Nonexponential and polynomial decay of a thermoelastic Bresse system in the absence of the second spectrum

In this paper, we consider a one-dimensional linear truncated thermoelastic Bresse system consisting of two hyperbolic equations and one elliptic equation coupled with a heat equation where the heat conduction is modeled by the Fourier thermal law. We establish the well posedness, using the semigroup theory and with the help of non classical differential operators. After that, we show that the system is not exponentially stable. Moreover, we prove a polynomial decay of the solution in the case of the equal speeds of wave propagation of the two hyperbolic equations and illustrate this latter result by some numerical tests.