Effective boundary conditions for the Fisher-KPP equation on a domain with 3-dimensional optimally aligned coatings

We investigate the Fisher-KPP equation on a three-dimensional domain surrounded by a thin layer whose diffusion rate is drastically different from that in the bulk. The bulk is isotropic, whereas the layer is anisotropic and "optimally aligned", meaning that 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-toNeumann mapping and the Fractional Laplacian. Moreover, we also examine the maximal interval during which these EBCs remain valid. We emphasize that each EBC can keep effective indefinitely, even as time approaches infinity. (c) 2024 Published by Elsevier Inc.