Boundary-layer profile of a singularly perturbed nonlocal semi-linear problem arising in chemotaxis

This paper is concerned with the stationary problem of an aero-taxis system with physical boundary conditions proposed by Tuval et al (2005 Proc. Natl Acad. Sci. 102 2277-82) to describe the boundary layer formation in the air-fluid interface in any dimensions. By considering a special case where fluid is free, the stationary problem is essentially reduced to a singularly perturbed nonlocal semi-linear elliptic problem. Denoting the diffusion rate of oxygen by ϵ > 0, we show that the stationary problem admits a unique classical solution of boundary-layer profile as ϵ → 0, where the boundary-layer thickness is of order ϵ. When the domain is a ball, we find a refined asymptotic boundary layer profile up to the first-order approximation of ϵ by which we find that the slope of the layer profile in the immediate vicinity of the boundary decreases with respect to (w.r.t.) the curvature while the boundary-layer thickness increases w.r.t. the curvature.