Integrated Local Energy Decay for the Damped Wave Equation on Stationary Space-Times

Abstract

We prove integrated local energy decay for the damped wave equation on stationary, asymptotically flat space-times in (1 + 3) dimensions. Local energy decay constitutes a powerful tool in the study of dispersive partial differential equations on such geometric backgrounds. By utilizing the geometric control condition to handle trapped trajectories, we are able to recover high frequency estimates without any loss. We may then apply known estimates from the work of Metcalfe, Sterbenz, and Tataru in the medium and low frequency regimes in order to establish local energy decay. This generalizes the integrated version of results established by Bouclet and Royer from the setting of asymptotically Euclidean manifolds to the full Lorentzian case.