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 $Σ_0 = { t = 0 }$ and that through the null conformal boundaries $\mathfrak{H}^+ \cup \scri^+$ (respectively, $\mathfrak{H}^- \cup \scri^-$). 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.

💡 Research Summary

This paper establishes a conformal (geometric) scattering theory for the defocusing cubic semilinear wave equation on the exterior region of a Schwarzschild black hole. The equation under study is
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