Geometry effects on the boundary-layer profiles of the Keller-Segel system

We consider the boundary-layer problem of a nonlocal semilinear elliptic equation in a bounded smooth domain of all dimensions with the Dirichlet boundary condition, which arises as the stationary problem of the Keller-Segel system with physical boundary conditions describing the boundary-layer formation driven by chemotaxis. Using the Fermi coordinates and delicate analysis with subtle estimates, we rigorously derive the asymptotic expansion of the boundary-layer profile and thickness in terms of the small diffusion rate with coefficients explicitly expressed by the domain geometric properties including mean curvature, volume and surface area. By these expansions, one can explicitly find the joint impact of the mean curvature, surface area and volume of the spatial domain on the boundary-layer steepness and thickness. This seems to be the first result revealing how the boundary-layer profiles depend on the domain geometries for chemotaxis models.