Stabilisation et asymptotique spectrale de l’équation des ondes amorties vectorielle

Abstract : In this thesis we are considering the vectorial damped wave equation on a compact and smooth Riemannian manifold without boundary. The damping term is a smooth function from the manifold to the space of Hermitian matrices of size n. The solutions of this équation are thus vectorial. We start by computing the best exponential energy decay rate of the solutions in terms of the damping term. This allows us to deduce a sufficient and necessary condition for strong stabilization of the vectorial damped wave equation. We also show the appearance of a new phenomenon of high-frequency overdamping that did not exists in the scalar case. In the second half of the thesis we look at the asymptotic distribution of eigenfrequencies of the vectorial damped wave equation. Were show that, up to a null density subset, all the eigenfrequencies are in a strip parallel to the imaginary axis. The width of this strip is determined by the Lyapunov exponents of a dynamical system defined from the damping term.
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Guillaume Klein. Stabilisation et asymptotique spectrale de l’équation des ondes amorties vectorielle. Equations aux dérivées partielles [math.AP]. Université de Strasbourg, 2018. Français. ⟨NNT : 2018STRAD050⟩. ⟨tel-01943093v2⟩

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