  This paper studies the stability of an abstract thermoelastic system with
Cattaneo's law, which describes finite heat propagation speed in a medium. We
introduce a parameters region containing coupling, thermal dissipation, and
possible inertial characteristics. The parameters region is partitioned into
distinct subregions based on the spectral properties of the infinitesimal
generator of the corresponding semigroup. By a careful estimation of the
resolvent, we obtain distinct polynomial decay rates for systems with
parameters located in different subregions. Furthermore, the optimality of
these decay rates is proved. Finally, we apply our results to several coupled
systems of partial differential equations.
