Homogenization Of Attractors For A Class Of Quasilinear Parabolic Equations

The study of nonlinear dynamics is a fascinating field which is at the very heart of the understanding of many important problems of the natural sciences. One of the important class of problems in nonlinear dynamics is the nonlinear diffusion problem. It comes from a variety of phenomena which exist widely in nature such as filtration, phase transition, biochemistry and dynamics of biological groups.In mathematics, many new ideas and new tools have been developed to solve the infinite dimensional dynamical system problems. In the terms of partial differential equation, the important part is to obtain uniform prior estimates to the solution of the initial-boundary problem for time t. It can help us to analyze whether the system has some asymptotic properties (e.g. invarant property and attractive property). Homogenization theory, as an asymptotic analysis method, is used to deal with 'the passage from microscopic description to macroscopic description of the behavior of the system'. The global attractor is one important object to describe the long time dynamics of an infinite system. Thus, to give the relation between the attractor of the nonhomogenized system and the homogenized system can in some sense unravel the inherent law of the system, which has theoretical and practical value .In recent years, B. Fiedler, M. Vishik [3] and M. Efendier, S. Zelik [7] have considered the attractor of the nonhomogenized and the homogenized linear and semi-linear reaction-diffusion system respectively. However, at least to author's knowledge, there are few papers that consider the case that the principal part is nolinear operator. The methods in [3] and [7] are not effective to deal with the case of nonlinear operator. With the help of the corrector (see [5]), the author has obtained the similar result as in [3] and [7] for the nonlinear equation.