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

TL;DR

This paper investigates how attractors of certain singularly perturbed parabolic equations converge to those of a reduced ODE, providing estimates of this convergence rate in the Hausdorff metric.

Contribution

It introduces a method to quantify the convergence rate of attractors for singularly perturbed parabolic problems using resolvent operator estimates.

Findings

Attractors converge to the reduced ODE attractor as perturbation parameter tends to zero.

The convergence rate can be explicitly estimated in the Hausdorff metric.

The approach applies to problems with large diffusion in parabolic equations.

Abstract

We exhibit a singularly perturbed parabolic problems for which the asymptotic behavior can be described by an one-dimensional ordinary differential equation. We estimate the continuity of attractors in the Hausdorff metric by rate of convergence of resolvent operator.