Exponential stability in viscoelastic and elastic systems with thermal memory

Some linear evolution problems arising in the three-dimensional theory of thermoviscoelasticity with memory are considered. Assuming mixed boundary conditions, the associated evolution systems are formulated in a history space setting. Making use of semigroup techniques, well-posedness results are discussed, as well as asymptotic behavior of solutions. When mechanical and thermal memory kernels decay exponentially, the longtime behavior of solutions is proved to be of the same type. Restricting our attention to a one-dimensional solid, the same decay is proved for a linearized thermoelastic model based on the Gurtin-Pipkin heat conduction law