Structure of solutions to the system of equations describing chemotactic aggregation of cellular slime molds

凯勒和塞格尔推导出了描述细胞黏菌向高浓度化学物质移动的趋化聚集的数学模型。因此，这种模型被称为凯勒-塞格尔系统。Chirresse和Percus猜想存在一个阈值数(8π)，使得如果初始值小于8π，解在时间上是全局存在的，反之，如果初始值大于8π，则可能发生趋化崩溃。我们的目标是解决这个猜想。我们研究R^2中的Keller-Segel系统，因为这个系统很少在洞空间中被研究。特别地，我们处理了自相似解，并且几乎正解了Chirresse和Percus猜想。为此，我们遇到了几个问题，并解决了这些问题。我们列举了以下问题：(1)确定自相似解的衰减阶(2)导出Lieuvile型定理(3)从两个椭圆型方程组化为一个椭圆型方程(4)用移动平面法证明解的径向对称性(5)确定解的整体白化(6)求解Chirresse和Percus猜想。

Keller and Segel derived the mathematical model describing chemotactic aggregation of cellular slime molds which move toward high cocentrations of chemical substance. So this model is called as the Keller-Segel system. Chirdress and Percus conjectured that there exists a threshold number (8π) such that if an initial value is smaller than 8π, the solutions exist globally in time, on the other hand if an initial value is greater than 8π, the chemotactic collapse can occure. Our objective is to solve this conjecture. We study the Keller-Segel system in R^2 because this system is rarely studied in hole space. Especially we treated the self-similar solution and solved almost positively the Chirdress and Percus conjectuer. For this purpose we encountered several problems and solved these. We enumerate these problems : (1)determine decay order of the self-similar solutions (2)derive the Lieuville type theorem (3)reduction to one elliptic equation from an system of two elliptic equations (4)show the radial symmetry of solutions by using the moving plane method (5)determine the global blanch of solutions (6)solve the Chirdress and Percus conjectuer.