Global Attractors And Fractal Dimensions For Nonlocal Benjamin-bona-mahony Equation

Infinite dimensional dynamical systems have rich actual background,the study hasbecome one of the central issues in the nonlinear sciences,and meanwhile,it is also animportant issue in the research of partial differential equations.Infinite dimensional dynamicalsystems are very important not only in pure mathematics but also in the realworld applications. The research on infinite dimensional dynamical systems has been receivingmore and more attentions. Many scholars study infinite dimensional dynamicalsystems in the world and have got some beautiful results.In this paper, we discuss the existence of global attractors for nonlocal Benjamin-Bona-Mahony equation and its fractal dimensions. In chapter 1, we introduce someimportant theorems about the global attractors, hausdorff dimensions, fractal dimensions,and the common existence theorems of the global attractors.In chapters 2 and 3, we give a detailed analysis of the global attractor and its fractaldimensions for a nonlocal Benjamin-Bona-Mahony equation. We extend the results ofA.O.Celebi et al in two directions. First, we impose uΩu2dx instead of fiF(u) andconsider the affect of the nonlocal term on the property of global attractor. Second,we impose Dirichlet boundary conditions instead of the periodic boundary conditions.Using the relevant estimates, we obtain that the semigroup S(t) of the initial boundaryproblem has a bounded absorbing set in H10 . Then we decompose the semigroup S(t)into S1(t) + S2(t), which satisfy the conditions of the existence theorem of the globalattractor respectively, therefore we get the existence of the global attractor. Finally, weprove that the fractal dimension of the global attractor is finite and give its estimate.In chapter 4, we deal with the existence of global attractor of the initial-boundaryvalue problem of the metabolic and nonlocal Benjamin-Bona-Mahony equation and itsfractal dimension by using the method which described in the chapters 2 and 3.