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

Pascal Hébrard, Emmanuel Humbert
2006 E S A I M: Control, Optimisation and Calculus of Variations
The energy in a square membrane Ω subject to constant viscous damping on a subset ω ⊂ Ω decays exponentially in time as soon as ω satisfies a geometrical condition known as the "Bardos-Lebeau-Rauch" condition. The rate τ (ω) of this decay satisfies τ (ω) = 2 min(−µ(ω), g(ω)) (see Lebeau [Math. Phys. Stud. 19 (1996) 73-109]). Here µ(ω) denotes the spectral abscissa of the damped wave equation operator and g(ω) is a number called the geometrical quantity of ω and defined as follows. A ray in Ω is
more » ... lows. 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 g(ω) is then defined as the upper limit (large time asymptotics) of the average trajectory length. We give here an algorithm to compute explicitly g(ω) when ω is a finite union of squares. Mathematics Subject Classification. 35L05, 93D15. Let Ω = [0, 1] 2 be the unit square of R 2 and let ω ⊂ Ω be a subdomain of Ω. We are interested here in the problem of uniform stabilization of solutions of the following equation where χ ω is the characteristical function of the subset ω. This equation has its origin in a physical problem. Consider a square membrane Ω. We study here the behaviour of a wave in Ω. Let u(x, t) be the vertical position of x ∈ Ω at time t > 0. We assume that we apply on ω a force proportional to the speed of the membrane at x. Then, u satisfies equation (0.1). To get more information on this subject, one can refer to [2, 7, 10] . With regard to the one-dimensional case, the reader can consult [3, 6]. We do not deal here with the existence of solutions. We assume that there exists a solution u. Let us define the energy of u at time t by E(t) = Ω |∇u| 2 + u 2 t dx.