Chain recurrence and structure of <inline-formula><tex-math id="M1">\begin{document}$ \omega $\end{document}</tex-math></inline-formula>-limit sets of multivalued semiflows

Abstract

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