Asymptotic decay for defocusing semilinear wave equations on Schwarzschild spacetimes

In this paper, we study the long time dynamics of solutions to the defocusing semilinear wave equation ${\mathrm{\square }}_g\phi =\mathrm{\mid }\phi {\mathrm{\mid }}^{p-1}\phi$ 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 $\mathrm{\mid }\phi \mathrm{\mid }\lesssim t^{-1+\epsilon }$ 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.