Study of solution space of nonlinear partial differential equations

1.研究了&lt；Planck常数维欧氏空间中的球或环上的Emden-Fowler方程，它是一类椭圆型偏微分方程组。设C是正交群0(&lt；Planck常数&gt；)的闭子群。如果u在G作用下是不变的，则方程的解u(X)称为G不变的。任何径向解都是G不变的。考虑了逆问题。群G是单位球面上的变换群，因为G是正交群的子群。证明了存在G不变的非径向解的充要条件是G在单位球面上不传递。用变分方法、泛函分析、Lie变换群和常微分的Sturm-Liouville理论证明了这一结果。我们研究了描述细胞黏液具有定向运动的数学模型的Keller-Segel方程。(1)考虑了一个抛物型系统，它是Keller-Segel方程的简化。当敏感函数为线性函数且空间维为2时，详细研究了爆破解的渐近性质。(2)证明了当L^1密度集中在原点时，径向解爆破。(3)以解的总质量L^1为参数。然后利用该参数得到了非常数平稳解的存在性和非存在性结果。

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.