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

Jose A. Carrillo, Jingyu Li, Zhi-An Wang, Wen Yang

Abstract

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.

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

Abstract

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 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 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 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 so that the singularity is accurately captured and hence properly handled to achieve our results.

Paper Structure

This paper contains 13 sections, 25 theorems, 302 equations, 3 figures.

Table of Contents

  1. introduction
  2. Main results and conjectures
  3. Existence and uniqueness
  4. Existence
  5. Uniqueness
  6. boundary-layer profile and thickness
  7. The radial case
  8. Asymptotic profile near the boundary
  9. Asymptotic profile as $p \to \infty$
  10. Proof of Theorem \ref{['radial']}
  11. Nonlinear stability of the radial boundary-layer steady state
  12. A preliminary result
  13. Nonlinear stability of $(U,V)(r)$ on $B_R(0)$

Key Result

Theorem 2.1

Let $\Omega$ be a bounded smooth domain in $\mathbb{R}^n~(n\ge1)$ with smooth boundary and let $m$ and $b$ be given positive constants. Then for any $\varepsilon>0$, the nonlocal problem 4 admits a unique positive classical solution $W_\varepsilon\in C^1(\overline{\Omega})\cap C^\infty(\Omega)$ and

Figures (3)

  • Figure 1: (a) A schematic of domain $\Omega_\rho$. (b) Illustration of radial boundary-layer thickness
  • Figure 2: Numerical simulations of steady-state boundary layer profiles of \ref{['KS']} in a disk (first row) and in an ellipse (second row) with the same area, where the parameter values are $p=5, \varepsilon=0.1, b=1$ and initial value $(u_0, w_0)=(1,1)$.
  • Figure 3: Numerical simulations of steady-state boundary layer profiles of \ref{['KS']} in a two dimensional general domain, where the parameter values are $p=5, \varepsilon=0.1, b=1$ and initial value $(u_0, w_0)=(1,1)$.

Theorems & Definitions (49)

  • Theorem 2.1
  • Definition 2.1
  • Theorem 2.2
  • Remark 2.1
  • Theorem 2.3
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4: Nonlinear stability of the radial steady state
  • proof : Proof of Theorem \ref{['th1.1']}
  • Lemma 4.1
  • ...and 39 more