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

9623390 PI将进一步发展动平面方法，研究以下三个问题：一些非线性微分方程解的先验估计、分类和渐近分析；给定的高斯(或标量)曲率问题；以及行波阵面的存在性、唯一性和稳定性。这个项目的一个重点是根据奇异解或爆炸解序列的出现时间、地点、方式和形式对它们进行分类。移动平面法是为这个目的和其他目的而开发和使用的主要工具。这种分析在后两个问题的研究中尤其有用，在这些问题中，PI已经得到了一些有趣的结果。自然现象的许多方面都是由支配它们的自然法则相互关联的，而且这些法则常常用微分方程式来描述。为了理解这些微分方程，有必要对所有可能的奇异解或爆破解序列进行分类。PI建议研究规定的曲率问题，这涉及一些非常具有挑战性的非线性偏微分方程组。事实上，这种类型的方程也出现在许多不同的科学领域。此外，奇异解或爆破解序列是完全理解这些方程的主要障碍。PI还将研究描述发展型方程解的长期行为的行波前线的存在、唯一性和稳定性。他主要对燃烧理论中的人口动力学、火焰传播和点火等问题感兴趣。

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.