Finite time blow up for critical wave equations in high dimensions

We prove that solutions to the critical wave equation below can not be global if the initial values are positive somewhere and nonnegative. This completes the solution to the famous blow up conjecture about critical semilinear wave equations of the form $\Delta u - \partial^2_t u + |u|^p = 0$ in dimensions $n \ge 4$. The lower dimensional case $n \le 3$ was settled many years earlier.