Exponential stability of a linear viscoelastic bar with thermal memory

In this paper we study a one-dimensional evolution problem arising in the theory of linear thermoviscoelasticity with hereditary heat conduction. Depending on the instantaneous conductivity Ko, both Coleman-Gurtin (Ko > O) and Gurtin-Pipkin (Ko = O) heat flow theories are involved. In any case, the exponential stability of the corresponding semigroup is proved for a class of memory functions including weakly singular kernels. In order to achieve the exponential decay of the energy, we assume that mechanical and thermal memory kernels decay exponentially for large time.