Analysis of the structure of solutions of nonlinear partial differential equations

非线性偏微分方程解的结构分析

• 批准号: 07454031
• 负责人: MATANO Hiroshi
• 金额: $ 0.96万
• 依托单位: University of Tokyo
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1995
• 资助国家: 日本
• 起止时间: 1995 至 1996
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-07454031/
• 关键词: nonlinear partial differential equations; qualitatuve theory; diffusion equations; infinite dimensional dynamical systems; parabolic equations; attractor; パターン形成; 定性的研究; 境界現象

Westudied the structure of the infinite dimensional dynamical systems defined by nonlinear parabolic equations in one space dimension and showed that their global attractors are always finite dimensional manifolds. This result implies that the essential features of the long-time behavior of solutions can be described by a finite system of ordinary differential equations, and is therefore important from the point of view of qualitative theory. It should be noted that the so-called inertial manifold theory, which has been well-known since mid 1980's as a tool for studying the finite dimensionality of attractors, does not apply to the equation treated in our research.2. We studied the behavior of solutions of degenerate diffusion equations and proved that any unstable equilibrium solution has an unstable manifold of infinite Hausdorff dimension. This result shows that there is essential difference between the dynamical structure of degenerate diffusion equations and that of nondegenerate equations. currently Matano is also studying the properties of traveling waves for nonlinear diffusion equations with spatially priodic coefficients and has obtained partial results.3. We studied a mathematical model which combines Maxwell equation and Schrodinger equation. We obtained new results on the uniqueness and global existence of solutions.4.Kusuoka, one of the investigaters of the research project, has been studying problems in mathematical finance with probabilistic method. He has obtained some interesting results.

我们研究了一维非线性抛物线方程定义的无限维动力系统的结构，并表明它们的全局吸引子总是有限维流形。这一结果意味着解的长期行为的基本特征可以通过常微分方程的有限系统来描述，因此从定性理论的角度来看很重要。需要指出的是，自20世纪80年代中期以来作为研究吸引子有限维性的工具而广为人知的所谓惯性流形理论并不适用于我们研究中处理的方程。2．我们研究了简并扩散方程解的行为，并证明任何不稳定平衡解都具有无限豪斯多夫维数的不稳定流形。这一结果表明，简并扩散方程的动力学结构与非简并扩散方程的动力学结构存在本质区别。目前Matano还在研究具有空间周期性系数的非线性扩散方程的行波性质，并取得了部分结果。 3．我们研究了麦克斯韦方程和薛定谔方程相结合的数学模型。我们在解的唯一性和全局存在性方面获得了新的结​​果。4.Kusuoka，该研究项目的研究者之一，一直在用概率方法研究数学金融问题。他获得了一些有趣的结果。

项目成果

Hiroshi Matano: "Dynamical structure of some nonlinear degenerate diffusion equations" J.Dynamics and Diff.Eq.(to appear).

Hiroshi Matano：“一些非线性简并扩散方程的动力学结构”J.Dynamics 和 Diff.Eq.（即将出现）。

俣野博: "The global attractor of semilinear parabdic equations on S'" Discrete and Continuous Dynamical Systems. 3. 1-24 (1997)

Hiroshi Matano：“S 上半线性 Parabdic 方程的全局吸引子”离散和连续动力系统。 3. 1-24 (1997)

堤誉志雄: "Space-time estimates for null gaugeforms and nonlinear Schrodinger equations" Differential and lntegral Equations. (掲載予定).

Yoshio Tsutsumi：“零规范形式和非线性薛定谔方程的时空估计”微分方程和积分方程（待出版）。

俣野 博: "Dynamical structure of some nonlinear degenerate diffusion equation" J.Dynamics and Diff.Eq.(掲載予定).

Hiroshi Matano：“一些非线性简并扩散方程的动态结构”J.Dynamics 和 Diff.Eq（待出版）。

俣野 博(陳旭彦): "Finite-point extinction and continuity of interfaces in a nonlinear diffusion equation with strong absorption" J.Reine Angew.Mathematik. 459. 1-36 (1995)

Hiroshi Matano (Xuhiko Chen)：“具有强吸收的非线性扩散方程中的有限点消光和界面连续性”J.Reine Angew.Mathematik 459. 1-36 (1995)