Smooth type II blow up solutions to the four dimensional energy critical wave equation

Details Smooth type II blow up solutions to the four dimensional energy critical wave equation Smooth type II blow up solutions to the four dimensional energy critical wave equation

2010-10-08

arxiv.org/abs/1010.1768v1

Mathematics

Analysis of PDEs

48 pages, 9 figures

Scientific paper

We exhibit $\mathcal C^{\infty}$ type II blow up solutions to the focusing energy critical wave equation in dimension $N=4$. These solutions admit near blow up time a decomposiiton $u(t,x)=1/l(t)(Q+e(t))(x/l(t)}}$ with $|e(t),\pa_t e(t)|_{\dot{H}^1\times L^2}<<1$ where $Q$ is the extremizing profile of the Sobolev embedding $\dot{H}^1\subset L^{2^*}$, and a blow up speed $l(t)=(T-t)e^{-\sqrt{|\log (T-t)|}(1+o(1))}$ as $t\to T.$

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