Effective Boundary Conditions for the Fisher-KPP Equation on a Domain with 3-dimensional Optimally Aligned Coating

TL;DR

This paper derives effective boundary conditions for the Fisher-KPP equation on a 3D domain with a thin, anisotropic, optimally aligned coating, revealing exotic boundary behaviors like fractional Laplacians that persist over time.

Contribution

It introduces novel effective boundary conditions for the Fisher-KPP equation on complex domains with thin anisotropic layers, including Dirichlet-to-Neumann and fractional Laplacian types.

Findings

Derived EBCs involve Dirichlet-to-Neumann and fractional Laplacian operators.

EBCs remain effective indefinitely, even as time approaches infinity.

The anisotropic layer significantly influences boundary behavior of solutions.

Abstract

We consider the Fisher-KPP equation on a three-dimensional domain surrounded by a thin layer whose diffusion rates are drastically different from that in the bulk. The bulk is isotropic, while the layer is considered to be anisotropic and ``optimally aligned", where the normal direction is always an eigenvector of the diffusion tensor. To see the effect of the layer, we derive effective boundary conditions (EBCs) by the limiting solution of the Fisher-KPP equation as the thickness of the layer shrinks to zero. These EBCs contain some exotic boundary conditions including the Dirichlet-to-Neumann mapping and the Fractional Laplacian. Moreover, we emphasize that each EBC keeps effective indefinitely, even as time approaches infinity.