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

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,Λ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(M,\Lambda )$$\end{document}. 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 r1-2Mr-Λ3r2∂∂r,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} r\sqrt{1-\frac{2M}{r}-\frac{\Lambda }{3}r^2}\frac{\partial }{\partial r}, \end{aligned}$$\end{document}where ∂r\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial _r$$\end{document} 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 Λ=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda =0$$\end{document}, 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).