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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-生长模型和盐诱导相变模型可以在具有对流项的反应扩散系统框架下进行处理。利用一般半群理论证明了这类系统局部解的存在性。利用先验估计和比较定理，证明了该系统在适当的泛函空间中存在整体解。此外，我们证明了指数吸引子的维数，这是一种性质支配的系统的动力学由于Temam等。al有限，表现出压缩性质。(2)除了反应扩散方程不是一个系统和所考虑区域的一维空间外，一般很难确定指数吸引子的维数。为了证明这一点，我们首先考虑了定态解和行波解的存在性及其稳定性。证明了二维平面上趋化生长模型的平面脉冲平稳解、平面行波解和径向对称脉冲平稳解。为了证明这些解的稳定性，我们必须估计系统的线性化特征值问题的特征值的分布。由于系统的扩散系数很小，所以用奇异极限分析法解决了这一问题。我们首先证明了对于小系数的特征值的主导项确定的稳定性是对应的系数的线性微分常方程由于奇异极限系统，这是由原来的反应扩散系统的减少。由这些结果可知，由于扩散系数的微小性，我们将得到超导相变模型的奇异极限系统。