Volume 12, Number 4, October 2006
|Page(s)||636 - 661|
|Published online||11 October 2006|
The geometrical quantity in damped wave equations on a square
Cartan, Université de Nancy
1, BP 239 54506 Vandœuvre-lès-Nancy Cedex, France; firstname.lastname@example.org; email@example.com
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
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