  We consider an elastic/viscoelastic transmission problem for the Bresse
system with fully Dirichlet or Dirichlet-Neumann-Neumann boundary conditions.
The physical model consists of three wave equations coupled in certain pattern.
The system is damped directly or indirectly by global or local Kelvin-Voigt
damping. Actually, the number of the dampings, their nature of distribution
(locally or globally) and the smoothness of the damping coefficient at the
interface play a crucial role in the type of the stabilization of the
corresponding semigroup. Indeed, using frequency domain approach combined with
multiplier techniques and the construction of a new multiplier function, we
establish different types of energy decay rate (see the table of stability
results below). Our results generalize and improve many earlier ones in the
literature and in particular some studies done on the Timoshenko system with
Kelvin-Voigt damping.
