Asymptotic behavior of the transmission Euler-Bernoulli plate and wave equation with a localized Kelvin-Voigt damping. (English) Zbl 1350.35031
Summary: Let a fourth and a second order evolution equations be coupled via the interface by transmission conditions, and suppose that the first one is stabilized by a localized distributed feedback. What will then be the effect of such a partial stabilization on the decay of solutions at infinity? Is the behavior of the first component sufficient to stabilize the second one? The answer given in this paper is that sufficiently smooth solutions decay logarithmically at infinity even the feedback dissipation affects an arbitrarily small open subset of the interior. The method used, in this case, is based on a frequency method, and this by combining a contradiction argument with the Carleman estimates technique to carry out a special analysis for the resolvent.
MSC:
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
| 35M32 | Boundary value problems for mixed-type systems of PDEs |
| 93D15 | Stabilization of systems by feedback |
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