Mathematical Sciences: Structural Stability and Floquet Theory for Parabolic Equations

本研究的主要目标是发展抛物型偏微分方程的局部和全局结构稳定性理论和时变线性抛物型方程的Floquet理论。结构稳定性最初是在常微分方程，特别是动力系统的研究中引入的。许多动力系统的概念已经适用于无限维系统，特别是在非紧空间上定义动力系统的偏微分方程。然而，由抛物方程生成的系统是不可逆的。这一特点造成了在有限维动力系统的研究中没有遇到的困难，必须开发新的方法来理解这些系统的本质。最近得到了标量反应扩散方程的一个理论。具体类方程，如卡恩-希利亚德方程和相场系统将考虑具体的分析。其他工作将研究时间周期抛物型方程周期解附近流动的结构稳定性。本着同样的精神，我们将努力把Floquet理论（过于技术性，在这里无法描述）从有限周期系统转移到类似类型的抛物型偏微分方程。

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.