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
(Mnm,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)≤CD/τ2, 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](ΣRτ)≤CDδ/τ4−2δ for any
δ>0, where
ΣRτ 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∣∣ΣRτ≤CD'δ/τ32−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
