Rates of decay to weak thermoelastic Bresse system

We consider the Bresse system with temperature and we show that there exist exponential stability if and only if the wave propagation is equal. We show that, in general, the system is not exponentially stable but that there exists polynomial stability with rates that depend on the wave propagations and the regularity of the initial data. Moreover, we introduce a necessary condition to dissipative semigroup decay polynomially. This result allows us to show some optimality to the polynomial rate of decay.