Decay of linear waves on higher dimensional Schwarzschild black holes

Dec, 2010
93 pages
Published in:
  • Anal.Part.Diff.Eq. 6 (2013) 3, 515-600
  • Published: Jul 11, 2013
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Abstract: (MSP)
We consider solutions to the linear wave equation on higher dimensionalSchwarzschild black hole spacetimes and prove robust nondegenerate energydecay estimates that are in principle required in a nonlinear stabilityproblem. More precisely, it is shown that for solutions to the wave equation□gϕ= 0 onthe domain of outer communications of the Schwarzschild spacetime manifold(ℳmn,g) (wheren≥ 3 is the spatialdimension, and m> 0is the mass of the black hole) the associated energy fluxE[ϕ](Στ) through a foliationof hypersurfaces Στ(terminating at future null infinity and to the future of the bifurcation sphere) decays,E[ϕ](Στ)≤CD∕τ2, whereC is a constantdepending on nand m, andD<∞ is a suitable higher-orderinitial energy on Σ0;moreover we improve the decay rate for the first-order energy toE[∂tϕ](ΣτR)≤CDδ∕τ4−2δ for anyδ> 0, whereΣτR denotes the hypersurfaceΣτ truncated at an arbitrarilylarge fixed radius R<∞ providedthe higher-order energy Dδon Σ0 is finite.We conclude our paper by interpolating between these two results to obtain the pointwiseestimate |ϕ|ΣτR≤CDδ′∕τ32−δ.In this work we follow the new physical-space approach to decay for the waveequation of Dafermos and Rodnianski (2010).
Note:
  • 93 pages, 7 figures
  • decay
  • wave equation
  • Schwarzschild black hole
  • spacetime,higher dimensions
  • mathematical general relativity
  • black hole: Schwarzschild
  • energy: decay
  • energy: flux
  • stability: nonlinear
  • black hole: higher-dimensional