Long-Time Behaviors Of Solutions For Some Classes Of Nonlinear Evolution Equations

In this thesis we deal with the existence and stability of periodic or almostperiodic solutions for some classes of nonlinear evolution equations related to PDEwith delay or not. Meanwhile, we consider the existence of global attractors, inertialmanifolds and approximate inertial manifolds for some equations among them. Themain results in this thesis consist of three parts.In the first part we consider mainly the problems about the existence and sta-bility of time-periodic or almost periodic solutions for the damped Sine-Gordenequations with delay and parabolic evolutionary equations with some delays. Thekey steps in the proof are constructing some suitable Lyapunov functionals, estab-lishing the priori bound for all possible periodic solutions and lastly proving theexistence of periodic solutions by Schaefer's fixed point theorem. This approach hasan advantage that only all possible periodic solutions are assumed to be uniformlybounded instead of all solutions in the traditional dissipative theory.In the second part we investigate the existence of inertial and approximateinertial manifolds for some classes of PDEs. Firstly, we deal with the long timebehavior of semi-linear parabolic equations with delay in the non-self adjoint case.Under some assumptions of delay and the spectral gap condition, the existence ofinertial manifolds is provided by Lyapunov-Perron method. Secondly, we considerthe long-time behavior of delayed semi-linear parabolic equations with quasi-periodicterms. Being used skew-product method, the non-autonomous systems are lifted intothe autonomous systems in the extended phase space and the existence of inertialmanifolds is presented under some assumptions about delay and the spectral gapcondition. Lastly, we deal with a class of nonautonomous evolutionary equationsand prove the existence of inertial manifolds with delay for the relevant autonomoussystems. Furthermore, the approximate inertial manifold is constructed for thenonautonomous equation based on inertial manifold with delay.In the last part we investigate the existence of global attractors for the timediscretization model from a class of reaction-diffudion equations with delay, andthe upper semicontinuity of the global attractors is obtained under some sufficientconditions. The difficulty caused by the delay in the continuous system is overcomesuccessfully during the discretization process.