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

Li 9225145这个项目集中在数学分析的三个方面的研究。总的主题是确定非线性方程和系统解的存在性、正则性、局部和整体行为以及对称性质。特别是，关于半线性椭圆型偏微分方程组的工作将继续进行，研究天体物理学中的Lane-Emden方程和Matukuma方程。这包括对正解的完整分类，以及确定所有解是否都是具有零值或无穷大极限值的径向解。第二个研究方向是各向异性曲线缩短。这具有更多的几何色彩，其中嵌入的平面曲线根据与法线向量成比例的速率演变。本文主要研究Minkowski几何中流形上曲线的一种广义设置。将努力确定这种流动何时可以缩减到不存在站立对称假设的程度。这项工作所代表的思想可以被视为相界演化的模型，例如正在生长的晶体的边界。这项工作的第三个方向涉及测值分支过程，包括由点催化剂引起的进化模型。*这项工作源于研究空间分布的出生和死亡过程的随机偏微分方程组。一个特别的目标是分析在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.