A blow-up result for a nonlinear wave equation on manifolds: the critical case

Abstract

We consider a inhomogeneous semilinear wave equation on a noncompact complete Riemannian manifold of dimension , without boundary. The reaction exhibits the combined effects of a critical term and of a forcing term. Using a rescaled test function argument together with appropriate estimates, we show that the equation admits no global solution. Moreover, in the special case when , our result improves the existing literature. Namely, our main result is valid without assuming that the initial values are compactly supported.