Mathematical Sciences: Nonlinear Partial Differential Equations and Systems

数学科学：非线性偏微分方程和系统

• 批准号: 9401333
• 负责人: Wei-Ming Ni
• 金额: $ 12.96万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-07-01 至 1998-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9401333&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Nonlinear; Partial; Differential

Ni 9401333 Work supported by this award focuses on the mathematical analysis of nonlinear partial differential equations, systems and their applications. The research continues several projects such as the study of spike-layers in mathematical biology. These arise in mathematical descriptions of the chemotactic aggregation stage of cellular slime molds. One is interested in solutions of the relevant equations which exhibit spiky patterns dependent on the equation coefficients. Such patterns have been observed experimentally. A second line of investigation concerns cross- diffusion in population dynamics. The primary question to be studied here is one of determining and understanding the steady states of such systems. Work will also be done measuring the stability of steady states of a class of semilinear heat equations in n-dimensional space and on the inhomogeneous semilinear elliptic or parabolic equations arising in infinite particle systems in probability. 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.

NI 9401333获此奖项支持的工作重点是非线性偏微分方程式、系统及其应用的数学分析。这项研究继续了几个项目，例如数学生物学中的尖峰层研究。这些出现在对细胞黏菌趋化聚集阶段的数学描述中。人们感兴趣的是相关方程的解，这些方程的解呈现出依赖于方程系数的尖峰图案。这样的模式已经在实验中观察到了。第二条线是关于种群动态中的交叉扩散。这里要研究的主要问题是确定和理解这类系统的稳态。还将对n维空间中一类半线性热方程的定态稳定性和无限粒子系统中的非齐次半线性椭圆型或抛物型方程的稳定性进行度量。偏微分方程式是建立物理世界数学模型的基础。数学分析的作用与其说是创建方程，不如说是提供有关解的定性和定量信息。这可能包括回答有关唯一性、平稳性和成长性的问题。此外，分析经常开发出近似解的方法和对这些近似的精度的估计。