The geometrical quantity in damped wave equations on a square

TL;DR

This paper investigates the decay rate of energy in a damped square membrane, introducing a new algorithm to explicitly compute the geometrical quantity related to wave trajectories, especially for unions of squares.

Contribution

It provides an explicit algorithm to compute the geometrical quantity for unions of squares, enhancing understanding of energy decay in damped wave equations.

Findings

Decay rate $ au( ext{omega})$ is linked to the geometrical quantity $g( ext{omega})$.

Algorithm for explicit computation of $g( ext{omega})$ for unions of squares.

Energy decays exponentially under the Bardos-Lebeau-Rauch condition.

Abstract

The energy in a square membrane $\Omega$ subject to constant viscous damping on a subset $\omega\subset \Omega$ decays exponentially in time as soon as $\omega$ satisfies a geometrical condition known as the "Bardos-Lebeau-Rauch" condition. The rate $\tau(\omega)$ of this decay satisfies $\tau(\omega)= 2 \min(-\mu(\omega), g(\omega))$ (see Lebeau [Math. Phys. Stud. 19 (1996) 73-109]). Here $\mu(\omega)$ denotes the spectral abscissa of the damped wave equation operator and $g(\omega)$ is a number called the geometrical quantity of $\omega$ and defined as follows. A ray in $\Omega$ is the trajectory generated by the free motion of a mass-point in $\Omega$ subject to elastic reflections on the boundary. These reflections obey the law of geometrical optics. The geometrical quantity $g(\omega)$ is then defined as the upper limit (large time asymptotics) of the average trajectory length. We give here an algorithm to compute explicitly $g(\omega)$ when $\omega$ is a finite union of squares.