Integrated Local Energy Decay for Waves with Time-Dependent Damping

We prove integrated local energy decay for solutions of the damped wave equation with time-dependent damping satisfying an appropriate generalization of the geometric control condition on asymptotically flat, stationary space-times. We first obtain a high frequency estimate, which we prove via a positive commutator estimate using an escape function explicitly constructed in terms of the damping around individual space-time trajectories. We combine the high frequency estimate with low and medium frequency results for the undamped problem, then we handle the damping term as a perturbation to obtain local energy decay.