## The relationship between pullback, forward and global attractors of nonautonomous dynamical systems.(English)Zbl 1054.34087

The authors study and compare various types of attractors for nonautonomous dynamical systems involving a cocycle state space driven by an autonomous dynamical system on a compact state space. In particular, they give conditions for a uniform pullback attractor to form a global attractor of the associated autonomous skew-product semi-dynamical system. They generalize Zubov’s theorem on the characterization of the asymptotic stability of a compact set with respect to an $$\alpha$$-condensing semi-dynamical system. The results are illustrated by several examples – in particular by a nonautonomous Navier-Stokes equation – that are generated by differential equations on a Banach space with a uniform dissipative structur induced by a monotone operator.

### MSC:

 34D45 Attractors of solutions to ordinary differential equations 35B41 Attractors 34D20 Stability of solutions to ordinary differential equations 34D40 Ultimate boundedness (MSC2000) 34G20 Nonlinear differential equations in abstract spaces 35B35 Stability in context of PDEs 35B40 Asymptotic behavior of solutions to PDEs 35Q30 Navier-Stokes equations 37C70 Attractors and repellers of smooth dynamical systems and their topological structure 37C75 Stability theory for smooth dynamical systems