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Abstract
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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
on
the domain of outer communications of the Schwarzschild spacetime manifold
(where
is the spatial
dimension, and
is the mass of the black hole) the associated energy flux
through a foliation
of hypersurfaces
(terminating at future null infinity and to the future of the bifurcation sphere) decays,
, where
is a constant
depending on
and
, and
is a suitable higher-order
initial energy on
;
moreover we improve the decay rate for the first-order energy to
for any
, where
denotes the hypersurface
truncated at an arbitrarily
large fixed radius
provided
the higher-order energy
on
is finite.
We conclude our paper by interpolating between these two results to obtain the pointwise
estimate
.
In this work we follow the new physical-space approach to decay for the wave
equation of Dafermos and Rodnianski (2010).
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Keywords
decay, wave equation, Schwarzschild black hole, spacetime,
higher dimensions, mathematical general relativity
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Mathematical Subject Classification 2010
Primary: 35L05, 35Q75, 58J45, 83C57
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Milestones
Received: 28 March 2011
Revised: 4 June 2012
Accepted: 20 December 2012
Published: 11 July 2013
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