Stability of the abstract thermoelastic system with singularity

In this paper, we analyze an abstract thermoelastic system, where the heat conduction follows the Cattaneo law. Zero becomes a spectrum point of the system operator when the coupling and thermal damping parameters of system satisfy specific conditions. We obtain the decay rates of solutions to the system with or without the inertial term. Furthermore, the decay rate of the system without inertial terms is shown to be optimal.