Mathematical Studies of singularities governed by nonlinear phenomena

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Towards understanding of nonlinear phenomena, we have analytically and complementarily numerically investigated singularities governed by these phenomena from different aspects in mathematics. Since the head investigator had moved from University of Tokyo to Hiroshima University from the second year of the research plan, the original members of investigators had to be altered but we have achieved the purpose of the proposed research. Mimura has been continuously studied pattern formation arising in reaction-diffusion systems. In particular, he has developed singular limit methods to understand dynamics of such patterns, Ueyama numerically studied this problem. Sakamoto has considered singular perturbation procedures to establish the theory of internal layers in higher dimensions. Matano has investigated qualitative properties of solutions of reaction-diffusion systems by using the theory of infinite dimensional dynamical systems. Nagai has discussed the qualitative properties of blow u … More p problem, which is one of the singularities arising in nonlinear diffusion systems. For the development of basic singularity theories in mathematics. Kohno, Shishikura and Kubo have respectively developed theories of geometry and complex dynamical systems. Oharu has developed the nonlinear semi group theory of evolutional equations. Tokihiro has discussed the methods of super discriminations and has revealed the relation between cell automaton and the related differential equations. Yamamoto has established the theoretical frame work of inverse problems which are one of the singular problems of partial differential equations. Yanagida and Kimura have studied analytically and numerically mean curvature equations. On the other hand, from application viewpoints. Inaba has considered epidemic models arising in mathematical demography. Mimura and Sakaguchi have given the theoretical implication on diversity of spatial pattern arising in biological systems by the analysis of mathematical models. Yamada and Hayashi have carried out large scale computer simulations to understand earth circulation phenomenon. The above results have been reported in several conferences inside and outside of Japan. Most of them were talked at the Applied Mathematics Meeting in Japan which was held every year and were published in their proceedings. Less