Long-time Behavior For Three Classes Of Dissipative Partial Differential Equations

In this doctoral dissertation,we study the long-time dynamical behavior for three classes of dissipative partial differential equations.We present a series of novel and profound results obtained by a priori estimation of solutions of the stochastic equations and the non-autonomous equation.This thesis consists of six chaptersIn Chapter One,firstly we recall briefly the background and classical results of infinite dynamical systems and the random dynamical systems.Also the background of the random attractors and pullback attractors is introduced.Secondly,we briefly introduce the background of the X-elliptic operator and recall the developing status of longtime behavior of stochastic parabolic equations.Thirdly,we introduce the development of the longtime behavior of stochastic damped wave equations.Nextly.we present respectively the background and the development for time-dependent damped hyperbolic equations with the damped coefficient has a positive lower bound and without.Finally,we show the motivations for studying the problems we consider in this thesisIn Chapter Two,We briefly introduce some basic definitions and theorems that will be needed in this thesis,including the definitions and properties of the random dynamical systems,the theorems for the existence of the random attractor and pullback attractors.Finally,we present some useful inequalitiesIn Chapter Three,we consider the longtime dynamics of a degenerate stochastic parabolic equation with X-elliptic operator.It is one of the most important topics to explore the properties of differential equations on manifolds.Parabolic equations are classical models that are applied to inspire the new concepts and verify new methods because of their concise form and abundant properties.Firstly,we transfer the stochastic differential equation into a deterministic one with parameter through using the Ornstein-Uhlenbeck process.Then by a priori estimate and Sobolev em-bedding,the existence of a(L2,L2)-attractor is obtained.Secondly,we prove that the(L2,L2)-attractor could attract every bounded set in L2 under L2+?-norm for any ?>0.Finally,the(L2,L2)-attractor is shown to attract every bounded set in L2 under the higher regular H-norm.To the best of my knowledge,this is the first time to extend the degenerate equation with X-elliptic operator to the stochastic case.The existence,high order integrability and regular attraction of the random attractor are new resultsIn Chapter Four,in order to be familiar with the random dynamical systems.the longtime behavior of the time-dependent damped stochastic wave equation with critical growth nonlinearity is considered.The damped wave equations are one of the classical models in infinite dimensional dynamical systems.Especially when the damping is weak,how to prove the compactness is one of the important topics in infinite dimensional dynamical systems.The longtime behavior of the time-dependent damped stochastic wave equation with critical growth nonlinearity is analyzed.Firstly,we take advantages of the Ornstein-Uhlenbeck process to transfer the stochastic equation into a deterministic one with a random parameter.The well-posedness is deduced through the Galerkin approximation.Secondly,the random absorbing set is obtained under the establishment of a priori estimates.Then the existence of the random attractor is proved since the asymptotic compactness is showed through the energy method.Thirdly,the fractal dimension of the random attractor is proved to be finite.Finally,we establish a random exponential pullback attractor due to the tempered absorbing set.We use more general conditions on the nonlinearity,under which we have also proved the finiteness of fractal dimension All of these are meaningful work to the theory of nonautonomous random dynamical systemsIn Chapter Five,based on the work in Chapter Four,the longtime behavior of a class of non-autonomous time-dependent damped wave equation with critical growth nonlinearity is considered.Instead of provided a positive lower bound,the coefficient function of the damping term could be negative here.The detail models are showed in quantum mechanics and electromagnetism.In order to overcome the difficult from the negative part,we propose some new conditions on the coefficient function.We establish new a priori estimates for positive damping and negative damping respectively,by which together with a generalized Gronwall's inequality we deduce the dissipation.A technique of operator decomposition is employed here to prove the asymptotic compactness.Finally,the existence of pullback attractor is obtained.To our best knowledge,this is the first explorative result for this equation All of the results in this chapter are newIn Chapter Six,based on the research in this thesis,the further research ques-tions are listed.