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。我们证明了该问题具有不同类型的奇异解，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值问题，目前还没有一种求解方法.我们发现了一种新的方法，它是现代偏微分方程方法与经典分析和代数方法的结合，有许多有趣的问题是我们的方法所适用的. Oseen螺旋流问题在Ikeda-Kondo-Okamoto-Yotsutani [CPAA 2003]中我们得到了完全的分岔图，Matsumoto-Murai-Yotsutani [比萨，2005]给出了在给定长度下确定具有最小能量的曲线问题的完全答案。2005]揭示了具有周期边界条件的一维Ginzburg-Landau方程的完全全局分支。

项目成果

On a limiting system in the Lotka-Volterra competition with cross-diffusion,

关于具有交叉扩散的 Lotka-Volterra 竞争的限制系统，

• 发表时间: 2004
• 期刊: Discrete and Continuous Dynamical Systems Vol.10, No.1
• 作者: Lou;Y.;Ni;W.-M.;Yotsutani;S
• 通讯作者: S

Global bifurcation structure of a one-dimensional Ginzburg-Landau model

一维 Ginzburg-Landau 模型的全局分叉结构

• 发表时间: 2005
• 期刊: J. Math. Physics Vol.46
• 作者: S.Kosugi;S.Kosugi;S.Kosugi;S.Kosugi
• 通讯作者: S.Kosugi

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。