Asymptotic Analysis and Applications of Transition Layers and Interfaces

在本研究计画中，我们从奇异极限分析的观点来研究反应扩散（对流）方程组。主要研究结果如下：1.在球对称多维区域上，研究了当抑制剂的反应速率较弱，而激活剂的扩散速率较小时，激活剂-抑制剂型反应扩散系统.结果表明，随着抑制剂扩散速率的减小，过渡层解出现对称破缺分岔。分叉发生的频率是无限的。在反应扩散方程的活化剂-抑制剂系统中，当两组分的反应速率相当且抑制剂的扩散速率较大时，随着活化剂扩散速率的减小，过渡层解出现对称破缺分岔.分叉发生的频率是无限的。此外，在平行于干涉方向上的典型波长 ...更多信息 ACE与活化剂扩散速率的平方根成比例。在三维有界区域中考虑Allen-Cahn方程，研究了界面与区域边界相交的过渡层定常解。结果表明，只有当界面是与区域边界成直角相交的极小曲面时，这种解才是可能的。此外，这种稳定的过渡层的稳定性是由一个椭圆边值问题的极小曲面上的Robin型边界条件。对于特定类型的区域，构造了定态过渡层解，并以可计算量的形式明确地表达了其稳定性条件.对双稳态反应扩散对流标量方程，通过渐近奇异摄动分析，导出了一个界面方程，它清楚地反映了对流对过渡层运动的影响。人们怀疑对流可以稳定静止的过渡层，而没有对流是不稳定的。少

In this research project, systems of reaction-diffusion (convection) equations are investigated from a viewpoint of singular limit analysis. The results obtained are summarized as follows.1. In spherically symmetric multidimensional domains, reaction-diffusion systems of activator-inhibitor type are studied when the reaction rate of inhibitor is weak and the diffusion rate of activator is small. It is shown that symmetry breaking bifurcations of transition layer solutions occur as the diffusion rate of inhibitor is decreased. The bifurcation takes place infinitely often.2. In activator-inhibitor systems of reaction-diffusion equations, when the reaction rates of both components are comparable and the diffusion rate of inhibitor is large, it is shown that symmetry breaking bifurcations of transition layer solutions occur as the diffusion rate of activator is decreased. The bifurcation takes place infinitely often. Moreover, the typical wave length in the direction parallel to the interf … More ace scales as proportional to the square root of the diffusion rate of activator.3. Allen-Cahn equation is considered in three dimensional bounded domains, and stationary transition layer solutions whose interface intersects the domain boundary are studied. It is found that such solutions are possible only when the interface is a minimal surface intersecting the domain boundary in right angle. Moreover, the stability of such stationary transition layers is determined by an elliptic boundary value problem on the minimal surface with Robin type boundary conditions. For specific types of domains, construction of stationary transition layer solutions are carried out and their stability conditions are explicitly expressed in the form of computable quantities.4. For scalar reaction-diffusion-convection equations of bi-stable type, asymptotic singular perturbation analysis is carried out to derive an interface equation which clearly displays the effects of convection on the motion of transition layers. It is suspected that the convection may stabilize stationary transition layers which without convection is know to be unstable. Less