Fick's las selects the Neumann boundary condition

TL;DR

This paper investigates how a boundary condition emerges along an interface in a reaction-diffusion system with heterogeneous diffusivity as the diffusivity parameter approaches zero, demonstrating the limit satisfies a Neumann boundary condition.

Contribution

It provides a rigorous analysis of the boundary condition formation in reaction-diffusion equations with heterogeneous diffusion laws as diffusivity tends to zero.

Findings

The limit satisfies the homogeneous Neumann boundary condition.

The approach applies heterogeneous diffusion laws to domain geometry effects.

The study connects diffusivity heterogeneity with boundary condition emergence.

Abstract

We study the appearance of a boundary condition along an interface between two regions, one with constant diffusivity $1$ and the other with diffusivity $\eps>0$, when $\eps\to0$. In particular, we take Fick's diffusion law in a context of reaction-diffusion equation with bistable nonlinearity and show that the limit of the reaction-diffusion equation satisfies the homogeneous Neumann boundary condition along the interface. This problem is developed as an application of heterogeneous diffusion laws to study the geometry effect of domain.