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Stable blow up dynamics for energy supercritical wave equations

We study the semilinear wave equation $\displaystyle \partial _t^2 \psi -\Delta \psi =\vert\psi \vert^{p-1}\psi $ for $ p > 3$ with radial data in three spatial dimensions. There exists an explicit solution which blows up at $ t=T>0$ given by $\displaystyle \psi ^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 leads 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)$.

Published in:

Transactions Of The American Mathematical Society, 366, 4, 2167-2189

Publisher:

Providence, American Mathematical Society

ISSN:

0002-9947

Keywords:

Laboratories:

Record created 2012-10-03, last modified 2018-12-03