Periodic damping gives polynomial energy decay

TL;DR

This paper proves that solutions to the damped Klein-Gordon equation with periodic damping exhibit polynomial energy decay, using a periodic observability estimate on the entire space.

Contribution

It establishes polynomial decay rates for energy in the damped Klein-Gordon equation with periodic damping, extending understanding of damping effects in unbounded domains.

Findings

Energy decays polynomially over time.

Periodic observability estimate is key to the proof.

Results apply to equations with positive mass and periodic damping.

Abstract

Let $u$ solve the damped Klein--Gordon equation $$ \big( \partial_t^2-\sum \partial_{x_j}^2 +m \text{Id} +\gamma(x) \partial_t \big) u=0 $$ on $\mathbb{R}^n$ with $m>0$ and $\gamma\geq 0$ bounded below on a $2 \pi \mathbb{Z}^n$-invariant open set by a positive constant. We show that the energy of the solution $u$ decays at a polynomial rate. This is proved via a periodic observability estimate on $\mathbb{R}^n.$