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Rate of convergence of attractors for semilinear singularly perturbed problems: scalar parabolic equations with localized large diffusion

Leonardo Pires, Alexandre Nolasco de Carvalho

Abstract

In this paper we study the asymptotic nonlinear dynamics of scalar semilinear parabolic problems reaction-diffusion type when the diffusion coefficient becomes large in a subregion which is interior to the domain. We obtain, under suitable assumptions, that the family of attractors behaves continuously and we exhibit the rate of convergence. An accurate description of localized large diffusion is necessary.

Rate of convergence of attractors for semilinear singularly perturbed problems: scalar parabolic equations with localized large diffusion

Abstract

In this paper we study the asymptotic nonlinear dynamics of scalar semilinear parabolic problems reaction-diffusion type when the diffusion coefficient becomes large in a subregion which is interior to the domain. We obtain, under suitable assumptions, that the family of attractors behaves continuously and we exhibit the rate of convergence. An accurate description of localized large diffusion is necessary.

Paper Structure

This paper contains 5 sections, 9 theorems, 111 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Elliptic Problem
  3. Rate of Convergence of Eigenvalues and Equilibria
  4. Rate of Convergence of Invariant Manifolds
  5. Rate of Convergence of Attractors

Key Result

Lemma 2.1

For $g\in L^2_{\Omega_0}$ with $\|g\|_{L^2}\leq 1$ and $\varepsilon\in [0,\varepsilon_0]$, let $u^\varepsilon$ be the solution of elliptic problem Then there is a constant $C>0$ independent of $\varepsilon$ such that

Figures (1)

  • Figure 1: Diffusion

Theorems & Definitions (18)

  • Lemma 2.1
  • proof
  • Corollary 2.2
  • proof
  • Theorem 2.3
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • ...and 8 more