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

Journal article

This paper investigates boundary-layer solutions of the singular Keller-Segel system (proposed in [19]) 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 ε → 0. For radially symmetric domains, we derive explicit expansions for the boundary-layer steepness and thickness in terms of the domain radius (for small ε > 0), 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 ε 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 ε → 0, 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.

Authors: Carrillo, JA; Li, J; Wang, Z-A; Yang, W

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Wiley
• Journal: Proceedings of the London Mathematical Society
• Volume: 132
• Issue: 2
• Article number: e70122
• Publication date: 2026-02-11
• Acceptance date: 2025-12-25
• DOI: 10.1112/plms.70122
• EISSN: 1460-244X
• ISSN: 0024-6115
• Language: English
• Keywords: math.AP; 35K57; 35Q92; 92D25