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About the structure of attractors for a nonlocal Chafee-Infante problem

Rubén Caballero, Alexandre Nolasco Carvalho, Pedro Marín-Rubio, José Valero

Abstract

In this paper, we study the structure of the global attractor for the multivalued semiflow generated by a nonlocal reaction-diffusion equation in which we cannot guarantee uniqueness of the Cauchy problem. First, we analyse the existence and properties of stationary points, showing that the problem undergoes the same cascade of bifurcations as in the Chafee-Infante equation. Second, we study the stability of the fixed points and establish that the semiflow is dynamically gradient. We prove that the attractor consists of the stationary points and their heteroclinic connections and analyse some of the possible connections.

About the structure of attractors for a nonlocal Chafee-Infante problem

Abstract

In this paper, we study the structure of the global attractor for the multivalued semiflow generated by a nonlocal reaction-diffusion equation in which we cannot guarantee uniqueness of the Cauchy problem. First, we analyse the existence and properties of stationary points, showing that the problem undergoes the same cascade of bifurcations as in the Chafee-Infante equation. Second, we study the stability of the fixed points and establish that the semiflow is dynamically gradient. We prove that the attractor consists of the stationary points and their heteroclinic connections and analyse some of the possible connections.
Paper Structure (13 sections, 37 theorems, 181 equations, 2 figures)

This paper contains 13 sections, 37 theorems, 181 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Setting of the problem
  3. Existence of solutions
  4. Existence and structure of attractors
  5. Fixed points
  6. Dependence on the parameters of the fixed points for the Chafee-Infante equation
  7. Nonlocal fixed points
  8. Lap number and some forbidden connections
  9. Morse decomposition
  10. Aproximations
  11. Instability
  12. Gradient structure
  13. Appendix

Key Result

Theorem 5

Assume conditions (A1), (A6) and (A9). Assume also the existence of constants $\beta,\gamma>0$ such that Then, for any $u_{0}\in H_{0}^{1}(\Omega)$ problem (problem1) has at least one strong solution.

Figures (2)

  • Figure 1: $a(d)$ non-decreasing
  • Figure 2: $a(d)$ whatever

Theorems & Definitions (51)

  • Definition 1
  • Definition 2
  • Remark 3
  • Remark 4
  • Theorem 5
  • Remark 6
  • Lemma 7
  • Corollary 8
  • Proposition 9
  • Proposition 10
  • ...and 41 more