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

描述细胞粘菌趋化聚集的方程组解的结构

基本信息

批准号：
13640216
负责人：
YOSHIDA Kiyoshi
金额：
$ 2.18万
依托单位：
HIROSHIMA UNIVERSIlY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2003
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640216/
关键词：
Keller-Segel system Chirdress-Percus conjecture global blanch locations of blow up points self-similar solution elliptic equations Lieuville type theorem radially symmetrical solution 粘菌形成方程式系楕円型方程式固有値問題大域分岐 Harnack型不等式大腸菌コロニ

项目摘要

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.

Keller和Segel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。这个模型被称为凯勒-西格尔系统。Chirdress和Percus推测存在一个阈值（8π），使得当初始值小于8π时，解在时间上全局存在，而当初始值大于8π时，则会发生趋化坍缩。我们的目标是解决这个猜想。我们研究R^2中的Keller-Segel系统，因为这种系统很少在孔空间中研究。特别地，我们处理了自相似解，几乎正解了Chirdress和Percus猜想。为此，我们遇到了几个问题，解决了这些问题。我们列举了这些问题：(1)确定自相似解的衰减阶数(2)推导出Lieuville型定理(3)由两个椭圆方程组成的方程组约简为一个椭圆方程(4)用移动平面法证明解的径向对称性(5)确定解的全局白化(6)求解Chirdress和Percus猜想。