Dynamics of patterns and their interactions for nonlinear evolutional equations

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In the period of this project, we completely classified the bifurcation structures of 1 dimensional traveling front solutions and analyzed the dynamics of traveling front solutions in inhomogeneous media from the dynamical system point of view. By the analysis, we can know how the solutions go through or reflect by obstacles with words of invariant manifold theory. The interaction of two front solutions is also treated. In general, the treatment of pulse solution is difficult. Applying the results of the interaction of two front solutions, we construct a pulse solution as the combination of two front solutions, which makes easier to treat pulse solutions. Second main result is to establish a general framework to analyze the boundary spike solutions in two dimensional domain. We proved a spike solution moves along the boundary towards the point with maximal curvature of the boundary. We also showed the existence of stable stationary solutions with two peaks in the neighborhood of a point on the boundary with maximal curvature. The results are applied to the Gierer-Meinhardt model and its shadow system. The extension of these results to higher dimensional spaces is still open. For the interfacial dynamics of reaction-diffusion systems in two dimensional spaces including spiral motion, the rigorous treatment has not been completed yet. But almost any necessary preparation in order to study such an interfacial problem such as visualization soft ware and the search program of the motion of a spiral core has been done. At the beginning of the project, we did not know explicit direction to solve this kind of interfacial problems except ambiguous expectations. At the end of the period of this project, now we have explicit ways to reach the goal. The results obtained during the period of the project are inherited to the next project as indispensable and important results.