The number of nonoscillatory solutions and the length and the fractal dimension for two-point boundary value problems

两点边值问题的非振荡解的数量以及长度和分形维数

• 批准号: 23740113
• 负责人: TANAKA Satoshi
• 金额: $ 1.66万
• 依托单位: Okayama University of Science
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Young Scientists (B)
• 财政年份: 2011
• 资助国家: 日本
• 起止时间: 2011 至 2013
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-23740113/
• 关键词: ２点境界値問題; 正値解; 球対称解; 一意性; フラクタル次元; 国際情報交換; 振動解; 解の個数; クロアチア; 国際研究者交流; 台湾

In this study, two-point boundary value problems for nonlinear ordinary differential equations were considered. The positive solution is a kind of nonoscillatory solutions. The symmetry-breaking for positive solutions was proved by finding the Morse index of positive solutions. Then the existence of positive symmetry solutions and positive asymmetry solutions was shown, and therefore, a result of the nonuniqueness of positive solutions was established. A sufficient condition is obtained for the existence of exact two positive solutions of problems with one dimensional p-Laplacian. The fractal dimension of radial oscillatory solutions for a class of elliptic partial differential equations was found.

在本研究中，考虑了非线性常微分方程的两点边值问题。正解是一种非振荡解。通过求正解的莫尔斯指数证明了正解的对称性破缺。然后证明了正对称解和正不对称解的存在性，从而建立了正解非唯一性的结果。给出了一维p-拉普拉斯问题的精确两个正解存在的充分条件。找到了一类椭圆偏微分方程径向振荡解的分形维数。

项目成果

Morse index and symmetry-breaking for positive solutions of one-dimensional Hénon type equations

• DOI: 10.1016/j.jde.2013.05.029
• 发表时间: 2013-10
• 期刊: Journal of Differential Equations
• 影响因子: 2.4
• 作者: Satoshi Tanaka

2点境界値問題の正値解の対称性の破れ

两点边值问题正解的对称性破缺

• 发表时间: 2014
• 作者: 永安聖 (講演内容にはGunther Uhlmann 氏;Jenn-Nan Wang 氏との共同研究;及びJishan Fan 氏;Kyoungsun Kim 氏;中村玄氏との共同研究を含む);宮尾 忠宏;S. Nishibata and M. Suzuki;J. Sugie and M. Onitsuka;田中 敏
• 通讯作者: 田中 敏

Fractal oscillations of self-adjoint and damped linear differential equations of second-order

• DOI: 10.1016/j.amc.2011.07.047
• 发表时间: 2011-11
• 期刊: Appl. Math. Comput.
• 作者: Mervan Pašić;Satoshi Tanaka

2階線形常微分方程式の振動解のフラクタル次元

二阶线性常微分方程振荡解的分形维数

• 发表时间: 2012
• 作者: T. Ogawa;H. Takeda,;田中 敏
• 通讯作者: 田中 敏

Fractal oscillations near the domain boundary of radially symmetric solutions of p-Laplace equations

p-拉普拉斯方程径向对称解域边界附近的分形振荡

• 发表时间: 2014
• 期刊: Contemporary Mathematics 601 : Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics II
• 作者: Y. Naito;M. Pasic;S. Tanaka;D. Zubrinic
• 通讯作者: D. Zubrinic