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Robustness of dynamically gradient multivalued dynamical systems

Rubén Caballero, Alexandre N. Carvalho, Pedro Marín-Rubio, José Valero

TL;DR

This work develops a robustness theory for dynamically gradient multivalued semiflows ($G$) and their Morse decompositions, showing that gradient structure persists under perturbations when attractors and weakly invariant sets remain close. The authors prove a general theorem: if a base semiflow $G_0$ has a finite Morse decomposition and a family of perturbed semiflows $G_$ has attractors $\uA_$ converging to $_0$, then for small perturbations the system remains dynamically gradient with a corresponding Morse structure. They then apply this to a family of Chafee-Infante reaction-diffusion problems approximating a differential inclusion, establishing that the weak-solution-induced m-semiflows are dynamically gradient and that the attractors and Morse sets converge under small perturbations. The results extend gradient-like dynamics to set-valued PDE contexts, enabling rigorous analysis of long-term behavior when uniqueness of solutions fails.

Abstract

In this paper we study the robustness of dynamically gradient multivalued semiflows. As an application, we describe the dynamical properties of a family of Chafee-Infante problems approximating a differential inclusion studied in [3], proving that the weak solutions of these problems generate a dynamically gradient multivalued semiflow with respect to suitable Morse sets.

Robustness of dynamically gradient multivalued dynamical systems

TL;DR

This work develops a robustness theory for dynamically gradient multivalued semiflows () and their Morse decompositions, showing that gradient structure persists under perturbations when attractors and weakly invariant sets remain close. The authors prove a general theorem: if a base semiflow has a finite Morse decomposition and a family of perturbed semiflows has attractors converging to , then for small perturbations the system remains dynamically gradient with a corresponding Morse structure. They then apply this to a family of Chafee-Infante reaction-diffusion problems approximating a differential inclusion, establishing that the weak-solution-induced m-semiflows are dynamically gradient and that the attractors and Morse sets converge under small perturbations. The results extend gradient-like dynamics to set-valued PDE contexts, enabling rigorous analysis of long-term behavior when uniqueness of solutions fails.

Abstract

In this paper we study the robustness of dynamically gradient multivalued semiflows. As an application, we describe the dynamical properties of a family of Chafee-Infante problems approximating a differential inclusion studied in [3], proving that the weak solutions of these problems generate a dynamically gradient multivalued semiflow with respect to suitable Morse sets.

Paper Structure

This paper contains 7 sections, 24 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Robustness of dynamically gradient m-semiflows
  4. An equivalent definition of dynamically gradient families.
  5. Application to a reaction-diffusion equation
  6. Stationary points
  7. Approximations

Key Result

Lemma 5

Let (K1)-(K2) be satisfied. Then every fixed point (resp. complete trajectory) of $\mathcal{R}$ is also a fixed point (resp. complete trajectory) of G. If $\mathcal{R}$ fulfills (K1)-(K4), then the fixed points of $\mathcal{R}$ and G coincide. Besides, a map $\gamma:\mathbb{R} \rightarrow X$ is a co

Theorems & Definitions (42)

  • Definition 1
  • Definition 2
  • Definition 3
  • Remark 4
  • Lemma 5
  • Theorem 6
  • Lemma 7
  • Definition 8
  • Remark 9
  • Definition 10
  • ...and 32 more