Chain recurrence and structure of ω-limit sets of multivalued semiflows

Abstract

We study properties of ω-limit sets of multivalued semiflows like chain recurrence or the existence of cyclic chains. First, we prove that under certain conditions the ω-limit set of a trajectory is chain recurrent, applying this result to an evolution differential inclusion with upper semicontinous right-hand side. Second, we give conditions ensuring that the ω-limit set of a trajectory contains a cyclic chain. Using this result we are able to check that the ω-limit set of every trajectory of a reaction-diffusion equation without uniqueness of solutions is an equilibrium.