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

Abstract

We obtain local energy decay as well as global Strichartz estimates for the solutions $u$ of the wave equation ${\partial }_t^{2}u-di{v}_x(a(t,x){\nabla }_{x}u)=0,t\in \mathbb{R},x\in {\mathbb{R}}^n,$ 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 $R_{\chi }(\theta )=\chi (\mathcal{U}(T,0)-e^{-i\theta }{)}^{-1}\chi$, where $\mathcal{U}(T,0)$ is the monodromy operator and $T$ the period of $a(t,x)$, admits an holomorphic continuation to $\{\theta \in \mathbb{C}:\text{Im}(\theta )\ge 0\}$, for $n\ge 3$ , odd, and to $\{\theta \in \mathbb{C}:\text{Im}(\theta )\ge 0,\theta \ne 2k\pi -i\mu ,k\in \mathbb{Z},\mu \ge 0\}$ for $n\ge 4$, even, and for $n\ge 4$ even $R_{\chi }(\theta )$ is bounded in a neighborhood of $\theta =0$.