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

Abstract

Consider the wave equation ${\partial }_t^{2}u-{\Delta }_{x}u+V(t,x)u=0$, 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 $\im(θ)>BT $ with $B>0$ sufficiently large. We establish a meromorphic continuation of $R(\theta )$ from which we deduce the asymptotic expansion of $e^{-(D+\epsilon )<x>}U(t,0)e^{-D<x>}f$, where $f\in {\dot{H}}^1({\mathbb{R}}^n)\times L^2({\mathbb{R}}^n)$, 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 $\{θ\in\C\ :\ \im(θ)\geq0\}$ and is bounded for $\theta \to 0$, we obtain local energy decay as well as global Strichartz estimates for the solutions of ${\partial }_t^{2}u-{\Delta }_{x}u+V(t,x)u=F(t,x)$.