Study of Nonlinear Partial Differential Equations from the View Point of Functional Analysis

从泛函分析的角度研究非线性偏微分方程

• 批准号: 01460004
• 负责人: MASUDA Kyuya
• 金额: $ 3.97万
• 依托单位: University of Tokyo
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for General Scientific Research (B)
• 财政年份: 1989
• 资助国家: 日本
• 起止时间: 1989 至 1990
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-01460004/
• 关键词: reaction-diffusion system; nonlinear heat equation; blow-up of solutions; bifurcation theory; Riemannian manifold; water wave; 表面張力方程式; 多様体上のベキレイ構造; 非線型反応拡散系; 非線型拡散方程式; 楕円型方程式; ラプラス変換; 境界値混合問題; 順序保存力学系

We have studied nonlinear partial differential equations from many aspects in the period 1989 May-1991 March. Masuda succeeded first in showing the existence of stable periodic solutions of reaction-diffusion systems of some type, the most important class of nonlinear parabolic equations. He also considered the heat equation in the form u=*u+a (x) u^p and showed any non-negative solution blows up in a finite time if a (x) is posive at some point in the domain considered. Matano considered the nonlinear heat equation in some one-dimensional space interval with Dirichlet boundary condition.He could show the remarkable result that any solutions with a continuous function as the initial data blows up at a (at most) finite number of points.Ishimura showed the existence and nonexistence of multi-valued solutions of nonlinear elliptic equations of the form *u=u^p. shouji made research into the permanent progressive waves and analyzed the bifurcation of progressive waves. Fukaya made research into nonlinear differential equations in differential geometry. He clarified the structure of small ball in the Riemannian manifold with sectional curvature less than one in absolute value.Kataoka succeeded in showing the hypoellipticity of degenerate elliptic equations.

1989年5月至1991年3月期间，我们从多个方面研究了非线性偏微分方程。Masuda首先成功地证明了某些类型的反应扩散系统的稳定周期解的存在性，这是最重要的一类非线性抛物方程。他还考虑了形式为u=*u+a(X)u^p的热方程，并证明了如果a(X)在所考虑的区域中的某一点为正，则任何非负解在有限时间内爆破。Matano研究了具有Dirichlet边界条件的一维空间中的非线性热方程，证明了任何以连续函数为初值的解在(至多)有限点处爆破。石村证明了形式为*u=u^P的非线性椭圆型方程多值解的存在性和不存在性。Shouji研究了永久行波，并分析了行波的分支。Fukaya对微分几何中的非线性微分方程进行了研究。他阐明了黎曼流形中截面曲率绝对值小于1的小球的结构。片冈成功地证明了退化椭圆型方程的亚椭圆性。

项目成果

深谷 賢治: "Hausdorff Covergence of Riemannian manifolds and its applications" Advanced Studies in Pure Math.18. 143-238 (1990)

Kenji Fukaya：“黎曼流形的豪斯多夫收敛及其应用”纯数学高级研究143-238（1990）。

増田 久弥: "Asymptotic behavior of sosolutions of reactionーdiffusion systems of Volterra type" J.Differential Equations.

Hisaya Masuda：“Volterra 型反应扩散系统解的渐近行为”J.Differential Equations。

Kyuya Masuda,: "Asymptotic behavior of solutions of reaction-diffusion systems of Volterra type" J. Differential Equations.

Kyuya Masuda，：“Volterra 型反应扩散系统解的渐近行为”J. 微分方程。

Naoyuki Ishimura,: "Nonlinear eigenvalue problem associated with the generalized capollarity equation" J. Fac. Sci. Univ. Tokyo Sect. IA,. 37. 457-466 (1990)

Naoyuki Ishimura，：“与广义毛极性方程相关的非线性特征值问题”J. Fac。

Toshiyuki Kobayashi: "Asymptotic behaviours of the null vaiety for a convex domain in a non-positively curved space form" Journal of the Faculty of the Science,the University of Tokyo. 36. 389-478 (1989)

Toshiyuki Kobayashi：“非正弯曲空间形式中凸域的零簇的渐近行为”，东京大学理学院学报。