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

TL;DR

This paper establishes long-term existence results for small data solutions to nonlinear wave equations on various asymptotically flat space-times, including Schwarzschild and Kerr, with sharp lifespan bounds in critical cases.

Contribution

It provides the first existence results up to exponential lifespan bounds for the critical three-dimensional case on several physically relevant space-times.

Findings

Lifespan bounds depend on energy and local energy estimates.

Sharp lower bounds for subcritical and critical cases in 3D and 4D.

First exponential lifespan result in the critical 3D case for multiple space-times.

Abstract

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.