Mathematical Sciences: Further Development and Applications of the Method of Moving Planes

数学科学：移动平面方法的进一步发展和应用

• 批准号: 9623390
• 负责人: Congming Li
• 金额: $ 6.86万
• 依托单位: University of Colorado at Boulder
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-06-01 至 1999-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9623390&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Further; Development; Applications

Abstract 9623390 Li The PI will develop further the method of moving planes and study the following three problems: the a priori estimates, classification, and asymptotic analysis of solutions to some nonlinear differential equations; the prescribed Gaussian (or scalar) curvature problem; and the existence, uniqueness, and stability of travelling fronts. One focusing point of this project is to classify singular solutions or sequences of blow-up solutions in terms of when, where, how, and in what form do they occur. The method of moving planes is the main tool to be developed and used for this and other purposes. The analysis is particularly useful in the study of the last two problems where the PI has already obtained some interesting results. Many aspects of natural phenomena are related to each other by the natural laws governing them and very often these laws are described by differential equations. To understand these differential equations, it is necessary to classify all possible singular solutions or sequences of blow-up solutions. The PI proposes to study the prescribing curvature problem which involves some very challenging nonlinear partial differential equations. In fact, this type of equations also appear in many different fields of sciences. Also, singular solutions or sequences of blow-up solutions are the major barriers to the complete understanding of these equations. The PI will also study the existence, uniqueness, and stability of travelling fronts which describe the long term behavior of solutions to evolution type equations. He is mainly interested in problems arising from population dynamics and from flame propagation and ignition in combustion theory.

摘要9623390 李 PI将进一步发展移动平面的方法， 以下三个问题：先验估计，分类， 某些非线性微分方程解的渐近分析; 规定的高斯（或标量）曲率问题;和存在性， 独特性和稳定性的旅行前线。其中一个焦点是 一个项目是分类奇异解或爆破解序列 在何时、何地、如何以及以何种形式发生。的方法 移动飞机是主要的工具来开发和使用这一点和其他 目的这种分析在研究最后两个问题时特别有用 PI已经获得了一些有趣的结果。 自然现象的许多方面都是由自然界相互联系的。 这些定律通常用微分方程来描述 方程为了理解这些微分方程，有必要 分类所有可能的奇异解或爆破解序列。 PI建议研究处方曲率问题，其中涉及 一些非常有挑战性的非线性偏微分方程其实这 这类方程也出现在许多不同的科学领域。还有， 奇异解或爆破解序列是主要障碍 完全理解这些方程。PI还将研究 存在性，唯一性和稳定性的旅行前线，描述了 发展型方程解的长期行为。他主要 感兴趣的问题所产生的人口动态和火焰 燃烧理论中的传播和点火。