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

Abstract

Consider the wave equation $\partial_t^2u-\Delta_xu+V(t,x)u=0$, where $x\in\R^n$ with $n\geq3$ 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(\theta)>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 $f\in \dot{H}^1(\R^n)\times L^2(\R^n)$, 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 $\{\theta\in\C\ :\ \im(\theta)\geq0\}$ and is bounded for $\theta\to0$, we obtain local energy decay as well as global Strichartz estimates for the solutions of $\partial_t^2u-\Delta_xu+V(t,x)u=F(t,x)$.