Well-posedness, exponential and polynomial decay for a Timoshenko system with logarithmic nonlinear damping: The cases of equal & nonequal-speed propagation

In this work, we consider a nonlinear Timoshenko system subject to a single logarithmic damping of the form $ \operatorname{sgn}(\psi_t)\, \ln\!\big(1+\lvert \psi_t \rvert\big) $ acting only in the second equation, corresponding to the rotation angle. We establish the global existence of strong solutions by employing the Galerkin method together with compactness arguments, thereby ensuring the well-posedness of the model. In addition, by applying the multiplier method, we derive, in the case of equal-speed propagation, an exponential decay estimate for the associated energy functional and, in addition, in the case of non-equal-speed propagation, we prove that the energy of the solutions is polynomially decaying to zero for large time. By the end, we present a numerical study and give some examples to validate the obtained exponential and polynomial decay results. These results underscore the stabilizing role of the nonlinear weak logarithmic feedback in Timoshenko systems and motivate further investigation of models governed by nonlinear logarithmic dampings.