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

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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.