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arXiv · 1012.5963

Decay of linear waves on higher dimensional Schwarzschild black holes

Volker Schlue
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Abstract

In this paper 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 \Box_gϕ=0 on the domain of outer communications of the Schwarzschild spacetime manifold (M^n_m, g) (where n >= 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) <= CD/τ^2, where C is a constant only 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_tϕ](Σ_τ^R) <= CD/τ^(4-2δ) 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_δon Σ_0 is finite. We conclude our paper by interpolating between these two results to obtain the pointwise estimate |ϕ|_{Σ_τ^R} <= (C D'_δ) / τ^(3/2-δ). In this work we follow the new physical-space approach to decay for the wave equation of Dafermos and Rodnianski.

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Volker Schlue. 2010-12-29. Decay of linear waves on higher dimensional Schwarzschild black holes. https://doi.org/10.2140/apde.2013.6.515

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