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.

中文翻译：

---

奇异Keller-Segel系统的边界峰值层解：存在与稳定性

我们利用对数奇异灵敏度在半空间中规定了物理零通量和Dirichlet边界条件的Keller-Segel系统的边界尖峰/层解的存在性和非线性稳定性。我们首先证明，在上述边界条件下，Keller-Segel系统接受唯一的边界尖峰层稳态，其中系统的第一溶液组分（细菌密度）以狄拉克质量集中在边界，第二溶液组分（细菌密度）集中在边界处。当化学扩散系数趋于零时，在边界附近形成边界层轮廓。然后我们表明，在适当的扰动下，边界尖峰层稳态是渐近非线性稳定的。据我们所知，这是具有奇异非线性消耗率的Keller-Segel系统的整体适定性的第一个结果。我们介绍了一种通过Cole-Hopf类型变换将奇异性降级为非线性非局部性的新策略，该非线性非局部性通过“采用反导数”技术来解决，即在分布函数的层次上工作。然后，我们仔细选择权重函数，以通过具有Hardy不等式的适当加权能量估计来证明我们的主要结果，该不等式完全捕获了系统的耗散结构。在分配功能级别上工作。然后，我们仔细选择权重函数，以通过具有Hardy不等式的适当加权能量估计来证明我们的主要结果，该不等式完全捕获了系统的耗散结构。在分配功能级别上工作。然后，我们仔细选择权重函数，以通过具有Hardy不等式的适当加权能量估计来证明我们的主要结果，该不等式完全捕获了系统的耗散结构。