Vol. 6, No. 3, 2013

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Decay of linear waves on higher-dimensional Schwarzschild black holes

Volker Schlue

Vol. 6 (2013), No. 3, 515–600

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 gϕ = 0 on the domain of outer communications of the Schwarzschild spacetime manifold (mn,g) (where n 3 is the spatial dimension, and m > 0 is the mass of the black hole) the associated energy flux E[ϕ](Στ) through a foliation of hypersurfaces Στ (terminating at future null infinity and to the future of the bifurcation sphere) decays, E[ϕ](Στ) CDτ2, where C is a constant depending on n and m, and D < is a suitable higher-order initial energy on Σ0; moreover we improve the decay rate for the first-order energy to E[tϕ](ΣτR) CDδτ42δ for any δ > 0, where ΣτR denotes the hypersurface Στ truncated at an arbitrarily large fixed radius R < provided the higher-order energy Dδ on Σ0 is finite. We conclude our paper by interpolating between these two results to obtain the pointwise estimate |ϕ|ΣτR CDδτ3 2 δ. In this work we follow the new physical-space approach to decay for the wave equation of Dafermos and Rodnianski (2010).

decay, wave equation, Schwarzschild black hole, spacetime, higher dimensions, mathematical general relativity
Mathematical Subject Classification 2010
Primary: 35L05, 35Q75, 58J45, 83C57
Received: 28 March 2011
Revised: 4 June 2012
Accepted: 20 December 2012
Published: 11 July 2013
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