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

Jose A. Carrillo (Corresponding Author), Jingyu Li, Zhi An Wang

Research output: Journal article publicationJournal articleAcademic researchpeer-review

18 Citations (Scopus)


We explore 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 anti-derivatives’, that is, 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.

Original languageEnglish
Pages (from-to)42-68
Number of pages27
JournalProceedings of the London Mathematical Society
Issue number1
Publication statusPublished - 1 May 2020


  • 35A01
  • 35B40
  • 35K57
  • 35Q92
  • 76D10
  • 92C17 (primary)

ASJC Scopus subject areas

  • General Mathematics


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