Geometric scattering for nonlinear wave equations on the Schwarzschild metric

In this paper, we establish a conformal scattering theory for defocusing semilinear wave equations on Schwarzschild spacetime. We combine the energy and pointwise decay results for solutions obtained in \cite{Yang} with a Sobolev embedding on spacelike hypersurfaces to derive two-sided energy estimates between the energy flux of solutions through the Cauchy initial hypersurface ${\mathrm{\Sigma }}_0=\{t=0\}$ and that through the null conformal boundaries ${\mathfrak{H}}^{+}\cup {\text{scri}}^{+}$ (respectively, ${\mathfrak{H}}^{-}\cup {\text{scri}}^{-}$). By combining these estimates with the well-posedness of the Cauchy and Goursat problems for nonlinear wave equations, we construct a bounded linear and locally Lipschitz scattering operator that maps past scattering data to future scattering data.