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

Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero

TL;DR

The paper extends Conley-type chain-recurrence theory to multivalued semiflows, proving that ω-limit sets of trajectories are chain recurrent under suitable compactness and regularity assumptions and applying this to differential inclusions with upper semicontinuous right-hand sides. It then characterizes when ω-limit sets contain cyclic chains and develops a structural description of these sets via stable/unstable sets and connections, including homoclinic structures. A constructive converse shows that certain chain-recurrent, quasi-invariant sets arise as ω-limits of perturbed inclusions, linking to global attractors and dissipativity. Finally, the results yield convergence to equilibria for reaction–diffusion equations without uniqueness when the number of equilibria is finite, illustrating the framework on Chafee–Infante-type problems. Together, the work clarifies the internal architecture of long-term dynamics in non-unique evolution equations and provides tools to assess convergence to equilibria in complex infinite-dimensional systems.

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.

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

TL;DR

The paper extends Conley-type chain-recurrence theory to multivalued semiflows, proving that ω-limit sets of trajectories are chain recurrent under suitable compactness and regularity assumptions and applying this to differential inclusions with upper semicontinuous right-hand sides. It then characterizes when ω-limit sets contain cyclic chains and develops a structural description of these sets via stable/unstable sets and connections, including homoclinic structures. A constructive converse shows that certain chain-recurrent, quasi-invariant sets arise as ω-limits of perturbed inclusions, linking to global attractors and dissipativity. Finally, the results yield convergence to equilibria for reaction–diffusion equations without uniqueness when the number of equilibria is finite, illustrating the framework on Chafee–Infante-type problems. Together, the work clarifies the internal architecture of long-term dynamics in non-unique evolution equations and provides tools to assess convergence to equilibria in complex infinite-dimensional systems.

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.
Paper Structure (5 sections, 17 theorems, 129 equations)

This paper contains 5 sections, 17 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Chain recurrence for multivalued semiflows
  3. Chain recurrence for differential inclusions
  4. Stable and unstable sets of compact quasi-invariant isolated sets
  5. Convergence to equilibria for reaction-diffusion equations without uniqueness

Key Result

Lemma 2

Let $\left( K1\right) -\left( K4\right)$ be satisfied. If the closure of the positive orbit of $\varphi\in\mathcal{K}$ is compact, then $\omega\left( \varphi\right)$ is non-empty, compact, connected, quasi-invariant and If $\phi$ is a complete trajectory such that the closure of the negative orbit is compact, then $\alpha(\phi)$ is non-empty, compact, connected, quasi-invariant and

Theorems & Definitions (24)

  • Remark 1
  • Lemma 2
  • Definition 3
  • Remark 4
  • Theorem 5
  • Corollary 6
  • Corollary 7
  • Theorem 8
  • Corollary 9
  • Theorem 10
  • ...and 14 more