LONG TIME EXISTENCE FOR SEMILINEAR WAVE EQUATIONS ON ASYMPTOTICALLY FLAT SPACE-TIMES

Abstract. We study the long time existence of solutions to semilinear wave equations of the form u = |u|p with small data, on a large class of (1 + n)-dimensional nonstationary asymptotically flat backgrounds, which models the 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. 1