Long-time shadow limit for a reaction–diffusion-ODE system

Abstract

Shadow systems are an approximation of reaction–diffusion-type models with the largest diffusion coefficient tending to infinity. They are often used to reduce the model and facilitate its analysis. In this paper we extend the already existing finite time interval analysis to a case of very long time intervals, i.e., time intervals scaled with the diffusion coefficient and tending to infinity for diffusion tending to infinity. The approach is presented for a linear system of a reaction–diffusion equation coupled to ordinary differential equations but it can be extended to a semi-linear case or a classical reaction–diffusion system. In addition to the convergence result, we provide error estimates in terms of a power of the inverse of the diffusion coefficient.