Stability of thermoelastic extensible Timoshenko system with nonlinear frictional damping of variable exponent-type

In this article, we consider a nonlinear extensible thermoelastic Timoshenko system with nonlinear frictional damping of the form (Formula presented.) having a time-dependent coefficient (Formula presented.) and a variable exponent (Formula presented.) modeling complex energy dissipation behaviors. We study the effects of both types of dissipative mechanisms as well as the extensibility on the energy estimates. First, an existence and uniqueness result are recalled. Then, using the multiplier method, we construct explicit formulas depending on both (Formula presented.) and m and the extensibility, which prove that the energy of the considered model decays toward zero in an exponential and polynomial manner. The result obtained generalizes the result of which can be considered a special case.