Well-posedness and asymptotic stability of a one-dimensional Timoshenko system with a logarithmic damping

In this paper, we study the energy decay of a Timoshenko system with Dirichlet boundary conditions and a nonlinear damping mechanism of logarithmic type. We establish the well-posedness and investigate the asymptotic behavior of the system under this weak dissipation. To the best of our knowledge, this is the first study that addresses a Timoshenko system associated with a logarithmic damping. One of the main difficulties of this problem lies in the logarithmic damping mechanism, which is non-homogeneous and lacks standard polynomial lower bounds. This makes the analysis considerably more involved compared to systems with classical linear damping. We establish the well-posedness of the system and prove that the total energy decays at a polynomial rate. The proof is based on the multiplier method and a Lyapunov-type functional involving logarithmic inequalities. The use of logarithmic damping constitutes the main novelty of our result, offering new insight into the long-time behavior of weakly damped hyperbolic systems.