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

TL;DR

This paper investigates the structure of omega-limit sets in multivalued semiflows, establishing conditions for chain recurrence and cyclic chains, and applies these results to reaction-diffusion equations without solution uniqueness.

Contribution

It provides new conditions under which omega-limit sets are chain recurrent or contain cyclic chains, with applications to differential inclusions and reaction-diffusion equations.

Findings

Omega-limit sets can be chain recurrent under certain conditions.

Omega-limit sets may contain cyclic chains.

Reaction-diffusion equations without solution uniqueness have omega-limit sets that are equilibria.

Abstract

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.