# Distance of attractors for thin domains

## TL;DR

This paper studies how attractors of a reaction-diffusion system behave as the domain shrinks to a line, providing precise convergence rates using inertial manifold estimates and Shadowing theory.

## Contribution

It offers new quantitative convergence rates for attractors in thin domains, extending previous inertial manifold estimates to this setting.

## Key findings

- Convergence rates for attractors as domain shrinks
- Application of Shadowing theory to reaction-diffusion equations
- Quantitative bounds on attractor distance in thin domains

## Abstract

In this work we consider a dissipative reaction-diffusion equation in a $d$-dimensional thin domain shrinking to a one dimensional segment and obtain good rates for the convergence of the attractors. To accomplish this, we use estimates on the convergence of inertial manifolds as developed previously in \cite{Arrieta-Santamaria-C0} and Shadowing theory.