Studies On Dynamics Of Solutions To Several Kinds Of Partial Functional Differential Equations

By using theory of partial functional differential equations,semigroup theory and theory of infinite dimensional dynamical systems,this thesis studies the long time behavior of solutions to several kinds of partial functional differential equations,including the existence of pullback attractors,their dimension and semi-continuity,as well as the polynomial and exponential stability of stationary solutions.The dissertation contains six chapters.In Chapter 1,we introduces the background of partial functional differential equations as well as the corresponding infinite dimensional dynamical systems.In addition,the recent results and the current research situation of partial functional differential equations and infinite dimensional dynamical systems are also presented.In the end,the main work of this dissertation is summarized.In Chapter 2,first of all,the classical Faedo-Galerkin approximations is applied to prove the existence and uniqueness of weak solutions to a non-autonomous stochastic p-Laplace equation,the uniform estimates and asymptotical compactness are used to obtain the existence of bi-spatial random attractors and semi-continuity.Then the Galerkin method and Aubin-Lions compactness are combined to show the existence and uniqueness of weak solutions to a p-Laplace equation with finite delay.Moreover,the existence and semi-continuity of pullback attractors is established by energy method.In Chapter 3,the existence and uniqueness of weak solutions to a Navier-Stokes equation with unbounded delay is analyzed by theory of functional differential equations.Then Lyapunov function etc are used to prove the local stability of stationary solutions.And by constructing suitable Lyapunov functionals,the asymptotic stability of stationary solutions is proved.Besides,under a special unbounded delay case,the polynomial stability of stationary solutions is obtained.Furthermore,It? formula is utilized to prove the existence and uniqueness of weak solutions to a stochastic NavierStokes equation with infinite delay.Then by constructing proper Lyapunov functionals,the asymptotical stability of stationary solutions is established.At the end,for the case of a special infinite delay,the polynomial stability of stationary solutions is proved.The energy method and compactness results are used to establish the existence and uniqueness of weak solutions to a kind of incompressible non-Newtonian fluids with delay in Chapter 4.And the uniform estimates and decomposition method are applied to prove the existence of pullback attractors.Additionally,by using Lax-Milgram theo-rem and Schauder fixed theorem,the existence and uniqueness of stationary solutions to a delayed incompressible non-Newtonian fluids is obtained.Finally,Razumikhin arguments etc are used to prove the exponential stability of stationary solutions.The existence and uniqueness as well as the continuous dependence on initial value of mild solutions to a fractional stochastic reaction-diffusion equation with memory is proved in Chapter 5 by using semigroup arguments.Moreover,the existence of random attractors with finite Hausdorff dimension is obtained.In Chapter 6,the main work of this thesis and put forward some unsolved problems are summarized.