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

TL;DR

This paper proves the meromorphic continuation of the resolvent for the wave equation with time-periodic potential, leading to detailed asymptotic behavior and decay estimates of solutions in odd and even dimensions.

Contribution

It establishes the meromorphic continuation of the Floquet resolvent for wave equations with time-periodic potentials, enabling precise long-time asymptotics and decay estimates.

Findings

Asymptotic expansion of wave solutions as t→∞

Exponential decay of energy in odd dimensions

Polynomial-logarithmic bounds in even dimensions

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 $\partial_t^2u-\Delta_xu+V(t,x)u=F(t,x)$.