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Exponential attractors for a nonlocal delayed reaction-diffusion equation on an unbounded domain

Wenjie Hu, Tomás Caraballo

Abstract

The main objective of this paper is to investigate exponential attractors for a nonlocal delayed reaction-diffusion equation on an unbounded domain. We first obtain the existence of a globally attractive absorbing set for the dynamical system generated by the equation under the assumption that the nonlinear term is bounded. Then, we construct exponential attractors of the equation directly in its natural phase space, i.e., a Banach space with explicit fractal dimension by combining squeezing properties of the system as well as a covering lemma of finite subspace of Banach spaces. Our result generalizes the methods established in Hilbert spaces and weighted spaces, and the fractal dimension of the obtained exponential attractor does not depend on the entropy number but only depends on some inner characteristic of the studied equation.

Exponential attractors for a nonlocal delayed reaction-diffusion equation on an unbounded domain

Abstract

The main objective of this paper is to investigate exponential attractors for a nonlocal delayed reaction-diffusion equation on an unbounded domain. We first obtain the existence of a globally attractive absorbing set for the dynamical system generated by the equation under the assumption that the nonlinear term is bounded. Then, we construct exponential attractors of the equation directly in its natural phase space, i.e., a Banach space with explicit fractal dimension by combining squeezing properties of the system as well as a covering lemma of finite subspace of Banach spaces. Our result generalizes the methods established in Hilbert spaces and weighted spaces, and the fractal dimension of the obtained exponential attractor does not depend on the entropy number but only depends on some inner characteristic of the studied equation.
Paper Structure (3 sections, 4 theorems, 68 equations)

This paper contains 3 sections, 4 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Exponential attractors

Key Result

Lemma 2.1

Let $\{S(t)\}_{t\geq 0}$ be defined by 2.1, then we have the following results. (i) $\|S(t) \phi\|\leq e^{-\mu t} \|\phi\|$ for all $\phi \in \mathbb{X}$, $t \in \mathbb{R}_{+}$. (ii) $\{S(t)\}_{t\geq 0}$ is an analytic and strongly continuous semigroup on $\mathbb{X}$. (iii) For all $t \in(0, \inft

Theorems & Definitions (7)

  • Lemma 2.1
  • Theorem 2.1
  • proof
  • Definition 3.1
  • Theorem 3.1
  • proof
  • Theorem 3.2