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

Abstract

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 ɛ > 0, we show that the stationary problem admits a unique classical solution of boundary-layer profile as ɛ → 0, where the boundary-layer thickness is of order ɛ. When the domain is a ball, we find a refined asymptotic boundary layer profile up to the first-order approximation of ɛ 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.