Stable blow up dynamics for energy supercritical wave equations

Author:

Donninger Roland,Schörkhuber Birgit

Abstract

We study the semilinear wave equation \[ t 2 ψ Δ ψ = | ψ | p 1 ψ \partial _t^2 \psi -\Delta \psi =|\psi |^{p-1}\psi \] for p > 3 p > 3 with radial data in three spatial dimensions. There exists an explicit solution which blows up at t = T > 0 t=T>0 given by \[ ψ T ( t , x ) = c p ( T t ) 2 p 1 , \psi ^T(t,x)=c_p (T-t)^{-\frac {2}{p-1}}, \] where c p c_p is a suitable constant. We prove that the blow up described by ψ T \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 leads to a solution which converges to ψ T \psi ^T as t T t\to T- in the backward lightcone of the blow up point ( t , r ) = ( T , 0 ) (t,r)=(T,0) .

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

Reference32 articles.

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