The asymptotic behavior of the linear transmission problem in viscoelasticity

We consider a transmission problem with localized Kelvin‐Voigt viscoelastic damping. Our main result is to show that the corresponding semigroup is not exponentially stable, but the solution of the system decays polynomially to zero as when the initial data are taken over the domain 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.