Local decay of waves on asymptotically flat stationary space-times

Abstract

In this article we study the pointwise decay properties of solutions to the wave equation on a class of stationary asymptotically flat backgrounds in three space dimensions. Under the assumption that uniform energy bounds and a weak form of local energy decay hold forward in time we establish a $t^{-3}$ local uniform decay rate for linear waves. This work was motivated by open problems concerning decay rates for linear waves on Schwarzschild and Kerr backgrounds, where such a decay rate has been conjectured by R. Price. Our results apply to both of these cases.

33 pages; minor corrections, updated references

References in corpus (12)

• A proof of Price's Law on Schwarzschild black hole manifolds for all angular momenta
• Hidden symmetries and decay for the Vlasov equation on the Kerr spacetime
• Improved decay for solutions to the linear wave equation on a Schwarzschild black hole
• Semilinear wave equations on the Schwarzschild manifold I: Local decay estimates
• A note on energy currents and decay for the wave equation on a Schwarzschild background
• Decay Rates for Spherical Scalar Waves in the Schwarzschild Geometry
• Local energy estimate on Kerr black hole backgrounds
• Strichartz estimates on Kerr black hole backgrounds
• Weighted-$L^{\infty }$ and pointwise space-time decay estimates for wave equations with potentials and initial data of low regularity
• Global parametrices and dispersive estimates for variable coefficient wave equations
• Semiclassical resolvent estimates in chaotic scattering
• Decay estimates for variable coefficient wave equations in exterior domains

Cited by in corpus (3)

• Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces
• The Wave Equation on Extreme Reissner-Nordström Black Hole Spacetimes: Stability and Instability Results
• Asymptotic Behavior of Massless Dirac Waves in Schwarzschild geometry