Boundary spike-layer solutions of the singular Keller-Segel system: existence, profiles and stability

This paper is concerned with the boundary-layer solutions of the singular Keller-Segel model proposed by Keller-Segel (1971) in a multi-dimensional domain, where the zero-flux boundary condition is imposed to the cell while inhomogeneous Dirichlet boundary condition to the nutrient. The steady-state problem of the Keller-Segel system is reduced to a scalar Dirichlet nonlocal elliptic problem with singularity. Studying this nonlocal problem, we obtain the unique steady-state solution which possesses a boundary spike-layer profile as nutrient diffusion coefficient $\varepsilon>0$ tends to zero. When the domain is radially symmetric, we find the explicit expansion for the slope of boundary-layer profiles at the boundary and boundary-layer thickness in terms of the radius as $\varepsilon>0$ is small, which pinpoints how the boundary curvature affects the boundary-layer profile and thickness. Furthermore, we establish the nonlinear exponential stability of the boundary-layer steady-state solution for the radially symmetric domain. The main challenge encountered in the analysis is that the singularity will arise when the nutrient diffusion coefficient $\varepsilon>0$ is small for both stationary and time-dependent problems. By relegating the nonlocal steady-state problem to local problems and performing a delicate analysis using the barrier method and Fermi coordinates, we can obtain refined estimates for the solution of local steady-state problem near the boundary. This strategy finally helps us to find the asymptotic profile of the solution to the nonlocal problem as $\varepsilon \to 0$ so that the singularity is accurately captured and hence properly handled to achieve our results.