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The effect of perturbations on the convergence of attractors for reaction-diffusion equations concerning variations of nonlinear boundary conditions

Flank D. M. Bezerra, Marcone C. Pereira, Leonardo Pires

Abstract

This paper presents estimates of the convergence of asymptotic dynamics of reaction-diffusion equations with nonlinear boundary conditions. We show how the convergence of the global attractors can be affected by the variations of diffusion coefficients, boundary conditions, and vector fields.

The effect of perturbations on the convergence of attractors for reaction-diffusion equations concerning variations of nonlinear boundary conditions

Abstract

This paper presents estimates of the convergence of asymptotic dynamics of reaction-diffusion equations with nonlinear boundary conditions. We show how the convergence of the global attractors can be affected by the variations of diffusion coefficients, boundary conditions, and vector fields.

Paper Structure

This paper contains 9 sections, 20 theorems, 143 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Functional setting
  3. Convergence of the resolvent operators
  4. Convergence of linear and nonlinear semigroups
  5. Convergence of local unstable manifolds
  6. Convergence of global attractors
  7. Scalar reaction-diffusion equations
  8. General theory of rate of convergence of attractors
  9. Shadowing theory and rate of convergence of attractors

Key Result

Lemma 2.1

The inner product in $X_\varepsilon^\frac{1}{2}$ given by gives a norm in $H^{1}(\Omega)$, equivalent to the usual one for all $\varepsilon \in [0, \varepsilon_0]$ and some $0<\varepsilon_0\leqslant 1$.

Figures (1)

  • Figure 1: Thin Domain

Theorems & Definitions (50)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • Remark 2.3
  • Theorem 2.4
  • proof
  • Theorem 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 40 more