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 $\partial_t^2 u-div_x(a(t,x)\nabla_xu)=0,\ t\in{\R},\ x\in{\R}^n,$ with time-periodic non-trapping metric $a(t,x)$ equal to $1$ outside a compact set with respect to $x$. We suppose that the cut-off resolvent $R_χ(θ)=χ(\mathcal U(T, 0)-e^{-iθ})^{-1}χ$, where $\mathcal U(T, 0)$ is the monodromy operator and $T$ the period of $a(t,x)$, admits an holomorphic continuation to $\{θ\in\mathbb{C}\ :\ \textrm{Im}(θ) \geq 0\}$, for $n \geq 3$ , odd, and to $\{ θ\in\mathbb C\ :\ \textrm{Im}(θ)\geq0,\ θ\neq 2kπ-iμ,\ k\in\mathbb{Z},\ μ\geq0\}$ for $n \geq4$, even, and for $n \geq4$ even $R_χ(θ)$ is bounded in a neighborhood of $θ=0$.

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