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 $$u(t,x)=\frac{1}{\lambda^{\frac{N-2}{2}}(t)}(Q+\e(t))(\frac{x}{\lambda(t)}) \mbox{with} \|\e(t),\pa_t\e(t)\|_{\dot{H}^1\times L^2}\ll1 $$ where $Q$ is the extremizing profile of the Sobolev embedding $\dot{H}^1\to L^{2^*}$, and a blow up speed $$\lambda(t)=(T-t)e^{-\sqrt{|\log (T-t)|}(1+o(1))} \mbox{as} t\to T.$$