Periodic damping gives polynomial energy decay

Let $u$ solve the damped Klein--Gordon equation $\big( \partial_t^2-\sum \partial_{x_j}^2 +m \text{Id} +γ(x) \partial_t \big) u=0$ on $\mathbb{R}^n$ with $m>0$ and $γ\geq 0$ bounded below on a $2 π\mathbb{Z}^n$-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 $\mathbb{R}^n.$

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