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

TL;DR

This paper establishes local energy decay and Strichartz estimates for wave equations with time-periodic, non-trapping metrics in even dimensions, under specific resolvent continuation assumptions.

Contribution

It introduces new decay and estimate results for wave equations with periodic metrics, extending understanding in even dimensions with spectral assumptions.

Findings

Proves local energy decay for wave equations with periodic metrics.

Establishes global Strichartz estimates under spectral conditions.

Provides resolvent bounds near zero frequency for even dimensions.

Abstract

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 $R_\chi(\theta)=\chi(\mathcal U(T, 0)-e^{-i\theta})^{-1}\chi$, where $\mathcal U(T, 0)$ is the monodromy operator and $T$ the period of $a(t,x)$, admits an holomorphic continuation to $\{\theta\in\mathbb{C}\ :\ \textrm{Im}(\theta) \geq 0\}$, for $n \geq 3$ , odd, and to $\{ \theta\in\mathbb C\ :\ \textrm{Im}(\theta)\geq0,\ \theta\neq 2k\pi-i\mu,\ k\in\mathbb{Z},\ \mu\geq0\}$ for $n \geq4$, even, and for $n \geq4$ even $R_\chi(\theta)$ is bounded in a neighborhood of $\theta=0$.