Asymptotic behavior for coupled systems of second order abstract evolution equations with one infinite memory

We are concerned with the asymptotic behavior for a coupled system of two second order abstract evolution equations with one infinite memory and the damping effect given by only one equation, which describes the dynamics of many viscoelastic systems and covers the well-known Timoshenko system. We present an optimal energy decay estimate under very weak conditions. More specifically, we prove that the energy of the system decays to zero at least at the rate of t -1 only under the condition that the related kernel functions are non-increasing and integrable. Moreover, we apply our abstract results to three classes of coupled systems of second-order partial differential equations arising in viscoelasticity. (C) 2019 Elsevier Inc. All rights reserved.