Bresse System in Thermoelasticity of Type III Acting on Shear Force

In this work we are considering the thermoelastic beam system where the oscillations are defined by the Bresse model and the heat conduction is given by Green and Naghdi theories. Our main result is to show that the corresponding semigroup is exponentially stable if and only if the wave speeds associated to the hyperbolic part of the system are equal. In the case of lack of exponential stability we show that the solution decays polynomially and we prove that the rate of decay is optimal. It is worth mentioning that this theory is very new and then few applicability studies have been developed for this theory. However, mathematical and physical analysis is needed to clarify its applicability.