The Long Time Behavior Of Several Kinds Of Nonautonomous Infinite Dimensional Dynamical Systems

This dissertation, consisting of four chapters, intends to describe the long time behavior of several kinds of non-autonomous infinite-dimensional dynamical systems, especially the existence of the inertial manifolds and approximate inertial manifolds.Chapter 1 is a brief introduction to the historical background and the significance of this research. Some important definitions are also introduced in this chapter.Chapter 2 is devoted to study the long time behavior of a class of non-autonomous reaction-diffusion equations with delays via Lyapunov-Perron method. When the time delays are small enough and the spectral gap conditions are satisfied, the existence of the inertial manifolds is proved, which extends the corresponding results of the autonomous systems.Chapter 3 describes the long-time behavior of a class of cascade systems with delays. By using the Lyapunov-Perron method, the existence of inertial manifolds is proved when the spectral gap conditions are satisfied and the time delays are small enough.Chapter 4 is a study of a class of non-autonomous reaction-diffusion equations with quasi-periodic terms. Firstly, with the aid of Skew-Product method, the non-autonomous systems are lifted to autonomous systems in the extended phase spaces, then the approximate inertial manifolds of the non-autonomous systems are constructed based on the inertial manifold with delays of the autonomous systems.