We consider a transmission problem with localized Kelvin‐Voigt viscoelastic damping. Our main result is to show that the corresponding semigroup (SA(t))t≥0 is not exponentially stable, but the solution of the system decays polynomially to zero as 1/t2 when the initial data are taken over the domain D(A). Moreover, we prove that this rate of decay is optimal. Finally, using a second order scheme that ensures the decay of energy (Newmark‐β method), we give some numerical examples which demonstrate this polynomial asymptotic behavior.

In this paper we study the transmission for a partially viscoelastic beam, that is, a beam which is composed of two components, elastic and viscoelastic. In the rotation angle of the filaments of the beam, ψ 1 (x, t) and ψ 2 (x, t), the dissipation is occasioned by the memory effect. In the transverse vibrations φ 1 (x, t) and φ 2 (x, t) we do not have dissipation and the system is purely elastic. For this type of beam we show the uniform stabilization, i.e., the rate of decay has directly relation with the velocity of the relaxation function.

We consider the hybrid laminated Timoshenko beam model. This structure is given by two identical layers uniform on top of each other, taking into account that an adhesive of small thickness is bonding the two surfaces and produces an interfacial slip. We suppose that the beam is fastened securely on the left while on the right it’s free and has an attached container. Using the semigroup approach and a result of Borichev and Tomilov, we prove that the solution is polynomially stable.

This work is devoted to the analysis of a fully-implicit numerical scheme for the critical generalized Korteweg-de Vries equation (GKdV with p = 4) in a bounded domain with a localized damping term. The damping is supported in a subset of the domain, so that the solutions of the continuous model issuing from small data are globally defined and exponentially decreasing in the energy space. Based in this asymptotic behavior of the solution, we introduce a finite difference scheme, which despite being one of the first order, has the good property to converge in L 4 -strong. Combining this strong convergence with discrete multipliers and a contradiction argument, we show that the smallness of the initial condition leads to the uniform (with This work was supported by grant 49.0987/2005-2 of the Cooperation CNPq/CONICYT (Brasil-Chile). M.S. has been also supported by Fondecyt Project #1070694, FONDAP and BASAL projects CMM, Universidad de Chile, and CI 2 MA, Universidad de Concepción. O.V.V. has been supported by Postdoctoral fellowship of LNCC (National Laboratory for Scientific Computation), and he is grateful for the dependences of the LNCC while he realized postdoctorate.

a b s t r a c tIn this paper we investigate the asymptotic behavior of solutions to the initial boundary value problem for a one-dimensional mixture of thermoviscoelastic solids. Our main result is to establish the exponential stability of the corresponding semigroup and the lack of exponential stability of the corresponding semigroup.