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Asymtotic Analysis of Transition Layers Intersecting the Boundary

与边界相交的过渡层的渐近分析

基本信息

批准号：
11640204
负责人：
SAKAMOTO Kunimochi
金额：
$ 1.86万
依托单位：
Hiroshima University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2000
项目状态：
已结题

项目摘要

1. Infinitely Many Bifurcations of Fine Modes. For reaction-diffusion systems of activator-inhibitor type, the existence of multidimensional radially symmetric transition layer solutions is established. Moreover, when the thickness of the layer goes to zero, it is shown that transition layers with non-radial symmetry and fine structures bifurcate at an infinitely many values of thickness of the layers. We proved the one-half power-law between the thickness and the wave length of bifurcating solutions.2. Interface Equation with Non-local Effects. As a distinguished limit of reaction-diffusion systems, interface equations involving the mean curvature and non-local effects are derived. The well-posedness of the latter equations is establislaed. When the domain geometry is simple, equilibrium solutions of the interface equations are constructed. It is shown that these equilibrium interfaces give rise to those of the original reaction-diffusion systems with stability properties inclusive.3. … More Geometric Variational Problem and Interface Equation. Interface equations of reaction-diffusion systems with balanced non-linearities are shown to be realized as a gradient system of geometric variational problems.4. Asymptotic Expansion of Interface Equation and Hierarchical Structure of Dynamics. By using the method of detailed asymptotic expansion, a rigorous treatment is given to the derivation procedure of interface equations for reaction-diffusion systems, which have not been emphasized in conventional studies. In this process, we have found that a reaction-diffusion system in general have several time scales and that each time scale gives rise to a different interface equation, thus providing us with a hierarchical viewpoint to the dynamics of reaction-diffusion systems.5. Internal Layers Intersecting the Boundary of Domain. For the Allen-Cahn Equation, the existence of internal layers intersecting the boundary of domain is established. We also established the relationship between the stability of the layers and the geometric properties of the boundary. The method employed here does not depend on the maximum principle and hence has a possible extension to deal with reaction-diffusion systems. Less

1. 精细模态的无限多分岔。对于活化剂-抑制剂型反应扩散体系，建立了多维径向对称过渡层解的存在性。此外，当层厚趋于零时，具有非径向对称和精细结构的过渡层在层厚的无限多个值处分叉。证明了分岔解的厚度与波长之间的1 / 2幂律。具有非局部效应的界面方程。作为反应扩散系统的一个特殊极限，导出了包含平均曲率和非局部效应的界面方程。建立了后两个方程的适定性。当区域几何结构较简单时，构造界面方程的平衡解。结果表明，这些平衡界面产生了原始反应扩散系统的平衡界面，其稳定性质包括：更多几何变分问题与界面方程具有平衡非线性的反应扩散系统的界面方程可以用几何变分问题的梯度系统来实现。界面方程的渐近展开与动力学的层次结构。利用详细渐近展开的方法，对反应扩散系统界面方程的推导过程进行了严格的处理，这是传统研究中没有强调的问题。在这个过程中，我们发现一个反应扩散系统通常有几个时间尺度，每个时间尺度产生一个不同的界面方程，从而为我们提供了一个层次的反应扩散系统动力学的观点。与域边界相交的内层。对于Allen-Cahn方程，建立了与区域边界相交的内层的存在性。我们还建立了层的稳定性与边界的几何性质之间的关系。这里采用的方法不依赖于最大原理，因此有可能扩展到处理反应扩散系统。少