Mathematical Studies of singularities governed by nonlinear phenomena

为了更好地理解非线性现象，我们从数学的不同方面对这些现象所支配的奇点进行了分析和互补的数值研究。由于从研究计划的第二年开始，首席研究员从东京大学转到广岛大学，因此不得不改变原来的研究人员，但我们已经达到了拟议研究的目的。Mimura一直在研究反应扩散系统中产生的模式形成。特别地，他开发了奇异极限方法来理解这种模式的动力学，Ueyama数值研究了这个问题。Sakamoto考虑了奇异摄动过程来建立高维的内层理论。Matano利用无限维动力系统理论研究了反应扩散系统解的定性性质。Nagai讨论了非线性扩散系统奇点之一的blow u…More p问题的定性性质。发展了数学中的基本奇点理论。Kohno， Shishikura和Kubo分别发展了几何理论和复杂动力系统。Oharu发展了演化方程的非线性半群论。Tokihiro讨论了超判别的方法，并揭示了元胞自动机与相关微分方程之间的关系。Yamamoto建立了逆问题的理论框架，逆问题是偏微分方程的奇异问题之一。柳田和木村研究了解析和数值平均曲率方程。另一方面，从应用程序的角度来看。Inaba考虑了数学人口学中产生的流行病模型。Mimura和Sakaguchi通过数学模型的分析给出了生物系统中空间格局多样性的理论含义。Yamada和Hayashi进行了大规模的计算机模拟，以了解地球环流现象。上述结果已在日本国内外的几次会议上报告。其中大部分是在每年在日本举行的应用数学会议上讨论的，并发表在会议论文集上。少

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