Finite time blow up for critical wave equations in high dimensions

We prove that solutions to the critical wave equation (1.1) with dimension n⩾4 can not be global if the initial values are positive somewhere and nonnegative. This completes the solution to the famous Strauss conjecture about semilinear wave equations of the form . The rest of the cases, the lower-dimensional case n⩽3, and the sub or super critical cases were settled many years earlier by the work of several authors.