Global Attractor For Fourth Order Parabolic Equations

As we all known, global attractor for the diffusion equation Belongs to the scope of infinite-dimensional dynamical system. Infinite-dimensional dynamical System is in-depth and development of the finite-dimensional dynamical systems, and it plays an irreplaceable role in physics and mechanics and so on.In mathematics, if we consider the problem in qualitative theory of partial differ-ential equations, the most important thing is to establish the solution of initial value condition's uniform a prior estimates if the time t is large enough. In fact, The most important of Infinite-dimensional dynamical system is to study the problem of the sys-tem's solution behavior. For the well-known partial differential equations, most of them can not find exact numerical solution. In order to realize the solution's behavior, especially the case t→∞, global attractor is introduced. We know that the existence of global attractor reflects the stability of the solution in a sense. In recent decades, the study of global attractor for parabolic equations has attracted many scholars' attention.Cahn-Hilliard equations, as an important class of higher order nonlinear diffusion equations, come from a variety of diffusion phenomena in the nature, for example, dif-fusion phenomena in phase transition, competition and exclusion of biological groups, diffusion of oil film over a solid surface. In recent years, lots of scholors pay their attention to the Cahn-Hilliard equation, for example, Yin and Liu [15], Yin [16], K. H. Kwek [24] and so on. The convective Cahn-Hilliard equation arises naturally as a continuous model for the formation of facets and corners in crystal growth. Here the convective Cahn-Hilliard equation which studied by us is put forward when scholars study the crystal growth. [2,7]. In the past, lots of scholars considered the convec-tive Cahn-Hilliard equation, and made a serious of result. It was K. H. Kwek [24] who first studied the equation for the case with convection. Based on the discontin-uous Galerkin finite element method, he proved the existence of classical solutions. Recently, Liu [14] proved the existence, uniqueness and asymptotic behavior of the convective Cahn-Hilliard equation in one space dimension. Gao and Liu [11] gave the existence of traveling wave solution for convective Cahn-Hilliard equation when B(u)=u. Eden and Kalantarov [28] considered the following equation with the peri-odic boundary value condition.They proved that the existence of compact attractor and the finite-dimensional inertial manifold for the equation.In this paper, we study the solutions of the existence of global attractor for the fol-lowing class of convective Cahn-Hilliard equation supplemented with periodic bound-ary value condition and initial value condition. where A(u)=γ2u3+γ1u2-u,B(u)=-(?)u4+(?)u2. Andγ,γ1,γ2 are constants,γ,γ2>0,Ωis a bounded domain in Rn (n≤2). After that, we consider a type of epitaxial growth equation whereΩ∈R is a bounded domain, and initial-boundary value condition: Eq. (1) arises in epitaxial growth of nanoscale thin films [10,26], where u(x,t) denotes the height from the surface of the film in epitaxial growth, the term D4u denotes the capillarity-driven surface diffusion, and D(│Du│p-2Du) correspond to the upward hop-ping of atoms. If the nonlinear relation│Du│p-2Du is replaced by the term of the form f(u)Du, we obtain the well known Cahn-Hilliard equation.In this paper, by using R. Temam's theorem on the existence of global attractor, some results on the existence of global attractor for the convective Cahn-Hilliard equa-tion and a fourth order parabolic equation arises in epitaxial growth of nanoscale thin films in H2 space are established.