Type II blow up manifolds for the energy supercritical wave equation

Abstract: We consider the non linear focusing wave equation $\partial_{tt}u-\Delta u-u|u|^{p-1}=0$ in large dimensions and for radially symmetric data, in the energy supercritical zone for p large enough. We construct finite time blow up solutions that concentrate the soliton profile $u \sim \frac{1}{\lambda^{\frac{2}{p-1}}}Q (\frac{r} {\lambda})$, $Q$ being a special stationary state of the equation. The blow up is of type II i.e all norms below scaling remain bounded. The scale shrinks at quantized blow up rates: we exhibit a countable family of asymptotic behaviors near blow up time for the scale lambda. For each rate we are able to prove that there exists a Lipschitz manifold with explicit codimension of initial data leading to such solutions. Our analysis adapts the robust energy method developed for the study of energy critical bubbles and the analogous result for the energy supercritical NLS problem. This work is also related to similar solutions for the non linear heat equation for which the existence was already known, but with some techniques that could not be adapted to the wave equation.