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 to describe the boundary layer formation in the air-uid interface in any dimensions. By considering a special case where uid 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 boundarylayer pro le as ε → 0, where the boundary-layer thickness is of order ε. When the domain is a ball, we nd a re ned asymptotic boundary layer pro le up to the rst-order approximation of ε by which we nd that the slope of the layer pro le 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.

Radial solutions to a chemotaxis-consumption model involving prescribed signal concentrations on the boundary

“…In the Dirichlet setting this article is concerned with we mention [20] where an asymptotic analysis of the vanishing diffusivity limit for v in the stationary system seems to confirm the potential of (1.1) to capture pattern dynamics. In the time-dependent problem (including fluid flow), solutions in R 2 ×[0, 1] were constructed in [22] if the signal consumption was strong, at least quadratic with respect to u, and in Ω ⊂ R 2 in [32], in both cases under a smallness condition on the initial data.…”

“…However, we will later show that α is in one-to-one correspondence with Ω u. (An alternative would be to compute α = m Ω e v and work with the nonlocal equation for v, see [20], where this approach was used for the special case of constant boundary value. )…”

Asymptotic stability of exogenous chemotaxis systems with physical boundary conditions

“…On the half line R + = (0, ∞), the existence and stability of the unique stationary solution (ū, v) of (1.1)-(1.2) with φ(v) = ln v was recently established in [5] for any m ≥ 0, where (ū, v) is of a boundary (spike, layer) profile as D > 0 is small. When φ(v) = v, the existence of stationary solutions to (1.1)-(1.2) with m = 1 was proved in [19] for all dimensions and the existence of global weak solutions was established in [36] in one dimension. The local existence of weak solutions to (1.3) on the water-drop shaped domain as in [32] with (1.2) and w| ∂Ω = 0 was proved in [26].…”

“…The results of [19] assert that the stationary problem (1.6) with D > 0 admits a unique nonconstant classical solution (ū, v) which is of a boundary layer profile as D > 0 is small. While for the case D = 0, the system (1.4) with (1.5b) clearly has a unique constant solution (M, 0).…”

Construction of unique mild solution and continuity of solution for the small initial data to 1-D Keller-Segel system

“…Miura [15] deals with the system on the no-flux and Neumann boundary conditions. In Lee, Wang and Yan [14], the system with the velocity field of a fiuid fiow governed by the incompressible Navier-Stokes equation is considered with zero-flux/Dirichlet/no-slip boundary conditions, and Winkler [20] is discussed about the small-mass solutions with no-flux/no-flux/Dirichlet boundary conditions in the system coupled to the Navier-Stokes equations. On the other hand, there is a little research (although, cf.…”