Singular limit system and pattern formation of some reaction-diffusion system

某种反应扩散系统的奇异极限系统和模式形成

基本信息

批准号：
10640143
负责人：
TSUJIKAWA Tohru
金额：
$ 2.37万
依托单位：
MIYAZAKI UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 2001
项目状态：
已结题

项目摘要

(1) A chemotaxis-growth model and an absorbate-induced phase transition model are able to treat in the framework of the reaction-diffusion system with advection terms. The existence of the local solution of these systems is proved in the general semi-group theory. By using the a priori estimate and the comparison theorem, it is shown that the global solutions of these system exist in the suitable functional space. Moreover, we prove that the dimension of the exponential attractor, which is some kind of property governed the dynamics of the system due to Temam et. al, is finite by showing the squeezing property.(2) It is generally difficult to determine the dimension of the exponential attractor except for the reaction diffusion equation which is not a system, and the one space dimension of the considered domain. For the first step to show it, we consider the existence of stationary solutions and traveling solutions and their stability. We prove these of the planar pulse stationary solutions, planar traveling front solutions and radial symmetric pulse stationary solutions of the chemotaxis-growth model in 2-dimensional plane. Showing the stability of these solutions, we must estimate the distribution of the eigenvalues of the linearized eigenvalue problem of the system. This problem is solved by the singular limit analysis because that the diffusion coefficient of the system is small. We first show that for the small coefficient t he dominant term of the eigenvalues determined the stability is corresponding to the coefficient of the linear differential ordinary equation due to the singular limit system, which is obtained by the reduction of the original reaction diffusion system. From these results, we will have the singular limit system of the absorbate-induced phase transition model because of the smallness of the diffusion coefficient.

（1）趋化增长模型和吸收型相变模型能够在具有对流项的反应扩散系统的框架中对待。这些系统的局部解决方案的存在在一般的半组理论中得到了证明。通过使用先验估计和比较定理，可以表明这些系统的全局解决方案存在于合适的功能空间中。此外，我们证明了指数吸引子的维度，这是某种属性控制了由于Temam等人的系统动力学。 Al，通过显示挤压特性是有限的。（2）通常很难确定指数吸引子的维度，除了不是系统的反应扩散方程，以及所考虑域的一个空间维度。为了展示它的第一步，我们考虑存在固定解决方案和旅行解决方案及其稳定性。我们证明了平面脉冲固定溶液，平面行进前溶液以及二维平面中趋化性增长模型的径向对称脉冲固定溶液。显示了这些解决方案的稳定性，我们必须估计系统线性特征值问题的特征值的分布。这个问题是通过奇异极限分析解决的，因为系统的扩散系数很小。我们首先表明，对于特征值的主要系数，由于奇异极限系统，特征值的主要项确定稳定性对应于线性微分方程方程的系数，这是通过还原原始反应扩散系统获得的。从这些结果中，由于扩散系数的较小性，我们将具有吸收物诱导的相变模型的奇异极限系统。