DECAY OF LINEAR WAVES ON HIGHER-DIMENSIONAL SCHWARZSCHILD BLACK HOLES

Abstract

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 on the domain of outer communications of the Schwarzschild spacetime manifold the associated energy flux through a foliation of hypersurfaces decays, E[φ](Σ(τ))≤ CD/τ², where C is a constant, and D is a suitable higher-order initial energy on Σ(0); moreover we improve the decay rate for the first-order energy. We conclude our paper by interpolating between these two results to obtain the pointwise est..