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 u− 2 t u+|u|p=0. 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. © 2005 Elsevier Inc. All rights reserved.