General decay of solution for a transmission problem in infinite memory-type thermoelasticity with second sound

In this article, we study the initial boundary value problem in one space variable for an elastic-thermoelastic bar with the elastic part being surrounded by two thermoelastic parts in the presence of an infinite memory term, for which the heat conduction in thermoelasticity is described by Cattaneo's law, removing the physical paradox of infinite propagation speed of signals. The main diculties in handing this problem are that the system does not have any frictional damping term, and that the dissipative effects of heat conduction induced by Cattaneo's law are usually weaker than those induced by Fourier's law. To overcome these diculties, we shall introduce the second-order energy and two extra functionals. Under appropriate hypothesis on the relaxation function, we establish a general decay result.