Asymptotic behavior for Timoshenko systems with fractional damping

This article deals with the asymptotic behavior of the solutions of a Timoshenko beam with a fractional damping. The damping acts only in one of the equations and depends on a parameter theta is an element of [0, 1]. Timoshenko systems with frictional or Kelvin-Voigt dampings are particular cases of this model. We prove that, for regular initial data, the semigroup of this system decays polynomially with rates that depend on theta and some relations between the structural parameters of the system. We also show that the decay rates obtained are optimal and the only possibility to obtain exponential decay is when theta = 0 and the wave propagation speeds of the equations coincide.