Mathematical Sciences: Studies on Nonlinear Partial Differential Equations

数学科学：非线性偏微分方程研究

• 批准号: 9418779
• 负责人: Henghui Zou
• 金额: $ 3.5万
• 依托单位: University of Alabama at Birmingham
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-07-15 至 1997-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9418779&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Studies; Nonlinear; Partial

9418779 Zou This award supports mathematical research on problems arising in two areas of nonlinear partial differential equations. The first is related to the analysis of the asymptotic behavior and classification of positive solutions of semilinear equations in n-dimensional space, focusing on the Lane-Emden equation. The problem has been studied exhaustively in various special cases. Extensions and generalizations of this work will also be carried out. Uniqueness of positive solutions, especially in the case of radial solutions will also be taken up. A new direction of research concerns existence and non-existence questions for semilinear systems such as the Fujita system. More applied studies will consider the thermistor problem, analyzing electrical resistors whose resistance increases rapidly with increasing temperature. Finally, work will be done on the existence problem for ground states for quasilinear elliptic quations and the Dirichlet problem for the Monge-Amper Equation Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations. ***

邹：本奖项支持在非线性偏微分方程两个领域中出现的问题的数学研究。第一部分是关于n维空间中半线性方程正解的渐近行为和分类的分析，重点是Lane-Emden方程。这个问题已在各种特殊情况下进行了详尽的研究。还将对这项工作进行扩展和概括。还将讨论正解的唯一性，特别是径向解的唯一性。一个半线性系统如Fujita系统的存在性和非存在性问题是一个新的研究方向。更多的应用研究将考虑热敏电阻问题，分析电阻随温度升高而迅速增加的电阻。最后，研究拟线性椭圆方程基态的存在性问题和Monge-Amper方程的Dirichlet问题。偏微分方程是物理世界数学建模的基础。数学分析的作用与其说是创建方程，不如说是提供关于解的定性和定量信息。这可能包括回答关于独特性、平滑性和增长性的问题。此外，分析常常发展出解的近似方法和对这些近似精度的估计。＊＊＊