Asymptotic behavior of nonlocal partial differential equations with long time memory

Abstract

In this paper, it is first addressed the well-posedness of weak solutions to a nonlocal partial differential equation with long time memory, which is carried out by exploiting the nowadays well-known technique used by Dafermos in the early 70’s. Thanks to this Dafermos transformation, the original problem with memory is transformed into a nondelay one for which the standard theory of autonomous dynamical system can be applied. Thus, some results about the existence of global attractors to the transformed problem are provided. Particularly, when the initial values have higher regularity, the solutions of both problems (the original and the transformed ones) are equivalent. Nevertheless, the equivalence of global attractors for both problems is still unsolved due to the lack of enough regularity of solutions in the transformed problem, it is therefore proved the existence of global attractors of the transformed problem. Eventually, it is highlighted how to proceed to obtain meaningful results about the original problem, without performing any transformation, but working directly with the original delay problem.