【作者】 王国联；

【导师】 张兴友；

【作者基本信息】 重庆大学 ， 应用数学， 2004， 硕士

【摘要】 非线性动力系统是理解许多重要自然科学的核心问题，它一直吸引着人们的注意力。无穷维动力系统中一类重要的问题是非线性扩散问题，它来源于自然界广泛存在的扩散现象，渗流理论、相变理论、生物化学以及生物群体动力学等领域都存在这种现象。 数学上，现已建立了无穷维动力系统的重要的理论与数值计算方法[1，13，14]。就偏微分方程而言，最关键的是要建立定解问题的解对时间大范围的一致先验估计，从而考察该解是否具有渐近状态(即不变性和吸引性等)。偏微分方程中的均匀化理论是处理“系统的局部性质的描述过渡到宏观性质的描述”的渐近分析方法。而吸引子是描述无穷维动力系统的重要物理量。因此，寻求系统均匀化之前的吸引子及其均匀化之后的系统的吸引子之间的关系，可以反映出系统的局部性质与整体性质之间的某种联系。这在理论及实际中都有着重要的意义。 在近几年中，B．Fiedler& M．Vishik[3]和M．Efendiev & S．Zelik[7]，分别针对线性和半线性的反应-扩散系统均匀化之前的吸引子及其均匀化之后的吸引子之间的关系进行了讨论。而对非线性系统吸引子之间的关系至今很少人涉及，就笔者所知，对于主部原型为p-Laplace算子的情形，还没有文章讨论过吸引子之间的误差估计。但对主部是非线性的情形，[3]和[7]所提供的方法已不在适用，笔者利用已有的结果，巧妙的以“纠正子”作为过渡进行了讨论，得到了主部为p-Laplace算子的微分算子的微分方程的吸引子与其均匀化后的吸引子之间的距离的一个明确的上界估计。更多还原

【Abstract】 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.更多还原