SIAM Journal on Mathematical Analysis, Volume 52, Issue 5, Page 4565-4615, January 2020.
This paper investigates the Keller--Segel model with quadratic cellular diffusion over a disk in $\mathbb R^2$ with a focus on the formation of its nontrivial patterns. We obtain explicit formulas of radially symmetric stationary solutions and such configurations give rise to the ring patterns and concentric airy patterns. These explicit formulas empower us to study the global bifurcation and asymptotic behaviors of the solutions, within which the cell population density has $\delta$-type spiky structures when the chemotaxis rate is large. The explicit formulas are also used to study the uniqueness and quantitative properties of nontrivial stationary radial patterns ruled by several threshold phenomena determined by the chemotaxis rate. We find that all nonconstant radial stationary solutions must have the cellular density be compactly supported unless for a discrete sequence of bifurcation values at which there exist strictly positive small-amplitude solutions. The hierarchy of free energy shows that in the radial class the inner ring solution has the least energy while the constant solution has the largest energy, and all these theoretical results are illustrated through bifurcation diagrams. A natural extension of our results to $\mathbb R^2$ yields the existence, uniqueness, and closed-form solution of the problem in this whole space. Our results are complemented by numerical simulations that demonstrate the existence of nonradial stationary solutions in the disk.

中文翻译：

---

二次扩散的Keller-Segel模型的平稳环和同心环解

SIAM数学分析杂志，第52卷，第5期，第4565-4615页，2020年1月。
本文研究了具有$ \ mathbb R ^ 2 $磁盘上的二次细胞扩散的Keller-Segel模型，重点研究了其非平凡模式的形成。我们获得了径向对称平稳解的显式公式，并且这种构造产生了环形图案和同心的通风图案。这些明确的公式使我们能够研究溶液的整体分叉和渐近行为，其中当趋化率大时，细胞群体密度具有$ \ delta $型尖峰结构。该显式公式还用于研究由趋化速率确定的几个阈值现象所决定的非平凡固定径向模式的唯一性和定量性质。我们发现，除非对于分叉值的离散序列存在严格的正小振幅解，否则所有非恒定径向固定解都必须具有紧密支持的细胞密度。自由能的层次结构表明，在径向类中，内环解决方案具有最小的能量，而恒解具有最大的能量，所有这些理论结果均通过分叉图进行了说明。将结果自然扩展为$ \ mathbb R ^ 2 $，可以得出在整个空间中问题的存在，唯一性和封闭形式的解决方案。我们的结果得到了数值模拟的补充，数值模拟表明了磁盘中非径向固定解的存在。