We consider the wave equation with Kelvin-Voigt damping in a bounded domain. The exponential stability result proposed by Liu and Rao [18] or Tébou [21] for that system assumes that the damping is localized in a neighborhood of the whole or a part of the boundary under some consideration. In this paper we propose to deal with this geometrical condition by considering a singular Kelvin-Voigt damping which is localized faraway from the boundary. In this particular case it was proved by Liu and Liu [16] the lack of the uniform decay of the energy. However, we show that the energy of the wave equation decreases logarithmically to zero as time goes to infinity. Our method is based on the frequency domain method. The main feature of our contribution is to write the resolvent problem as a transmission system to which we apply a specific Carleman estimate.2010 Mathematics Subject Classification. 35A01, 35A02, 35M33, 93D20. 2 H 1 0 (Ω)×L 2 (Ω) , ∀ t ≥ 0. Simple formal calculations give E(u, 0) − E(u, t) = − d t 0 ω |∇∂ t u(x, s)| 2 dx ds, ∀t ≥ 0,

In this paper, we consider the longitudinal and transversal vibrations of the transmission Euler-Bernoulli beam with Kelvin-Voigt damping distributed locally on any subinterval of the region occupied by the beam and only in one side of the transmission point. We prove that the semigroup associated with the equation for the transversal motion of the beam is exponentially stable, although the semigroup associated with the equation for the longitudinal motion of the beam is polynomially stable. Due to the locally distributed and unbounded nature of the damping, we use a frequency domain method and combine a contradiction argument with the multiplier technique to carry out a special analysis for the resolvent.

We consider a beam and a wave equations coupled on an elastic beam through transmission conditions. The damping which is locally distributed acts through one of the two equations only; its effect is transmitted to the other equation through the coupling. First we consider the case where the dissipation acts through the beam equation. Using a recent result of Borichev and Tomilov on polynomial decay characterization of bounded semigroups we provide a precise decay estimates showing that the energy of this coupled system decays polynomially as the time variable goes to infinity. Second, we discuss the case where the damping acts through the wave equation. Proceeding as in the first case, we prove that this system is also polynomially stable and we provide precise polynomial decay estimates for its energy. Finally, we show the lack of uniform exponential decay of solutions for both models.Consider a clamped elastic beam of length L. One segment of the beam is made of a viscoelastic material with Kelvin-Voigt constitutive relation. The longitudinal and transversal vibration of the beam can be described by the following equationsfor t ∈ (0, +∞) w ′ 1 (l, t) = 0 for t ∈ (0, +∞) w ′′′ 1 (l, t) + w ′ 2 (l, t) = 0 for t ∈ (0, +∞) w 1 (0, t) = w ′ 1 (0, t) = w 2 (L, t) = 0 for t ∈ (0, +∞) w 1 (x, 0) = w 0 1 (x),ẇ 1 (x, 0) = w 1 1 (x) on (0, l) w 2 (x, 0) = w 0 2 (x),ẇ 2 (x, 0) = w 1 2 (x) on (l, L),

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.

In this paper we will study the asymptotic behaviour of the energy decay of a transmission plate equation with locally distributed Kelvin-Voigt feedback. Precisly, we shall prove that the energy decay at least logarithmically over the time. The originality of this method comes from the fact that using a Carleman estimate for a transmission second order system which will be derived from the plate equation to establish a resolvent estimate which provide, by the famous Burq's result [Bur98], the kind of decay mentionned above.