Non-autonomous nonlocal partial differential equations with delay and memory

Abstract

The paper addresses a kind of non-autonomous nonlocal parabolic equations when the external force contains hereditary characteristics involving bounded and unbounded delays. First, well-posedness of the problem is analyzed by the Galerkin method and energy estimations in the phase space Cρ(X). Moreover, some results related to strong solutions are proved under suitable assumptions. The existence of stationary solutions is then established by a corollary of the Brower fixed point theorem. By constructing appropriate Lyapunov functionals in terms of the characteristic delay terms, a deep analysis on stability and attractiveness of the stationary solutions is established. Furthermore, the existence of pullback attractors in L2(Ω), with bounded and unbounded delays, is shown. We emphasize that, to prove the existence of pullback attractors in the unbounded delay case, a new phase space, Eγ, has to be constructed.