Global Attractor For Some Types Of Fourth Order Nonlinear Evolution Equations

There are many nonlinear problems in a lot of fields of science, such as Mathematics, Physics, Astronomy, Biology, Space Science, Environmen-tal Science and Meteorology. A large amount of nonlinear problems can be expressed by nonlinear evolution equations. Since 1980s, as the raise of the higher order nonlinear differential equation, more and more Mathemati-cians begin to focus on the research of higher-order nonlinear evolutions. There are many classical results on higher-order nonlinear evolution equa-tions.One of the important problems on the theory of nonlinear higher-order evolution equations is:If the existence and uniqueness of the global solu-tions for the equation can be proved, then what the asymptotic behavior of global solutions for large time is? Note that one of the important problem-s for the study of long time behavior for higher-order nonlinear evolution equations is to consider the asymptotic behavior for all of the global so-lutions which are corresponded by the initial dates for all bounded sets in some function spaces when the time tend to infinity. For example:the ex-istence of global attractor for the equations and the structure for the global attractor and so on(see[1]).In this paper, we consider the existence of global attractor for three types of fourth-order nonlinear evolution equations which had a broad range of physical background.Firstly, in chapter 2, we consider the global attractor for the follow-ing initial-boundary value problems of a fourth-order nonlinear evolution equations which is derived to model the formation of facets and corners in the course of kinetically controlled crystal growth. First of all, using Leray-Schauder fixed point theorem and some a prior estimates, we obtain the existence and uniqueness for the above initial-boundary value problem. Then, by using the properties of the semigroup S(t) which is associated by the initial-boundary value problem (1), we proved the existence of bounded set in H2 space and obtained the unifor-m compact for S (t). Hence, the existence of global attractor for initial-boundary value problem (1) in Sobolev space Furthermore, using the result on the existence of global attractor in u, the properties of fractional space (see[2]) and iterative technique, we obtain the existence of global attractor for problem (1) in the space Hk(0?k<+?).Secondly, in chapter 3, we consider the existence of global attractor for the following Marangoni convection equation which is supplied with periodic boundary value condition: Similar as chapter 2, we first use Leray-Schauder fixed point theorem and some a prior estimates to obtain the existence and uniqueness for the above equation. Then, by using the properties of the semigroup S (t), we proved the existence of bounded set in H2 space and obtained the uniform compact for S (t). Hence, the existence of global attractor for equation (1) in Sobolev space Furthermore, using the result on the existence of global attractor in u, the properties of fractional space (see[2]) and iterative technique, we obtain the existence of global attractor for equation (2) in the space Hk(0? k<+?).Here, we study the existence of global attractor for the 2D equation (2) with periodic boundary value condition in fractional space Hk(0< k< +?). And in chapter 2, we consider the existence of global attractor for the ID initial-boundary value problem (1) in fractional space Hk(0?k<+?). Through comparative analysis and the process of proof, we can find that a-long with the arise of dimensions, the problems have more realistic signif-icance, are closer to the real models and more hard. On the other hand, the nonlinear term for problem (2) is-?·(|??|2??)+??·(????)+??|??|2. It is easy to check that this nonlinear term is stronger than the nonlinear term of the problem (1), is harder to control. Here, we use Nirenberg's inequal-ity, Sobolev's embedding theorem and other mathematical tools to get the a prior estimates for equation (2) in the surface spaces.Thirdly, in chapter 4, we consider the existence of global attractor for the initial-boundary value problem of the following fourth-order nonlinear evolution equation describing epitaxial thin film growth: The epitaxial thin film growth model equation is a type of classical fourth-order nonlinear evolution equations. Many authors (for example, Winkler, Yagi, Kohn, Tang Tao and so on) have paid much attentions on the prop-erties of solutions and numerical solutions for this equation. In [3,4], Xi-aopeng Zhao et. al. have studied the existence of global attractor for prob- lem (3) in 1D and 2D case. Because of the restrict of nonlinear term, it is difficulty to obtain the H2-norm a prior estimates in the higher dimensional case. Here, using Lyapunov energy functional, we obtain the H2-norm es-timates in nD case, where n?3. Furthermore, using the result on Temam's classical book[5], we obtain the existence of global attractor in H2 space for initial-boundary value problem (3).