  In their paper "Stability to weak dissipative bresse system", Alabau et al.
studied the exponential and polynomial stability of the Bresse system with one
globally distributed dissipation law. Our goal is to extend their results, by
taking into consideration the important case when the dissipation law is
locally distributed and to improve the polynomial energy decay rate. We then
study the energy decay rate of the Bresse system with one locally internal
distributed dissipation law acting on the equation about the shear angle
displacement. Under the equal speed wave propagation condition, we show that
the system is exponentially stable. On the contrary, we establish a new
polynomial energy decay rate.
