Bi-space pullback attractors for closed processes (Q386123)

The pullback attractor of a nonautonomous dynamical system (for instance in its process formulation) can be considered as a natural extension of the classical global attractor occurring in a time-invariant setting. However, pullback attractors capture the past of a system and not necessary its forward behavior. The present paper deals with processes consisting of closed operators. Under such a closedness assumption (see the paper for a precise definition), as well as asymptotic compactness (in a pullback sense), it is shown that a pullback attractor exists. First, an abstract result, valid for processes on complete metric spaces \(V\), is formulated. Given a further complete metric space \(W\), under appropriate closedness and asymptotic compactness between \(V\) and \(W\), it is shown that a corresponding pullback attractor exists. It can be characterized as limit set of an absorbing set. Second, as an application, a reaction-diffusion equation with time-dependent inhomogeneity is considered, where the spaces \(V=L^2(\Omega)\) and \(W=H_0^1(\Omega)\) turn out to be suitable.