Local energy decay for the wave equation with a time-periodic non-trapping metric and moving obstacle

TL;DR

This paper proves local energy decay for the wave equation with a time-periodic, non-trapping metric and moving obstacle by analyzing the Floquet operator and its resolvent, under certain conditions.

Contribution

It establishes the meromorphic continuation of the resolvent of the Floquet operator and provides conditions for local energy decay in a time-periodic setting with moving obstacles.

Findings

Meromorphic continuation of the cut-off resolvent of the Floquet operator.

Conditions for local energy decay in the presence of a time-periodic non-trapping metric.

Analysis of wave propagation with moving obstacles in a periodic domain.

Abstract

Consider the mixed problem with Dirichelet condition associated to the wave equation $\partial_t^2u-\Div_{x}(a(t,x)\nabla_{x}u)=0$, where the scalar metric $a(t,x)$ is $T$-periodic in $t$ and uniformly equal to 1 outside a compact set in $x$, on a $T$-periodic domain. Let $\mathcal U(t, 0)$ be the associated propagator. Assuming that the perturbations are non-trapping, we prove the meromorphic continuation of the cut-off resolvent of the Floquet operator $\mathcal U(T, 0)$ and establish sufficient conditions for local energy decay.