Global Strichartz estimates for the wave equation with time-periodic potentials

We obtain global Strichartz estimates for the solutions u of the wave equation (∂2 t − x + V (t, x))u = F(t,x) for time-periodic potentials V (t,x) with compact support with respect to x. Our analysis is based on the analytic properties of the cut-off resolvent Rχ (z) = χ(U(T ) − zI )−1ψ1, where U(T ) = U(T, 0) is the monodromy operator and T >0 the period of V (t,x). We show that if Rχ (z) has no poles z ∈ C, |z| 1, then for n 3, odd, we have a exponential decal of local energy. For n 2, even, we obtain also an uniform decay of local energy assuming that Rχ (z) has no poles z ∈ C, |z| 1, and Rχ (z) remains bounded for z in a small neighborhood of 0. © 2005 Elsevier Inc. All rights reserved