Morawetz estimates without relative degeneration and exponential decay on Schwarzschild-de Sitter spacetimes

Abstract

We use a novel physical space method to prove relatively non-degenerate integrated energy estimates for the wave equation on subextremal Schwarzschild-de Sitter spacetimes with parameters $(M,\Lambda)$. These are integrated decay statements whose bulk energy density, though degenerate at highest order, is everywhere comparable to the energy density of the boundary fluxes. As a corollary, we prove that solutions of the wave equation decay exponentially on the exterior region. The main ingredients are a previous Morawetz estimate of Dafermos-Rodnianski and an additional argument based on commutation with a vector field which can be expressed in the form $r\sqrt{1-\frac{2M}{r}-\frac{\Lambda}{3}r^2}\frac{\partial}{\partial r}$, where $\partial_r$ here denotes the coordinate vector field corresponding to a well chosen system of hyperboloidal coordinates. Our argument gives exponential decay also for small first order perturbations of the wave operator. In the limit $\Lambda=0$, our commutation corresponds to the one introduced by Holzegel-Kauffman.