Asymptotic Analysis and Applications of Transition Layers and Interfaces

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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