Local energy decay for the wave equation with a time-periodic non-trapping metric and moving obstacle

consider the mixed problem with dirichelet condition associated to the wave equation , where the scalar metric periodic in t and uniformly equal to 1 outside a compact set in x, on a t-periodic domain. let be the associated propagator. assuming that the perturbations are non-trapping, we prove the meromorphic continuation of the cut-off resolvent of the floquet operator and we establish sufficient conditions for local energy decay.