A Vector Field Method for Non-Trapping, Radiating Spacetimes

Abstract

We study the global decay properties of solutions to the linear wave equation in 1+3 dimensions on time-dependent, weakly asymptotically flat spacetimes. Assuming non-trapping of null geodesics and a local energy decay estimate, we prove that sufficiently regular solutions to this equation have bounded conformal energy. As an application we also show a conformal energy estimate with vector fields applied to the solution as well as a global $L^{\infty }$ decay bound in terms of a weighted norm on initial data. For solutions to the wave equation in these dynamical backgrounds, our results reduce the problem of establishing the classical pointwise decay rate t^{-3/2} in the interior and t^{-1} along outgoing null cones to simply proving that local energy decay holds.

37 pages. V2: typos corrected; changed definition of the higher norms in order to improve readability. V3: References updated. Published in the Journal of Hyperbolic Differential Equations

References in corpus (5)

• On the quasilinear wave equations in time dependent inhomogeneous media
• Decay for solutions of the wave equation on Kerr exterior spacetimes III: The full subextremal case |a| < M
• Angular Regularity and Strichartz Estimates for the Wave Equation
• A Vector Field Method for Radiating Black Hole Spacetimes
• The Null Condition and Global Existence for Nonlinear Wave Equations on Slowly Rotating Kerr Spacetimes

Cited by in corpus (4)

• A Vector Field Method for Radiating Black Hole Spacetimes
• The $r^p$-weighted energy method of Dafermos and Rodnianski in general asymptotically flat spacetimes and applications
• A Sufficient Condition for Blowup of the Nonlinear Klein-Gordon Equation with Positive Initial Energy in FLRW Spacetimes
• Semilinear wave equations on accelerated expanding FLRW spacetimes