Local energy decay in even dimensions for the wave equation with a time-periodic non-trapping metric and applications to Strichartz estimates

We obtain local energy decay as well as global Strichartz estimates for the solutions $u$ of the wave equation $\partial_t^2 u-div_x(a(t,x)\nabla_xu)=0,
t\in{\R},\ x\in{\R}^n,$withtime-periodicnon-trappingmetric$a(t,x)$ equal to $1$ outside a compact set with respect to $x$. We suppose that the cut-off resolvent $R_{\chi }(\theta )=\chi (\mathcal{U}(T,0)-e^{-i\theta }{)}^{-1}\chi$, where $\mathcal{U}(T,0)$ is the monodromy operator and $T$ the period of $a(t,x)$, admits an holomorphic continuation to ${\theta\in$\mathbb{C}$\ :
$\text{Im}$(\theta) \geq 0}$,for$n \geq 3$,odd,andto${ \theta\in\mathbb C\ :\ $\text{Im}$(\theta)\geq0,\ \theta\neq 2k\pi-i\mu,
k\in$\mathbb{Z}$,\ \mu\geq0}$for$n \geq4$,even,andfor$n \geq4$even$R_\chi(\theta)$isboundedinaneighborhoodof$\theta=0$.