Mathematical Sciences: Semiliner Partial Different EquationsCurve Shortening in Minkowski Geometry and Branching Processes in Probability

本项目主要研究数学分析的三个方面。一般的主题是确定非线性方程和系统解的存在性、正则性、局部和全局行为以及对称性。特别是，对半线性椭圆偏微分方程的研究将继续进行，研究天体物理学中的Lane-Emden方程和Matukuma方程。这包括正解的完全分类，并确定是否所有解都是极限为零或无穷大的径向解。第二项研究考虑了各向异性曲线缩短。这有一种更几何的味道，其中嵌入的平面曲线根据与法向量成比例的速率进化。本文主要研究闵可夫斯基几何流形上曲线的广义集。人们将努力确定这种流动何时可以缩小到不存在常设对称假设的程度。这项工作所代表的思想可以被视为相边界演化的模型，如生长晶体的边界。本研究的第三个方向涉及测量值分支过程，包括由点催化剂引起的进化模型。***这项工作的根源是在研究空间分布的出生和死亡过程中产生的随机偏微分方程。一个特定的目标是分析多种催化剂存在下的非平衡反应。

Li 9225145 This project concentrates on three areas of research in mathematical analysis. The general theme is that of determining existence, regularity, local and global behavior and symmetry properties of solutions of nonlinear equations and systems. In particular, work on semilinear elliptic partial differential equations will continue, examining the Lane-Emden equation and Matukuma equation of astrophysics. This includes a complete classification of positive solutions, and to determine whether all solutions are radial with limiting values of zero or infinity. A second line of investigation considers anisotropic curve shortening. This has a more geometric flavor in which embedded plane curves evolve according to a rate proportional to the normal vector. The present work focuses on a generalized setting of curves on manifolds in a Minkowski geometry. Efforts will be made to determine when such flows can shrink to a point when standing symmetry hypotheses are not present. The ideas represented by this work can be viewed as models for the evolution of phase boundaries such as the boundaries of growing crystals. The third direction this works takes involves measure-valued branching processes, including the model on the evolution caused by point catalysts. *** The work has its roots in stochastic partial differential equations arising in the study of spatially distributed birth and death processes. A particular goal is to analyze nonequilibrium reactions in the presence of multiple catalysts.