Some results related to 2D Navier-Stokes equations when the external force contains hereditary characteristics involving unbounded delays are analyzed. First, the existence and uniqueness of solutions is proved by Galerkin approximations and the energy method. The existence of stationary solution is then established by means of the Lax-Milgram theorem and the Schauder fixed point theorem. The local stability analysis of stationary solutions is studied by several different methods: the classical Lyapunov function method, the Razumikhin-Lyapunov technique and by constructing appropriate Lyapunov functionals. Finally, we also verify the polynomial stability of the stationary solution in a particular case of unbounded variable delay. Exponential stability in this infinite delay setting remains as an open problem.

Some results concerning a stochastic 2D Navier-Stokes system when the external forces contain hereditary characteristics are established. The existence and uniqueness of solutions in the case of unbounded (infinite) delay are first proved by using the classical technique of Galerkin approximations. The local stability analysis of constant solutions (equilibria) is also carried out by exploiting two approaches. Namely, the Lyapunov function method and by constructing appropriate Lyapunov functionals. The asymptotic stability and hence, the uniqueness of equilibrium solution are obtained by constructing Lyapunov functionals. Moreover, some sufficient conditions ensuring the polynomial stability of the equilibrium solution in a particular case of unbounded variable delay will be provided. Exponential stability for other special cases of infinite delay remains as an open problem.

In this paper we investigate the well-posedness and dynamics of a fractional stochastic integro-differential equation describing a reaction process depending on the temperature itself. Existence and uniqueness of solutions of the integro-differential equation is proved by the Lumer-Phillips theorem. Besides, under appropriate assumptions on the memory kernel and on the magnitude of the nonlinearity, the existence of random attractor is achieved by obtaining first some a priori estimates. Moreover, the random attractor is shown to have finite Hausdorff dimension.