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The energy in a square membrane subject to constant viscous damping on a subset $\omega \subset \Omega$ decays exponentially in time as soon as satisfies a geometrical condition known as the “Bardos-Lebeau-Rauch” condition. The rate $\tau \left(\omega \right)$ of this decay satisfies $\tau \left(\omega \right)=2min(-\mu (\omega ),g(\omega \left)\right)$ (see Lebeau [ (1996) 73–109]). Here $\mu \left(\omega \right)$ denotes the spectral abscissa of the damped wave equation operator and $g\left(\omega \right)$ is a number called the geometrical quantity of and defined as follows. A ray in is the trajectory generated by the free motion of...