Global solution structure and the stability of nonlocal nonlinear second order boundary value problems with definite integrals

非局部非线性二阶定积分边值问题的全局解结构与稳定性

• 批准号: 15540220
• 负责人: YOTSUTANI Shoji
• 金额: $ 2.3万
• 依托单位: Ryukoku University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2003
• 资助国家: 日本
• 起止时间: 2003 至 2005
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540220/
• 关键词: cross-diffusion; nonlinear elliptic equation; nonlocal; nonlinear boundary value problem; periodic boundary condition; Ginzburg-Landau equation; positive radial solution; singular solution; 反応拡散方程式; 球対称解; 標準形

First, Lou-Ni-Yotsutani [DCDS 2004] investigated a limiting equation to a cross-diffusion equation that appears in mathematical biology. We showed that it has different kinds of singular solutions and revealed the structure of all solutions.This problem is a nonlocal nonlinear elliptic boundary problem, for which no method was known to solve it. We discovered a new method, which are the combination of the modern method of PDE and classical analysis and algebra.There are a lot of interesting problems for which our method is applicable.A problem of the Oseen's spiral flow is one of them, for which we obtained the complete bifurcation diagram in Ikeda-Kondo-Okamoto-Yotsutani [CPAA 2003].Matsumoto-Murai-Yotsutani [Pisa, 2005] gave the complete answer for a problem to determine curves with the least energy under the given length.Second, Kosugi-Morita-Yotsutani [CPAA 2005, J.Math.Phy. 2005] have revealed the complete Global bifurcation branches one dimensional Ginzburg-Landau equations with periodic boundary conditions.

首先，Lou-Ni-Yotsutani [DCDS 2004] 研究了数学生物学中出现的交叉扩散方程的极限方程。我们证明了它具有不同种类的奇异解，并揭示了所有解的结构。该问题是一个非局部非线性椭圆边界问题，目前还没有解决它的方法。我们发现了一种新方法，它是现代偏微分方程方法与经典分析和代数的结合。我们的方法适用于很多有趣的问题。Oseen 螺旋流问题就是其中之一，它我们在 Ikeda-Kondo-Okamoto-Yotsutani [CPAA 2003] 中获得了完整的分叉图。Matsumoto-Murai-Yotsutani [Pisa, 2005] 给出了完整的分叉图确定给定长度下具有最小能量的曲线问题的答案。第二，Kosugi-Morita-Yotsutani [CPAA 2005，J.Math.Phy。 2005]揭示了具有周期性边界条件的完整的全局分岔分支一维Ginzburg-Landau方程。

项目成果

Existence and global stability of traveling curved front in the Allen-Cahn equations

Allen-Cahn 方程中行进曲锋的存在性及其全局稳定性

• 发表时间: 2005
• 期刊: J.Differential Equations 213
• 作者: H.Ninomiya;M.Taniguchi
• 通讯作者: M.Taniguchi

E.Yanagida, S.Yotsutani: "Recent Topics on Nonlinear Partial Differential Equations : Structure of Radial Solutions for Semilinear Elliptic Equations"Amer.Math.Soc.Transl., Series : Selected Papers on Analysis and Differential Equations. 2・211. 121-137 (2

E.Yanagida、S.Yotsutani：“非线性偏微分方程的最新主题：半线性椭圆方程的径向解的结构”Amer.Math.Soc.Transl.，系列：分析和微分方程论文选集 2·211。 -137（2

J.-S.Guo, Y.Morita: "Entire solutions of reation-diffusion equations and an application to discrete diffusive equations"Discrete and Continuous Dynamical Systems. to appear. (2004)

J.-S.Guo，Y.Morita：“反应扩散方程的完整解及其在离散扩散方程中的应用”离散和连续动力系统。

Existence and global stability of traveling curved fronts in the Allen-Cahn equations

• DOI: 10.1016/j.jde.2004.06.011
• 发表时间: 2005-06
• 期刊: Journal of Differential Equations
• 影响因子: 2.4
• 作者: H. Ninomiya;M. Taniguchi

Recent topics on nonlinear partial differential equations : Selected Papers on Analysis and Differential Equations

非线性偏微分方程的最新主题：分析和微分方程论文精选

• 发表时间: 2003
• 期刊: Amer.Math.Soc.Transl.Series. 2・211
• 作者: E.Yanagida;S.Yotsutani
• 通讯作者: S.Yotsutani