Stabilization of two coupled wave equations with a localized singular Kelvin-Voigt damping

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Abstract We consider two wave equations coupled through a singular Kelvin-Voigt damping mechanism in a bounded domain. We are interested in investigating stability issues for this system.We prove the polynomial stability of the semigroup if the damping region is big enough, and logarithmic stability of the semigroup if the damping region is an arbitrarily small nonempty open subset of the domain under consideration. The main features of our proofs: i) frequency domain approach and, ii) flow multipliers combined with extra auxiliary elliptic systems in the case of polynomial stability, or iii) Carleman estimate in the case of logarithmic stability. A numerical analysis of the spectrum of the one dimensional space semi-discretized system using mixed finite element method indicates that uniform (with respect to the mesh size) exponential decay is not to be expected. This latter result leads us to conjecture that our first polynomial stability result cannot be improved to an exponential stability one.

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