Mathematical Sciences: Structural Stability and Floquet Theory for Parabolic Equations

这项研究的主要目标是发展一个当地的 抛物型偏微分方程的整体结构稳定性理论 微分方程和Floquet理论 线性抛物方程 结构稳定性是第一位的 在研究常微分方程时， 特别是动力系统。 动力学的许多概念 系统已经适应于无限维系统， 特别是偏微分方程， 非紧空间上的动力系统 然而，这些系统 由抛物方程产生的是不可逆的。 这 这一特点造成了研究中没有遇到的困难 有限维动力系统的新方法， 来理解这些系统的本质。 一个理论 标量反应扩散方程最近被 得到了 特殊类型的方程，如Cahn-Hilliard方程 方程和相场系统将被考虑为特定的 分析. 其他工作将研究流动的结构稳定性 时间周期抛物方程的近周期解 本着同样的精神，将努力把 Floquet理论（太技术性，无法在这里描述）从有限 抛物型偏微分方程 相似的类型。

The main goal of this research is the development of a local and global structural stability theory for parabolic partial differential equations and a Floquet theory for time dependent linear parabolic equations. Structural stability was first introduced in the study of ordinary differential equations, in particular for dynamical systems. Many concepts of dynamical systems have been adapted to infinite dimensional systems, especially to partial differential equations which define dynamical systems on non-compact spaces. However, those systems generated by parabolic equations are not reversible. This characteristic creates difficulties not encountered in the study of finite dimensional dynamical systems and new methods must be developed to understand the nature of these systems. A theory for scalar reaction-diffusion equations has recently been obtained. Specific classes of equations such as the Cahn-Hilliard Equation and Phase-Field System will be considered for specific analysis. Other work will study structural stability of flows near periodic solutions for time-periodic parabolic equations. In the same spirit, efforts will be made to carry over the Floquet theory (too technical to describe here) from finite periodic systems to parabolic partial differential equations of similar type.