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

We study properties of !-limit sets of multivalued semiflows like chain recurrence or the existence of cyclic chains. First, we prove that under certain conditions the omega-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 omega-limit set of a trajectory contains a cyclic chain. Using this result we are able to check that the omega-limit set of every trajectory of a reaction-diffusion equation without uniqueness of solutions is an equilibrium.