Rate of Convergence of Attractors for Singularly Perturbed Semilinear Problems

TL;DR

This paper studies how attractors of certain singularly perturbed parabolic problems converge to those of simpler systems, providing estimates on the rate of convergence using resolvent operators, with applications in homogenization and diffusion analysis.

Contribution

It introduces a method to estimate the convergence rate of attractors in singularly perturbed parabolic problems using resolvent operator convergence.

Findings

Convergence of attractors can be quantified in the Hausdorff metric.

Application to spatial homogenization demonstrates practical relevance.

Large diffusion effects near specific points are analyzed.

Abstract

We exhibit a class of singularly perturbed parabolic problems which the asymptotic behavior can be described by a system of ordinary differential equation. We estimate the convergence of attractors in the Hausdorff metric by rate of convergence of resolvent operators. Application to spatial homogenization and large diffusion except in a neighborhood of a point will be considered.