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

We 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 $\exp(c\epsilon^{-2\sqrt{2}})$, for many space-times including the nontrapping exterior domain, nontrapping asymptotically Euclidean space and Schwarzschild space-time.