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

TL;DR

This paper investigates how the attractors of scalar semilinear parabolic reaction-diffusion problems behave as the diffusion coefficient becomes large in a localized interior region, providing convergence rates and continuity results.

Contribution

It offers a detailed analysis of the convergence rate of attractors in singularly perturbed problems with localized large diffusion, which was previously less understood.

Findings

Attractors depend continuously on the diffusion parameter.

Explicit convergence rates of attractors are established.

Localized large diffusion significantly influences the asymptotic dynamics.

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.