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

Consider the wave equation ${\mathrm{\partial }}_t^{2}u-{\mathrm{\Delta }}_{x}u+V(t,x)u=0$, where $x\in {\mathbb{R}}^n$ with $n\ge 3$ and $V(t,x)$ is $T$-periodic in time and decays exponentially 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 of the Floquet operator $U(T,0)$ defined for $\text{im}(\theta )>BT$ with $B>0$ sufficiently large. We establish a meromorphic continuation of $R(\theta )$ from which we deduce the asymptotic expansion of $e^{-(D+ϵ)<x>}U(t,0)e^{-D<x>}f$, where $f\in \dot{H}^1(\R^n)\times L^2(\R^n)$,as$t\to+\infty$ with a remainder term whose energy decays exponentially when $n$ is odd and a remainder term whose energy is bounded with respect 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 ${\theta\in\C\ :
\im(\theta)\geq0}$andisboundedfor$\theta\to0$, we obtain local energy decay as well as global Strichartz estimates for the solutions of ${\mathrm{\partial }}_t^{2}u-{\mathrm{\Delta }}_{x}u+V(t,x)u=F(t,x)$.