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

Abstract: 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 $t\to T.$