Study of solution space of nonlinear partial differential equations

非线性偏微分方程解空间的研究

基本信息

批准号：
08640223
负责人：
KAJIKIYA Ryuji
金额：
$ 1.47万
依托单位：
Nagasaki Institute of Applied Science
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1996
资助国家：
日本
起止时间：
1996 至 1998
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-08640223/
关键词：
nonlinear elliptic equation group invariant solution variational method blow-up parabolic system chemotaxis parabelic system 走化性生物モデル

项目摘要

1. We study the Emden-Fowler equation, which is one of partial differential equations of elliptic type, in a ball or annulus of <planck's constant>-dimensional Euclid space. Let C be a closed subgroup of the orthogonal group 0(<planck's constant>). A solution mu(x) of the equation is called G invariant if mu is invariant under G action. Any radial solution becomes G invariant. The converse problem is considered. The group G is a transformation group on the unit sphere because G is a subgroup of the orthogonal group. We prove that there exists a G invariant non-radial solution if and only if G is not transitive on the unit sphere. This result is proved by using variational method, functional analysis, Lie transformation group and Sturm-Liouville theory of ordinary differential2. We study the Keller-Segel equation which is a mathematical model to describe a cellular slime having the oriented movement.(1)A parabolic system which is a simplification of the Keller-Segel equation is considered. When a sensitive function is linear and the space dimension is two, the asymptotic behavior of a blow-up solution is investigated in detail.(2)It is proved that a radial solution blows up as its L^1 density concentrates at the origin.(3)The L^1 total mass of a solution is chosen as a parameter. Then the existence and non existence results of non-constant stationary solutions are obtained by using the parameter.

1. 我们研究<普朗克常数>维欧几里得空间的球体或环体中的Emden-Fowler方程，它是椭圆型偏微分方程之一。令 C 为正交群 0（<普朗克常数>）的闭子群。如果 mu 在 G 作用下不变，则方程的解 mu(x) 称为 G 不变。任何径向解都变得 G 不变。考虑相反的问题。群 G 是单位球面上的变换群，因为 G 是正交群的子群。我们证明，当且仅当 G 在单位球面上不具有传递性时，存在 G 不变非径向解。利用变分法、泛函分析、李变换群和常微分的Sturm-Liouville理论证明了这一结果。我们研究了Keller-Segel方程，它是描述具有定向运动的细胞粘液的数学模型。(1)考虑了Keller-Segel方程的简化抛物线系统。当敏感函数为线性且空间维数为2时，详细研究了爆炸解的渐近行为。(2)证明了径向解由于其L^1密度集中在原点而发生爆炸。 (3)选择解的L^1总质量作为参数。然后利用该参数得到非常数平稳解的存在性和不存在性结果。