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.

References in corpus (3)

• Stable blow up dynamics for the 1-corotational energy critical harmonic heat flow
• Isolatedness of characteristic points at blow-up for a semilinear wave equation in one space dimension
• Stable blow up dynamics for energy supercritical wave equations

Cited by in corpus (6)

• Construction of type II blow-up solutions for the energy-critical wave equation in dimension 5
• Strongly interacting blow up bubbles for the mass critical NLS
• On the stability of type II blowup for the 1-corotational energy supercritical harmonic heat flow
• On strongly anisotropic type II blow up
• Stable type II blowup for the 7 dimensional 1-corotational energy supercritical harmonic map heat flow
• Blow up dynamics for the hyperbolic vanishing mean curvature flow of surfaces asymptotic to Simons cone