Mathematical Sciences: Stability for Reaction-diffusion Equations

这个项目将遵循应用数学问题的两条研究路线。首先是理解相变现象的某些数学模型。这些问题具有变分结构，涉及对具有小扩散系数的椭圆型或抛物型方程的微分解的研究。第二类的工作将集中在进化系统，不一定是变型的，并保持秩序。一个主要目标是确定所有解的渐近性质。将这些研究联系在一起的一个共同主题是，这些问题处理的是反应扩散机制，分析中考虑了稳定性因素。从应用的角度来看，第一个项目试图将某些可观察的模式建模为稳定的平衡或不稳定的平衡流形；第二种方法试图排除强迫系统类稳定混沌的可能性，并建立非常规则的行为准则。

This project will follow two lines of investigation into problems of applied mathematics. The first is that of understanding certain mathematical models for phase transition phenomena. The problems have a variational structure and involve the study of distinguished solutions for elliptic or parabolic equations with small diffusion coefficients. The second category of work will concentrate on evolution systems that are not necessarily of variational type and which preserve order. One main objective is to determine asymptotic properties of all solutions. A common theme which ties the research together is that the problems deal with reaction-diffusion mechanisms and that the analysis employs stability considerations. From the applications point of view, the first project seeks to model certain observable patterns either as stable equilibria or as unstable manifolds of equilibria; the second seeks to exclude the possibility of stable chaos for classes of forced systems and to establish very regular behavior criteria.