STABILIZATION OF TWO COUPLED WAVE EQUATIONS WITH A LOCALIZED SINGULAR KELVIN-VOIGT DAMPING
Résumé
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.
Mots clés
2020 Mathematics Subject Classification. 35A01 35A02 35L05 35M33 93D20 Carleman estimate simultaneous stabilization coupled wave equations Singular Kelvin-Voigt damping
2020 Mathematics Subject Classification. 35A01
35A02
35L05
35M33
93D20 Carleman estimate
simultaneous stabilization
coupled wave equations
Singular Kelvin-Voigt damping
Origine | Fichiers produits par l'(les) auteur(s) |
---|