Mathematical Research Letters

Volume 24 (2017)

Number 2

Periodic damping gives polynomial energy decay

Pages: 571 – 580

DOI: http://dx.doi.org/10.4310/MRL.2017.v24.n2.a15

Author

Jared Wunsch (Department of Mathematics, Northwestern University, Evanston, Illinois, U.S.A.)

Abstract

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

Published 24 July 2017