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

We obtain global Strichartz estimates for the solutions u of the wave equation 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−1(T)−zI)ψ1, 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.