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Existence and characterization of attractors for a nonlocal reaction-diffusion equation having an energy functional

Rubén Caballero, Pedro Marín-Rubio, José Valero

Abstract

In this paper we study a nonlocal reaction-diffusion equation in which the diffusion depends on the gradient of the solution. We prove first the existence and uniqueness of regular and strong solutions. Second, we obtain the existence of global attractors in both situations under rather weak assumptions by the defining a multivalued semiflow (which is a semigroup in the particular situation when uniqueness of the Cauchy problem is satisfied). Third, we characterize the attractor either as the unstable manifold of the set of stationary points or as the stable one when we consider solutions only in the set of bounded complete trajectories.

Existence and characterization of attractors for a nonlocal reaction-diffusion equation having an energy functional

Abstract

In this paper we study a nonlocal reaction-diffusion equation in which the diffusion depends on the gradient of the solution. We prove first the existence and uniqueness of regular and strong solutions. Second, we obtain the existence of global attractors in both situations under rather weak assumptions by the defining a multivalued semiflow (which is a semigroup in the particular situation when uniqueness of the Cauchy problem is satisfied). Third, we characterize the attractor either as the unstable manifold of the set of stationary points or as the stable one when we consider solutions only in the set of bounded complete trajectories.
Paper Structure (10 sections, 47 theorems, 207 equations)

This paper contains 10 sections, 47 theorems, 207 equations.

Table of Contents

  1. Introduction
  2. Existence of solutions
  3. Existence and structure of attractors
  4. Regular solutions
  5. The case of uniqueness
  6. Abstract theory of attractors for multivalued semiflows
  7. The case of non-uniqueness
  8. Strong solutions
  9. Attractor in the phase space $H_{0}^{1}\left( \Omega\right) \cap L^{p}\left( \Omega\right)$
  10. Attractor in the phase space $H_{0}^{1}\left( \Omega\right)$

Key Result

Lemma 4

Let $u\in L^{p}\left( \varepsilon,T;X\right)$, $\dfrac{du}{dt}\in L^{q}\left( \varepsilon,T;X^{\ast}\right)$ for all $0<\varepsilon<T$, where $X$ is a reflexive and separable Banach space and $X^{\ast}$ denotes its dual space. Assume that $\beta\in W^{1,\infty }(\mathbb{\varepsilon},T;[\alpha\left

Theorems & Definitions (65)

  • Definition 1
  • Definition 2
  • Remark 3
  • Lemma 4
  • Lemma 5
  • Remark 6
  • Definition 7
  • Theorem 8
  • Theorem 9
  • Theorem 10
  • ...and 55 more