Long-time shadow limit for reaction-diffusion-ODE systems

TL;DR

This paper investigates the long-time accuracy of shadow systems as approximations for reaction-diffusion-ODE systems, providing uniform error estimates and conditions for their validity over extended periods.

Contribution

It introduces a method to derive uniform error estimates for shadow systems over long times, including conditions for their validity and examples demonstrating their applicability.

Findings

Uniform error estimates are established for long-time shadow system approximations.

Stronger assumptions are necessary for global-in-time error control.

Examples illustrate the applicability and limitations of the theoretical results.

Abstract

Shadow systems are an approximation of reaction-diffusion-type problems obtained in the infinite diffusion coefficient limit. They allow reducing complexity of the system and hence facilitate its analysis. The quality of approximation can be considered in three time regimes: (i) short-time intervals taking account for the initial time layer, (ii) long-time intervals scaling with the diffusion coefficient and tending to infinity for diffusion tending to infinity, and (iii) asymptotic state for times up to $T = \infty$. In this paper we focus on uniform error estimates in the long-time case. Using linearization at a time-dependent shadow solution, we derive sufficient conditions for control of the errors. The employed methods are cut-off techniques and $L^p$-estimates combined with stability conditions for the linearized shadow system. Additionally, we show that the global-in-time extension of the uniform error estimates may fail without stronger assumptions on the model linearization. The approach is presented on example of reaction-diffusion equations coupled to ordinary differential equations (ODEs), including classical reaction-diffusion system. The results are illustrated by examples showing necessity and applicability of the established conditions.