Asymtotic Analysis of Transition Layers Intersecting the Boundary

1.抑制许多精细模式的分叉。对于活化-抑制型反应扩散方程组，建立了多维径向对称过渡层解的存在性。此外，当该层的厚度趋于零时，它示出，过渡层的非径向对称性和精细结构分叉在无限多个值的层的厚度。证明了分叉解的厚度与波长之间的二分之一幂律关系.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。建立了后一类方程的适定性。当区域的几何形状是简单的，平衡解的界面方程的构造。结果表明，这些平衡界面是原反应扩散系统的平衡界面，具有稳定性. ...更多信息 几何变分问题与界面方程。具有平衡非线性项的反应扩散方程组的界面方程可实现为几何变分问题的梯度系统.界面方程的渐近展开与动力学的层次结构。利用详细渐近展开法，对反应扩散方程组界面方程的推导过程进行了严格的处理，这在以往的研究中是不被重视的.在这个过程中，我们发现反应扩散系统一般具有多个时间尺度，每个时间尺度产生不同的界面方程，从而为我们研究反应扩散系统的动力学提供了一个层次的观点.与域边界相交的内部层。对于Allen-Cahn方程，证明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和边界的几何性质之间的关系。这里采用的方法不依赖于极大值原理，因此可以扩展到处理反应扩散系统。少

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