Periodic damping gives polynomial energy decay

Let $u$ solve the damped Klein–Gordon equation\[\left ( \partial^2_t - \sum{\partial^2_{xj}} + m \mathrm{Id}+\gamma (x) \partial_t \right ) u = 0\]on $\mathbb{R}^n$ with $m \gt 0$ 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 a solution decays at a polynomial rate. This is proved via a periodic observability estimate on $\mathbb{R}^n$.

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Received 14 December 2015

Accepted 11 August 2016

Published 24 July 2017