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emden-fowler方程，这是椭圆类型的部分微分方程之一，在<planck的常数>维二维欧几里得空间中。令C为正交组0的封闭子组（<Planck的常数>）。如果MU在G动作下不变，则方程的溶液MU（X）称为G不变。任何径向溶液都会变不变。考虑了相反的问题。 G组是单位球体上的转换组，因为G是正交组的子组。我们证明，只有当G在单位球体上不传递G时，就存在G不变的非辐射解。通过使用变分方法，功能分析，谎言转化组和普通差分理论的Sturm-liouville理论证明了这一结果。我们研究了凯勒 - 塞格方程，这是一个数学模型，用于描述具有方向运动的细胞粘液。（1）抛物线系统是简化凯勒 - 塞格方程的简化。当灵敏的函数是线性的，空间维度为两个时，详细研究了爆炸溶液的渐近行为。（2）证明，径向溶液将其l^1密度浓缩在原点上爆炸。（3）溶液的l^1总质量被选为一个参数。然后，使用参数获得了非固定固定溶液的存在和非存在结果。