Stable blow up dynamics for energy supercritical wave equations

Abstract

We study the semilinear wave equation \[ \partial_t^2 ψ-Δψ=|ψ|^{p-1}ψ\] for $p>3$ with radial data in three spatial dimensions. There exists an explicit solution which blows up at $t=T>0$ given by \[ ψ^T(t,x)=c_p (T-t)^{-\frac{2}{p-1}} \] where $c_p$ is a suitable constant. We prove that the blow up described by ${\psi }^T$ is stable in the sense that there exists an open set (in a topology strictly stronger than the energy) of radial initial data that lead to a solution which converges to ${\psi }^T$ as $t\to T-$ in the backward lightcone of the blow up point $(t,r)=(T,0)$.

References in corpus (6)

• From resolvent bounds to semigroup bounds
• Global dynamics away from the ground state for the energy-critical nonlinear wave equation
• Construction of a multi-soliton blow-up solution to the semilinear wave equation in one space dimension
• Blowup behaviour for the nonlinear Klein--Gordon equation
• Global dynamics of the nonradial energy-critical wave equation above the ground state energy
• Stable self-similar blowup in energy supercritical Yang-Mills theory

Cited by in corpus (2)

• Type II blow up manifolds for the energy supercritical wave equation
• Nondispersive decay for the cubic wave equation