Some well-posedness and stability results for abstracthyperbolic equations with infinite memory and distributed time delay

In this paper, we consider a class of second order abstract linear hyperbolic equations withinfinite memory and distributed time delay. Under appropriate assumptions onthe infinite memory and distributed time delay convolution kernels, weprove well-posedness and stability of the system. Ourestimation shows that the dissipation resulting from the infinite memory alone guarantees the asymptotic stability of the system in spite of the presence of distributed time delay. The decay rate of solutions is found explicitly in terms of the growth at infinity of the infinite memory and the distributed time delay convolution kernels. An application of our approach to the discrete time delay case is also given.