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; 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.

向西介绍了由一个空间维度在非线性抛物线方程定义的无限尺寸动态系统的结构，并表明它们的全球吸引子始终是有限的维歧管。该结果意味着，解决方案的长期行为的基本特征可以通过普通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。应当指出的是，自1980年代中期以来一直以来一直在研究吸引子的有限维度的工具，该理论一直众所周知，不适用于我们的研究中处理的方程式。2。我们研究了退化扩散方程的溶液的行为，并证明了任何不稳定的平衡溶液都具有无限的Hausdorff尺寸的不稳定歧管。该结果表明，退化扩散方程的动力结构与非排效方程的动力结构之间存在本质差异。目前，Matano还正在研究具有空间序数系数的非线性扩散方程的行进波的性质，并获得了部分结果。3。我们研究了一个结合了麦克斯韦方程和施罗宾格方程的数学模型。我们获得了有关解决方案的独特性和全球存在的新结果。4.kusuoka是研究项目的调查者之一，一直在研究使用概率方法研究数学金融方面的问题。他获得了一些有趣的结果。