Asymptotic behavior of solutions for Timoshenko beam system with viscoelastic damping, non-linear delay term with variable-exponent non-linearity

In this work, we investigate the effect of incorporating different types of damping into a viscoelastic Timoshenko beams system, namely, nonlinear internal damping, time delayed nonlinear damping and a nonlinear feedback with variable exponent in addition to a viscoelastic memory term. Using Galerkin techniques combined with the energy method, within the framework of Sobolev spaces with variable exponent, we establish the global well-posedness of solutions, under assumptions that link the delayed and the immediate damping intensities. Furthermore, we provide an examination of the general decay behavior over time of the solutions, where we draw conclusions about the energy decay rate of the system without imposing a predetermined decay rate.