## The geometrical quantity in damped wave equations on a square

Institut Élie
Cartan, Université de Nancy
1, BP 239 54506 Vandœuvre-lès-Nancy Cedex, France; pascal_hebrard@ds-fr.com; humbert@iecn.u-nancy.fr

Received:
14
November
2003

Revised:
19
July
2004

Revised:
13
June
2005

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

Mathematics Subject Classification: 35L05 / 93D15

Key words: Damped wave equation / mathematical billards.

*© EDP Sciences, SMAI, 2006*