SIAM Journal on Applied Mathematics, Volume 80, Issue 1, Page 232-261, January 2020.
We show that the Keller--Segel model in one dimension with Neumann boundary conditions and quadratic cellular diffusion has an intricate phase transition diagram depending on the chemosensitivity strength. Explicit computations allow us to find a myriad of symmetric and asymmetric stationary states whose stability properties are mostly studied via free energy decreasing numerical schemes. The metastability behavior and staircased free energy decay are also illustrated via these numerical simulations.

中文翻译：

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具有体积排阻的Keller-Segel模型的相变和凸点解

SIAM应用数学杂志，第80卷，第1期，第232-261页，2020年1月。
我们显示，一维具有诺伊曼边界条件和二次细胞扩散的Keller-Segel模型根据化学敏感性具有复杂的相变图。强度。显式计算使我们能够找到大量对称和不对称的稳态，其稳定性主要通过自由能下降数值方案来研究。通过这些数值模拟还说明了亚稳行为和阶梯式自由能衰减。