On The Global Attractor For Infinite Dimensional Dynamical Systems

In this doctoral dissertation,we are mainly concerned with the following two kinds of problems.First,in theory,we give a new estimation method of fractal dimension of the global attractor and provide a method of proving the existence of exponential attractors.Secondly,we discuss the well posedness of Kirchhoff wave equation with strong damping and the existence of global attractor.The existing dimension estimation methods can be roughly divided into two categories according to the applicable objects,that is,the one suitable for s-moothing semigroups and the other suitable for non smooth Lipschitz continuous semigroups.The first method is based on the differentiability of semigroups.We use the linearization of equations to estimate the dimensions of attractors.The second one relies on the semigroup decomposition and the asymptotic smoothness or weak smoothness of the solution.In this situation we estimate the coverage number by using the contraction operator and the compact perturbation.In the third chapter,we first briefly introduce the known methods,and then give a more general and generalizable method of estimating the fractal dimension.On this basis,we have established a new result of the existence of exponential attractor.The two methods introduced previously can be regarded as a special case of the conclusion of this article.In Chapter 4 and Chapter 5,we consider the Kirchhoff wave model with strong damping in a bounded smooth domain ? of RN.Here h ? L2(?)is a external force term,f(u)is a given source term,and ?,? are two nonlinear functions.In Chapter 4 we study the problem(0.0.2)in the non-degenerate case,that is,the case of ?(s)>0.We first prove the well-posedness of the problem in the space(H2(?)?H01(?))×H01(?)with the nonlinear term which is polynomial growth p?N/(N-4)+.Due to the influence of strong damping terms,the problem(0.0.2)has partial regularity,that is,ut,utt has the properties similar to the solution of the parabolic equation.However,there has no higher regularity for the strong solution of u(x,t)itself.We combine the method of ?-limit compactness and the quasi-stable estimate to overcome the difficult and obtain the existence of the global attractor.On the other hand,we prove that there exits a H01(?)×L2(?)-H01(?)×H01(?)global attractor for the problem(0.0.2)with critical source term,which attracts every H01(?)×L2(?)-bounded set with respect to the H01(?)×H01(?)norm.In Chapter 5,we study the problem(0.0.2)in the degenerate case,i.e.,the?(s)?0.For the degenerate Kirchhoff equation in the bounded domain with Dirichlet boundary condition,the conclusion about the attractor has not yet been seen.Under the condition of ?(s)? 1,we obtain the existence of global attractor for the problem(0.0.1)in H01(?)× L2(?)in the critical case.