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Consider the wave equation $\partial_t^2u-\Delta_xu+V(t,x)u=0$, where$x\in\R^n$ with $n\geq3$ and $V(t,x)$ is $T$-periodic in time and decaysexponentially in space. Let $ U(t,0)$ be the associated propagator and let$R(\theta)=e^{-D<x>}(U(T,0)-e^{-i\theta})^{-1}e^{-D<x>}$ be the resolvent ofthe Floquet operator $U(T,0)$ defined for $\im(\theta)>BT $ with $B>0$sufficiently large. We establish a meromorphic continuation of $R(\theta)$ fromwhich we deduce the asymptotic expansion of$e^{-(D+\epsilon)<x>}U(t,0)e^{-D<x>}f$, where $f\in \dot{H}^1(\R^n)\timesL^2(\R^n)$, as $t\to+\infty$ with a remainder term whose energy decaysexponentially when $n$ is odd and a remainder term whose energy is bounded withrespect to $t^l\log(t)^m$, with $l,m\in\mathbb Z$, when $n$ is even. Then,assuming that $R(\theta)$ has no poles lying in $\{\theta\in\C\ :\\im(\theta)\geq0\}$ and is bounded for $\theta\to0$, we obtain local energydecay as well as global Strichartz estimates for the solutions of$\partial_t^2u-\Delta_xu+V(t,x)u=F(t,x)$.

On the meromorphic continuation of the resolvent for the wave equation with time-periodic perturbation and applications