Exponential and Polynomial Decay Rates for a Shear Thermoelastic Bresse System: Dissipation is via the Axial Displacement

In this paper, we consider a one‐dimensional linear shear thermoelastic Bresse system which consists of two hyperbolic equations and one elliptic equation coupled, through the axial displacement equation, with a heat equation of Fourier type. By introducing some nonclassical differential operators and using the semigroup theory, we prove the existence and the uniqueness of solutions. After that, we show that the system is exponentially stable if and only if the speeds of wave propagation of the hyperbolic equations are equal. In the case of nonequal speeds, we establish a polynomial decay. Moreover, we give a full numerical study and illustrate our theoretical stability results through some numerical tests.