In the description of the long-time behavior of solutions to nonautonomous differential equations the notion of a pullback attractor plays a similar role as the global attractor in autonomous dynamical systems. We present the theorem on the existence of a pullback attractor if the evolution process is a family of closed operators. The abstract result is formulated in the context of the smoothin...

The existence of a pullback attractor is established for the singularly perturbed FitzHughNagumo system defined on the entire space Rn when external terms are unbounded in a phase space. The pullback asymptotic compactness of the system is proved by using uniform a priori estimates for far-field values of solutions. Although the limiting system has no global attractor, we show that the pullback...

This paper is concerned with a generalized 2D parabolic equation with a nonautonomous perturbation −∆ut + α ∆ut + μ∆ u+∇ · −→ F (u) +B(u, u) = ǫg(x, t). Under some proper assumptions on the external force term g, the upper semicontinuity of pullback attractors is proved. More precisely, it is shown that the pullback attractor {Aǫ(t)}t∈R of the equation with ǫ > 0 converges to the global attract...

Converse Lyapunov theorems are presented for nonautonomous systems modelled as skew product flows. These characterize various types of stability of invariant sets and pullback, forward and uniform attractors in such nonautonomous systems. MSC subject classification: 37B25, 37B55, 93D30

This article concerns the asymptotic behavior of solutions to the two-dimensional Navier-Stokes equations with both non-autonomous deterministic and stochastic terms defined on unbounded domains. First we introduce a continuous cocycle for the equations and then prove the existence and uniqueness of tempered random attractors. We also characterize the structures of the random attractors by comp...