Long-time existence for semilinear wave equations on asymptotically flat space-times

Abstract

ABSTRACTWe study the long-time existence of solutions to nonlinear wave equations with power-type nonlinearity (of order p) and small data, on a large class of (1+n)-dimensional nonstationary asymptotically flat backgrounds, which include the Schwarzschild and Kerr black hole space-times. Under the assumption that uniform energy bounds and a weak form of local energy estimates hold forward in time, we give lower bounds of the lifespan when n = 3,4 and p is not bigger than the critical one. The lower bounds for three-dimensional subcritical and four-dimensional critical cases are sharp in general. For the most delicate three-dimensional critical case, we obtain the first existence result up to , for many space-times including the nontrapping exterior domain, nontrapping asymptotically Euclidean space and Schwarzschild space-time.