A general decay result for a memory-type Timoshenko-thermoelasticity system with second sound

In this paper we consider a system of Timoshenko-thermoelasticity with second sound in the presence of an infinite history term. We prove, under suitable conditions on the coefficients and the relaxation function, a general decay result from which the exponential and polynomial decay estimates are only special cases. We also establish a lack of exponential decay result and discuss the optimality for some situation.