Blow-up of solutions of nonlinear wave equations in three space dimensions

Let u(x,t) be a solution, uA|u|^p for xIR₃, t0 where is the d'Alembertian, and A, p are constants with A>0, 12. It is shown that the support of u is contained in the cone 0tt₀–|x–x⁰|, if the initial data u(x,0), u_t(x,0) have their support in the ball |x–x⁰|t₀. In particular global solutions of u=A|u|^p with initial data of compact support vanish identically. On the other hand for A>0, p>1+2 global solutions of u=A|u|^p exist, if the initial data are of compact support and u is sufficiently small in a suitable norm. For p=2 the time at which u becomes infinite is of order u^–2.Dedicated to Hans Lewy and Charles B. Morrey, Jr.The research for this paper was performed at the Courant Institute and supported by the Office of Naval Research under Contract No. N00014-76-C-0301. Reproduction in whole or part is permitted for any purpose of the United States Government.