Boundary spike‐layer solutions of the singular Keller–Segel system: existence and stability
José A. Carrillo, Jingyu Li, Zhi‐An Wang
Abstract
We explore the existence and nonlinear stability of boundary spike/layer solutionsof the Keller-Segel system with logarithmic singular sensitivity in the half space, where thephysical zero-flux and Dirichlet boundary conditions are prescribed. We first prove that, underabove boundary conditions, the Keller-Segel system admits a unique boundary spike-layer steadystate where the first solution component (bacterial density) of the system concentrates at theboundary as a Dirac mass and the second solution component (chemical concentration) formsa 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 stableunder appropriate perturbations. As far as we know, this is the first result obtained on theglobal well-posedness of the singular Keller-Segel system with nonlinear consumption rate. Weintroduce 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 toprove our main results by suitable weighted energy estimates with Hardy’s inequality that fullycaptures the dissipative structure of the system.