Decay of linear waves on higher-dimensional Schwarzschild black holes

Schlue, Volker

doi:10.2140/apde.2013.6.515

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Decay of linear waves on higher-dimensional Schwarzschild black holes

Volker Schlue

Vol. 6 (2013), No. 3, 515–600

DOI: 10.2140/apde.2013.6.515

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 $□_{g} ϕ = 0$ on the domain of outer communications of the Schwarzschild spacetime manifold $(ℳ_{m}^{n}, g)$ (where $n \geq 3$ is the spatial dimension, and $m > 0$ is the mass of the black hole) the associated energy flux $E [ϕ] (Σ_{τ})$ through a foliation of hypersurfaces $Σ_{τ}$ (terminating at future null infinity and to the future of the bifurcation sphere) decays, $E [ϕ] (Σ_{τ}) \leq C D ∕ τ^{2}$ , where $C$ is a constant depending on $n$ and $m$ , and $D < \infty$ is a suitable higher-order initial energy on $Σ_{0}$ ; moreover we improve the decay rate for the first-order energy to $E [\partial_{t} ϕ] (Σ_{τ}^{R}) \leq C D_{δ} ∕ τ^{4 - 2 δ}$ for any $δ > 0$ , where $Σ_{τ}^{R}$ denotes the hypersurface $Σ_{τ}$ truncated at an arbitrarily large fixed radius $R < \infty$ provided the higher-order energy $D_{δ}$ on $Σ_{0}$ is finite. We conclude our paper by interpolating between these two results to obtain the pointwise estimate $| ϕ |_{Σ_{τ}^{R}} \leq C D_{δ}^{'} ∕ τ^{\frac{3}{2} - δ}$ . In this work we follow the new physical-space approach to decay for the wave equation of Dafermos and Rodnianski (2010).

Keywords

decay, wave equation, Schwarzschild black hole, spacetime, higher dimensions, mathematical general relativity

Mathematical Subject Classification 2010

Primary: 35L05, 35Q75, 58J45, 83C57

Milestones

Received: 28 March 2011

Revised: 4 June 2012

Accepted: 20 December 2012

Published: 11 July 2013

Authors

Volker Schlue
Department of Pure Mathematics and Mathematical Statistics University of Cambridge Cambridge, CB3 0WB United Kingdom	Department of Mathematics University of Toronto 40 St. George Street, Room 6120 Toronto, ON M5S 2E4 Canada

Vol. 6, No. 3, 2013