Asymtotic Analysis of Transition Layers Intersecting the Boundary

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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