters $\lambda (t)$ and potentials $V(r)$ with a resonance at zero.