On The Existence Of Pullback Attractors For Nonautonomous Infinite Dimensional Dynamical Systems

In this doctoral dissertation,we mainly consider the existence of pullback attractors of nonautonomous infinite dimensional dynamical systems.We present two methods to verify the property of pullback D-asymptotic compactness of cocycles generated by nonautonomous dynamical systems,which is an essential condition to show the existence of pullback attractors.We also apply our methods to concrete PDEs,and get a series of new and meaningful results.This thesis consists of five chapters.In Chapter one,the background and major results on the theory of dynamical systems,the main results of the theory of pullback attractors and this doctoral dissertation are introduced.In Chapter two,some preliminary results and definitions that we will used in this thesis are presented.In Chapter three,combining the concept of closed cocycles and norm-to-weak cocycles,we prove some abstract results as to the existence of pullback attractors of nonautonomous dynamical systems.We give two different(sufficient) conditions for verifying the pullback D-asymptotic compactness.We also consider the existence of pullback attractors in bi-spaces.In Chapter four,using the general results in Chapter three,we prove the existence of(L²(Ω),H₀¹(Ω))-pullback attractors for nonautonomous reactiondiffusion equations u_t-Δu+f(u)=g(x,t)in bounded domains,and(L²(Rⁿ), L^p(Rⁿ))-pullback attractors in unbounded domains with nonlinearities having polynomial growth of arbitrary order p≥2.In Chapter five,we prove the existence of pullback attractors generated by weak solutions and strong solutions of nonautonomous wave equations u_tt+ηu_t-Δu+f(u)=g(x,t)with critical and subcritical nonlinearity,respectively.