Well-posedness and stability for an abstract evolution equation with history memory and time delay in Hilbert space

Abstract

In this paper, we consider a general model of an abstract evolution equation with history memory and time delay with a nonlinear source term in the Hilbert space. First, we consider a linear evolution system with history memory and time delay. Under appropriate assumptions, we prove the well-posedness by using semigroup arguments and the stability results of the system by defining a suitable Lyapunov functional under a suitable condition on the time delay. We obtain the unique dissipation, given by history memory, is strong enough to exponentially stabilize the system. Second, we consider a semilinear evolution equation where the nonlinear source term and the nonlinear dissipative term verify some Lipschitz continuity conditions. In this later, we establish the well-posedness and the exponential decay results by a direct proof using Duhamel's formula and a perturbation method. In both cases, some examples are given to illustrate our abstract results. Our systems generalize the earlier problems in the literature.