Uniform stabilization for the transmission problem of the Timoshenko system with memory

Abstract

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.

The stability number of the Timoshenko system with second sound

“…Therefore χ is an important number that characterizes the asymptotic behavior of the solutions to Timoshenko system. This was proved to memory viscoelastic constitutive laws in [1,3,4,6,8,12]; to thermoelastic constitutive laws with Fourier law and also to thermoelastic dissipation of type III in [10,12,23]; to Timoshenko system with boundary dissipation in [15,16] and also with locally distributed dissipation [18][19][20]. Of course, since Timoshenko system is a two-by-two system of hyperbolic equations, if there exist two dissipative mechanisms, we always get the exponential stability, no matter if the velocities of propagations are equal or not, see [13,14,21].…”

“…Then we get K = p 2 + δ 2 μ 2 −τ κμ 2 / 1 + τ d/ 1 + iβλ (1 − 3 τ κ/ 1 )μ 2 + 3 τ d/ 1 + iβ 3 λ . (3.31) Let us denote χ 2 := 1 − 3 τ κ 1 . If χ 1 = 0, then we have that χ 2 = − 3 τ κ 1 χ 1 = 0.…”

Decay Property for the Timoshenko System With Fourier's Type Heat Conduction

“…We refer to [12,13,16] for frictional dissipation case, [5,14,15] for thermal dissipation case, and [1,2,8,9] for memory-type dissipation case.…”

Stabilization for the transmission problem of the Timoshenko system in thermoelasticity with two concentrated masses

“…Moreover, they proved the polynomial stability by using an extension of a result Borichev and Tomilov . Also, we refer the readers for some other results on the transmission problems and on the thermoelasticity. …”