Best exponential decay rate of energy for the vectorial damped wave\n equation

2017/07/25 by Guillaume Klein, Klein, Guillaume · 1 citation
Computer Science · Engineering · Mathematics · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Optimization and Control (math.OC) #Quantum chaos and dynamical systems #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.1707.07893

openalex publication_date 2017/07/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The energy of solutions of the scalar damped wave equation decays uniformly\nexponentially fast when the geometric control condition is satisfied. A theorem\nof Lebeau [leb93] gives an expression of this exponential decay rate in terms\nof the average value of the damping terms along geodesics and of the spectrum\nof the infinitesimal generator of the equation. The aim of this text is to\ngeneralize this result in the setting of a vectorial damped wave equation on a\nRiemannian manifold with no boundary. We obtain an expression analogous to\nLebeau's one but new phenomena like high frequency overdamping arise in\ncomparison to the scalar setting. We also prove a necessary and sufficient\ncondition for the strong stabilization of the vectorial wave equation.\n

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• On the Observability Inequality of Coupled Wave Equations: the Case without Boundary

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