## Preprint 2012-26

## Margareth Alves, Jaime Muñoz-Rivera, Mauricio Sepúlveda, Octavio Vera:

### Exponential and the lack of exponential stability in transmission problems with localized Kelvin-Voigt dissipation

### Abstract:

In this paper we consider the transmission problem of a material composed by three components, one of them is a Kelvin-Voigt viscoelastic material, the second is an elastic material (no dissipation) and the third is an elastic material inserted with a frictional damping mechanism. The main result of this paper is that the rate of decay will depend of the position of each component. When the viscoelastic component is not in the middle of the material, then there exists exponential stability of the solution. Instead, when the viscoelastic part is in the middle of the material, then there is not exponential stability. In this case we show that the decay is polynomial as $1/t^{2}.$ Moreover we show that the rate of decay is optimal over the domain of the infinitesimal generator. Finally using a second order scheme that ensures the decay of energy (Newmark-$beta$ method), we give some numerical examples which demonstrate these asymptotic behavior.

This preprint gave rise to the following definitive publication(s):

**Margareth ALVES, Jaime MUñOZ-RIVERA, Mauricio SEPúLVEDA, Octavio VERA: ***Exponential and the lack of exponential stability in transmission problems with localized Kelvin-Voigt dissipation*. SIAM Journal on Applied Mathematics, vol. 74, 2, pp. 345-365, (2014).