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

In this paper, we consider a class of second order abstract linear hyperbolic equations with infinite memory and distributed time delay. Under appropriate assumptions on the infinite memory and distributed time delay convolution kernels, we prove well-posedness and stability of the system. Our estimation 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.