Global Attractors And Fractal Dimensions For Non-classical Reaction-diffusion Equation With Memory

Infinite dimensional dynamical systems are the development of finite dimensional dynamical systems, the finite dimensional dynamical systems have been researched for more than fifty years. Global attractor, inertial manifolds, fractal dimensions,hausdoff dimensions of some nonlinear evolution equations with dissipative effect have been re-searched all-around. At present, some basic theorems of infinite dimensional dynamical systems have already established. As the importance in practical applications, many academicians are attracted to reserch these systems and they have got many useful re-sults.In this paper, we discuss the existence of global attractors and their fractal dimen-sions for a non-classical reaction-diffusion equation.In the first chapter, we introduce some important theorems about the global attrac-tors, then we discuss the global attractors of a non-classical reaction-diffusion equation.In the second and third chapters, we give a detail analysis to the existence of global attractors and their fractal dimensions for the non-classical reaction-diffusion. We extend the analysis of Zhong Cheng Kui et al in two directions. First, in this paper we use the periodic boundary conditions in Rn space instead of the Dirichlet conditions on the bounded domainΩ, which are discussed widely, we overcome the difficulty that is hard to deal with. Second, we give the estimate of the fractal dimension of the global attractor for the non-classical reaction-diffusion equation, this makes the results for the non-classical reaction-diffusion equation more substantial and perfect. For many infinite dimensional dynamical systems, only the global attractors are discussed. By verifying two conditions of the fractal dimension theorem gave in the reference, we obtain that the fractal dimension of the global attractor is finite.