U.S.-Polish Cooperative Research: Lyapunov Exponents and Spectrum for Random and Nonautonomous Parabolic Equations

这是一个美国-波兰合作研究项目，重点研究给定抛物系统的谱与其Lyapunov指数之间的联系。主要研究人员是奥本大学的沈文贤博士和弗罗茨瓦夫理工大学的雅努兹·米耶尔津斯基教授。本研究的目的是研究非自治/随机抛物型方程的谱/Lyapunov指数及其应用。在自然界中，许多系统都受时间依赖性的影响。这种依赖可能既不是周期性的，也不是准周期性的，也不是几乎周期性的。具有这种时间依赖性的系统更适合用非自治或随机方程来描述。因此，研究一般的非自治随机方程具有重要的意义。本文的研究重点是发展非自治/随机抛物方程的线性理论，即谱/Lyapunov指数理论。这样的理论将对非线性方程有重要的应用。其中最明显的是它在分析非线性抛物型方程某些特解的稳定性方面所起的作用。这个数学研究项目实现了将美国和中东欧的顶尖专家聚集在一起，在平等、互惠和互利的基础上，在共同利益和能力强的领域结合互补努力和能力的计划目标。

This is a U.S.-Polish cooperative research project that focuses on the links between the spectrum of a given parabolic system and its Lyapunov exponent. The principal investigators are Dr. Wenxian Shen from Auburn University and Professor Janusz Mierczynski from Wroclaw University of Technology. The objective of the proposed research is to investigate spectrum/Lyapunov exponents for nonautonomous/random parabolic equations and to study their applications. In nature, many systems are subject to time dependence. Such dependence may be neither periodic nor quasi-periodic or almost periodic. Systems with such time dependence are characterized more appropriately by nonautonomous or random equations. It is therefore of great importance to study general nonautonomous and random equations. The proposed research focuses on the development of linear theory, namely spectrum/Lyapunov exponents theory, for nonautonomous/random parabolic equations. Such theory would have important applications to nonlinear equations. The most obvious of these is its role played in the analysis of the stability of certain special solutions of nonlinear parabolic equations. This project in mathematics research fulfills the program objectives of bringing together leading experts in the U.S. and Central/Eastern Europe to combine complementary efforts and capabilities in areas of strong mutual interest and competence on the basis of equality, reciprocity, and mutuality of benefit.