Morse decompositions with infinite components for multivalued semiflows (Q526387)

The authors consider strict multivalued semiflows of the form \(G:[0,+\infty)\times X\to {\mathcal P}(X)\), where \(X\) is a complete metric space and \({\mathcal P}(X)\) is the collection of all non-empty subsets of \(X\). This class of multivalued semiflows was defined in [\textit{V. S. Melnik} and \textit{J. Valero}, Set-Valued Anal. 16, No. 4, 507--509 (2008; Zbl 1172.35331)], and a similar multivalued concept was used, e.g., in [\textit{A. L. Zuev}, Ukr. Mat. Zh. 58, No. 5, 629--637 (2006; Zbl 1126.34040); translation in Ukr. Math. J. 58, No. 5, 709--717 (2006)]. If \(\mathcal A\) is the global attractor and \(\emptyset =A_0 \subset A_1 \subset \cdots \subset A_\infty = \mathcal A\) is a sequence of local attractors, then the Morse decomposition on \(\mathcal A\) is defined as a disjoint family of weakly invariant sets \({\mathcal M}=\{M_j\}_{j\in \mathbb N}\cup \{M_\infty\}\), where \(M_j=A_j \cap A^*_{j-1}\) and \(A^*_{j-1}\) is the repeller associated with \(A_{j-1}\). The family \(\mathcal M\) is called ordered with respect to a Lyapunov function \({\mathcal V}: X\to \mathbb R\) if \({\mathcal V}(x)\equiv L_j\) for all \(x\in M_j\), and \(L_1 \leq L_2 \leq \cdots \leq L_n \leq \cdots <L_\infty\). The main result of this paper states the equivalence between the following three properties: 1)~\(G\) is dynamically gradient with respect to \(\mathcal M\); 2)~\(\mathcal M\) generates a Morse decomposition; 3)~there exists a Lyapunov function such that \(\mathcal M\) is ordered. A particular version of the above result is formulated for the case of continuous semigroups in a complete metric space. This theory is applied to an example of reaction-diffusion inclusion with a countable set of fixed points.