Bi-space pullback attractors for closed processes

Abstract

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 smoothing properties of the process and for pullback attractors attracting a given universe, i.e. a chosen class of possibly time-dependent families of sets. We also present an application of the result to reaction-diffusion equations.