The Research On The Asymptotic Properties Of Solutions For Two Classes Of Partial Differential Equations

This thesis focuses on the asymptotic attractors of solutions to Two Classes of Partial Differential Equations.The existence of exponential attractors for autonomous Cahn-Hilliard equation was proved.Meanwhile,by proving the existence of bounded absorbing set,the existence of exponential attractors of Cahn-Hilliard equation is proved as well.Finally,the dimension of asymptotic attractors for Autonomous Cahn-Hilliard equation on bounded domains is estimated.This thesis consists of four parts:Firstly,the background and state-of-the-art of the two Classes of Partial Differential Equations in infinite dimensional dynamical systems were reviewed.Consecutively,several basic concepts and methodologies of attractors were presented.Finally,the main research topics and the mostly used function spaces and inequalities were given.Secondly,the exponential attractors of autonomous Cahn-Hilliard equation on bounded domains were studied.By solving the semigroup Lipschitz and the property that discrete semigroup satisfies Squeezing,the existence theorem of exponential attractor for autonomous Cahn-Hilliard equation in Space L~2(?) was derived.The third part studies the existence of exponential attractors of Navier-Stokes-Voight function.By method presented in the second part,the existence of the exponential attractors was obtained.The last part of this thesis,the asymptotic attractor of Cahn-Hilliard equation with periodic boundary conditions was studied.First,the solution sequence is constructed by using orthogonal decomposition.Then,it is proved that the solution sequence does not go away from the global attractor by using mathematical induction.Furthermore,the estimation of dimension of the asymptotic attractors was given.