Finite time blow up for critical wave equations in high dimensions

摘要

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 Delta u - partial derivative(t)(2)u + vertical bar u vertical bar(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. (c) 2005 Elsevier Inc. All rights reserved.