Boundary spike‐layer solutions of the singular Keller–Segel system: existence and stability

Jose A Carrillo,Jingyu Li,Zhi‐An Wang

doi:10.1112/plms.12319

Abstract

We exploit the existence and nonlinear stability of boundary spike/layer solutions of the Keller-Segel system with logarithmic singular sensitivity in the half space, where the physical zero-flux and Dirichlet boundary conditions are prescribed. We first prove that, under above boundary conditions, the Keller-Segel system admits a unique boundary spike-layer steady state where the first solution component (bacterial density) of the system concentrates at the boundary as a Dirac mass and the second solution component (chemical concentration) forms a boundary layer profile near the boundary as the chemical diffusion coefficient tends to zero. Then we show that this boundary spike-layer steady state is asymptotically nonlinearly stable under appropriate perturbations. As far as we know, this is the first result obtained on the global well-posedness of the singular Keller-Segel system with nonlinear consumption rate. We introduce a novel strategy of relegating the singularity, via a Cole-Hopf type transformation, to a nonlinear nonlocality which is resolved by the technique of "taking antiderivatives", i.e. working at the level of the distribution function. Then, we carefully choose weight functions to prove our main results by suitable weighted energy estimates with Hardy's inequality that fully captures the dissipative structure of the system.

Highlights

In their seminal work 16, Keller and Segel proposed the following singular chemotaxis system ut = uxx − χ[u(ln w)x]x, wt = εwxx − uwm, (1.1)to describe the propagation of traveling bands of chemotactic bacteria observed in the celebrated experiment of Adler 1, where u(x, t) denotes the bacterial density and w(x, t) the oxygen/nutrient concentration. ε 0 is the chemical diffusion coefficient, χ > 0 denotes the chemotactic coefficient and m 0 the oxygen consumption rate
To describe the propagation of traveling bands of chemotactic bacteria observed in the celebrated experiment of Adler 1, where u(x, t) denotes the bacterial density and w(x, t) the oxygen/nutrient concentration. ε 0 is the chemical diffusion coefficient, χ > 0 denotes the chemotactic coefficient and m 0 the oxygen consumption rate
The prominent feature of the Keller–Segel system (1.1) is the use of a logarithmic sensitivity function ln w, which was experimentally verified later in 14. This logarithm results in a mathematically unfavorable singularity which, has been proved to be necessary to generate traveling wave solutions that were the first kind results obtained for the Keller–Segel system (1.1)

Summary

Introduction

The prominent feature of the Keller–Segel system (1.1) is the use of a logarithmic sensitivity function ln w, which was experimentally verified later in 14 This logarithm results in a mathematically unfavorable singularity which, has been proved to be necessary to generate traveling wave solutions (cf 27) that were the first kind results obtained for the Keller–Segel system (1.1). When m = 1, a Cole–Hopf type transformation was cleverly used to remove the singularity, which led to a lot of interesting analytical works, for instance the stability of traveling waves (cf 3, 4, 6, 13, 18, 22–25), global well-posedness and/or asymptotic behavior of solutions (see 5, 8, 20, 21, 26, 28, 32, 37, 42, 43 in onedimensional bounded or unbounded space and 7 9 19, 20 33 35 40, 41 in multidimensional spaces) and boundary layer solutions 10–12. We shall develop some new ideas to establish the existence, uniqueness and stability of steady states to the Keller–Segel system (1.1)–(1.3) with m 0.

Boundary spike-layer steady states

Reformulation of the problem

Case m 1

U xx χV U x