Decay of linear waves on higher dimensional Schwarzschild black holes

Abstract: In this paper we consider solutions to the linear wave equation on higher dimensional Schwarzschild black hole spacetimes and prove robust nondegenerate energy decay estimates that are in principle required in a nonlinear stability problem. More precisely, it is shown that for solutions to the wave equation \Box_g\phi=0 on the domain of outer communications of the Schwarzschild spacetime manifold (M^n_m, g) (where n >= 3 is the spatial dimension, and m > 0 is the mass of the black hole) the associated energy flux E\phi through a foliation of hypersurfaces (\Sigma_\tau) (terminating at future null infinity and to the future of the bifurcation sphere) decays, E\phi <= CD/\tau^2, where C is a constant only depending on n and m, and D < \infty is a suitable higher order initial energy on \Sigma_0; moreover we improve the decay rate for the first order energy to E\partial_t\phi <= CD/\tau^4-2\elta) for any \delta > 0 where \Sigma_\tau^R denotes the hypersurface (\Sigma_\tau) truncated at an arbitrarily large fixed radius R < \infty provided the higher order energy D_\delta on \Sigma_0 is finite. We conclude our paper by interpolating between these two results to obtain the pointwise estimate |\phi|{\Sigma\tau^R} <= (C D'_\delta) / \tau^3/2-\elta). In this work we follow the new physical-space approach to decay for the wave equation of Dafermos and Rodnianski.