A note on the wave equation on black hole spacetimes with small non-decaying first order terms

Abstract

We present an elementary physical space argument to establish local integrated decay estimates for the perturbed wave equation ${\square }_g\varphi =\epsilon {\beta }^a{\partial }_a\varphi$ on the exterior of the Schwarzschild geometry $(\mathcal{M},g)$. Here $\beta$ is a regular vectorfield on $\mathcal{M}$ decaying suitably in space but not necessarily in time. The proof is formulated to cover also perturbations of the Regge--Wheeler equation.

7 pages

References in corpus (4)

• Semilinear wave equations on the Schwarzschild manifold I: Local decay estimates
• Errata for ``Global existence and scattering for the nonlinear Schrodinger equation on Schwarzschild manifolds'', ``Semilinear wave equations on the Schwarzschild manifold I: Local Decay Estimates'', and ``The wave equation on the Schwarzschild metric II: Local Decay for the spin 2 Regge Wheeler equation''
• Local energy estimate on Kerr black hole backgrounds
• The spin $\pm$1 Teukolsky equations and the Maxwell system on Schwarzschild