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The geometrical quantity in damped wave equations on a square

2006
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E S A I M: Control, Optimisation and Calculus of Variations
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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

doi:10.1051/cocv:2006015
fatcat:ew6meckorzhgnjvdzwkmjivg4y