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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)趋化生长模型和吸收物诱导相变模型能够在具有平流项的反应扩散系统框架中进行处理。这些系统局部解的存在性在一般半群论中得到了证明。通过先验估计和比较定理，表明这些系统的全局解存在于合适的函数空间中。此外，我们证明了指数吸引子的维数，这是由于 Temam 等人控制系统动力学的某种属性。 al, 是有限的，通过显示挤压性质。(2) 除了反应扩散方程不是一个系统以及所考虑的域的一空间维数之外，一般很难确定指数吸引子的维数。为了展示这一点，我们首先考虑静止解和移动解的存在性及其稳定性。我们证明了二维平面趋化生长模型的平面脉冲平稳解、平面行进前沿解和径向对称脉冲平稳解。为了证明这些解的稳定性，我们必须估计系统线性化特征值问题的特征值分布。由于系统的扩散系数较小，该问题通过奇异极限分析来解决。我们首先证明，对于小系数，决定稳定性的特征值的主项对应于奇异极限系统的线性微分常方程的系数，这是通过原始反应扩散系统的约简得到的。从这些结果中，由于扩散系数很小，我们将得到吸收物诱导相变模型的奇异极限系统。