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 decomposition

$$u\left(t,x\right)=\frac{1}{{\lambda }^{\left(N-2\right)∕2}\left(t\right)}\left(Q+\epsilon \left(t\right)\right)\left(\frac{x}{\lambda \left(t\right)}\right),with\parallel \epsilon \left(t\right),{\partial }_t\epsilon \left(t\right){\parallel }_{{Ḣ}^1\times L^2}\ll 1,$$

where $Q$ is the extremizing profile of the Sobolev embedding ${Ḣ}^1\to L^{2^{\ast }}$, and a blow-up speed

$$\lambda \left(t\right)=\left(T-t\right)e^{-\sqrt{|log\left(T-t\right)|}\left(1+o\left(1\right)\right)}\text{ as }t\to T.$$