  In this paper, we investigate the stabilization of a linear Bresse system
with one singular local frictional damping acting in the longitudinal
displacement, under fully Dirichlet boundary conditions. First, we prove the
strong stability of our system. Next, using a frequency domain approach
combined with the multiplier method, we establish the exponential stability of
the solution if and only if the three waves have the same speed of propagation.
On the contrary, we prove that the energy of our system decays polynomially
with rates $t^{-1}$ or $t^{-\frac{1}{2}}$.
