Asymptotic decay for defocusing semilinear wave equations on Schwarzschild spacetimes

Abstract

In this paper, we study the long time dynamics of solutions to the defocusing semilinear wave equation ${\square }_g\varphi =|\varphi {|}^{p-1}\varphi$ on the Schwarzschild black hole spacetimes. For $\frac{1+\sqrt{17}}{2}<p<5$ and sufficiently smooth and localized initial data, we show that the solution decays like $|\varphi |\lesssim t^{-1+\epsilon }$ in the domain of outer communication. The proof relies on the $r$-weighted vector field method of Dafermos-Rodnianski together with the Strichartz estimates for linear waves by Marzuola-Metcalfe-Tataru-Tohaneanu.

arXiv comment33 pages, comments are welcome!