Morawetz Estimates Without Relative Degeneration and Exponential Decay on Schwarzschild–de Sitter Spacetimes

Abstract

AbstractWe 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 $$(M,\Lambda )$$ ( M , Λ ) . 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 $$\begin{aligned} r\sqrt{1-\frac{2M}{r}-\frac{\Lambda }{3}r^2}\frac{\partial }{\partial r}, \end{aligned}$$ r 1 - 2 M r - Λ 3 r 2 ∂ ∂ r , where $$\partial _r$$ ∂ 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 $$\Lambda =0$$ Λ = 0 , our commutation corresponds to the one introduced by Holzegel–Kauffman (A note on the wave equation on black hole spacetimes with small non-decaying first-order terms, 2020. arXiv:2005.13644).