DECAY OF LINEAR WAVES ON HIGHER-DIMENSIONAL SCHWARZSCHILD BLACK HOLES

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 square(g)phi = 0 on the domain of outer communications of the Schwarzschild spacetime manifold. (M-m(n), g) (where n >= 3 is the spatial dimension, and m > 0 is the mass of the black hole) the associated energy flux E[phi](Sigma(tau)) through a foliation of hypersurfaces Sigma(tau) (terminating at future null infinity and to the future of the bifurcation sphere) decays, E[phi](Sigma(tau)) <= CD/tau(2), where C is a constant depending on n and m, and D < infinity is a suitable higher-order initial energy on Sigma(0); moreover we improve the decay rate for the first-order energy to E[partial derivative(t)phi](Sigma(R)(tau)) <= CD delta/tau(4-28) for any delta > 0, where Sigma(R)(tau) denotes the hypersurface Sigma(tau) truncated at an arbitrarily large fixed radius R < infinity 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 vertical bar phi vertical bar Sigma(R)(tau) <= CD'(delta)/tau(3/2-delta). In this work we follow the new physical-space approach to decay for the wave equation of Dafermos and Rodnianski (2010).