The influence of the physical coefficients of a Bresse system with one singular local viscous damping in the longitudinal displacement on its stabilization

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 $t^{-\frac{1}{2}}$.