Minimality properties of set-valued processes and their pullback attractors

TL;DR

This paper investigates the minimality conditions for the existence of pullback attractors in multivalued dynamical systems, emphasizing weak assumptions and providing examples to demonstrate the sharpness of these conditions.

Contribution

It establishes existence results for pullback attractors under minimal assumptions, including weak continuity and closed graph conditions, with illustrative examples and counterexamples.

Findings

Pullback attractors exist without continuity assumptions.

Weak closed graph conditions suffice for invariance.

Examples include reaction-diffusion and heat equations.

Abstract

We discuss the existence of pullback attractors for multivalued dynamical systems on metric spaces. Such attractors are shown to exist without any assumptions in terms of continuity of the solution maps, based only on minimality properties with respect to the notion of pullback attraction. When invariance is required, a very weak closed graph condition on the solving operators is assumed. The presentation is complemented with examples and counterexamples to test the sharpness of the hypotheses involved, including a reaction-diffusion equation, a discontinuous ordinary differential equation and an irregular form of the heat equation.