Analytic regularity for a singularly perturbed system of reaction-diffusion equations with multiple scales: proofs

TL;DR

This paper develops a comprehensive asymptotic analysis for a coupled reaction-diffusion system with two small parameters, providing explicit error bounds and derivative estimates that account for overlapping boundary layers across all parameter regimes.

Contribution

It introduces a full asymptotic expansion with explicit error bounds for a singularly perturbed system with two scales, covering all parameter ranges and analyzing boundary layer interactions.

Findings

Constructed asymptotic expansions with error bounds for all parameter ranges.

Derived explicit derivative growth estimates for the expansion terms.

Analyzed boundary layer interactions in coupled reaction-diffusion systems.

Abstract

We consider a coupled system of two singularly perturbed reaction-diffusion equations, with two small parameters $0< \epsilon \le \mu \le 1$, each multiplying the highest derivative in the equations. The presence of these parameters causes the solution(s) to have \emph{boundary layers} which overlap and interact, based on the relative size of $\epsilon$ and $% \mu$. We construct full asymptotic expansions together with error bounds that cover the complete range $0 < \epsilon \leq \mu \leq 1$. For the present case of analytic input data, we derive derivative growth estimates for the terms of the asymptotic expansion that are explicit in the perturbation parameters and the expansion order.