Boundary spike‐layer solutions of the multi‐dimensional singular Keller–Segel system: Existence, profiles and stability

Abstract This paper investigates boundary‐layer solutions of the singular Keller–Segel system (proposed in Keller and Segel [J. Theor. Biol. 30 (1971), 377–380]) in multi‐dimensional domains, which describes cells' chemotactic movement toward the concentration gradient of the nutrient they consume, subject to a zero‐flux boundary condition for the cell density and a Dirichlet boundary condition for the nutrient. The steady‐state problem of the system reduces to a scalar nonlocal Dirichlet elliptic problem with a singularity. By analyzing this nonlocal problem, we establish the existence of a unique steady‐state solution which forms a boundary spike‐layer profile as the nutrient diffusion coefficient . For radially symmetric domains, we derive explicit expansions for the boundary‐layer steepness and thickness in terms of the domain radius (for small ), which quantifies the influence of radius on the profile and thickness. Additionally, we prove the nonlinear exponential stability of this boundary‐layer steady state in radially symmetric domains. The key challenge in our analysis is the emergence of a singularity for small in both stationary and time‐dependent problems. To address this, we reduce the nonlocal steady‐state problem to a local one and conduct a refined analysis via the barrier method and Fermi coordinates, yielding sharp estimates for the local steady‐state solution near the boundary. This approach enables us to determine the asymptotic profile of the nonlocal problem's solution as , accurately capturing and properly resolving the singularity to establish our main results. For the time‐dependent problem in radially symmetric domains, we employ a variable transformation to eliminate the singularity, ultimately proving the nonlinear stability of the unique steady‐state solution. Our analysis leverages the equation governing the radial mass distribution function relative to the steady state, along with delicate time‐weighted energy estimates.