Asymptotic Analysis and Applications of Transition Layers and Interfaces

在本研究项目中，从奇异极限分析的角度研究了反应扩散（对流）方程组。所得结果总结如下：1。在球对称多维域中，研究了缓蚀剂反应速率弱、活化剂扩散速率小时活化剂-缓蚀剂型反应扩散体系。结果表明，随着缓蚀剂扩散速率的降低，过渡层溶液出现对称破缺分岔。分岔无限频繁地发生。在反应扩散方程的活化剂-缓蚀剂体系中，当两组分的反应速率相当且缓蚀剂的扩散速率较大时，随着活化剂扩散速率的减小，过渡层溶液出现对称破缺分岔。分岔发生得无限频繁。此外，平行于界面方向的典型波长与激活剂扩散速率的平方根成正比。在三维有界区域中考虑Allen-Cahn方程，研究界面与区域边界相交的平稳过渡层解。发现只有当界面为与区域边界成直角相交的最小曲面时，才有这样的解。此外，这种平稳过渡层的稳定性由具有Robin型边界条件的最小曲面上的椭圆边值问题来确定。对于特定类型的区域，进行了平稳过渡层解的构造，并以可计算量的形式明确表示了其稳定性条件。对于双稳定型标量反应-扩散-对流方程，通过渐近奇异摄动分析，导出了一个界面方程，该方程清晰地反映了对流对过渡层运动的影响。我们怀疑，对流可以使静止过渡层稳定下来，而没有对流的过渡层是不稳定的。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