Sufficient Conditions for the Existence and Uniqueness of Maximal Attractors for Autonomous and Nonautonomous Dynamical Systems.

The theory of compact global attractors for dynamical systems relies on the existence of a bounded absorbing set. In this paper, suppressing this condition, we present sufficient conditions to ensure the existence and uniqueness of maximal attractors for dynamical systems (both autonomous and nonautonomous). Such attractors are, in general, unbounded. For semigroups satisfying these conditions, when a Lyapunov function exists, we also present the characterization of the unique maximal attractor as the unstable set of the critical elements (not necessarily fixed points). As an example we present a semilinear parabolic equation to illustrate the theory in the autonomous case. For the nonautonomous setting, applying the general theory, we present an abstract evolution equation and a concrete parabolic equation as examples. [ABSTRACT FROM AUTHOR]