Study on the nonlinear phenomena with moving boundaries

本研究项目主要涉及涉及运动边界的非线性现象的热门研究课题中的下两个课题。1.相分离的研究。1.对江口教授提出的用两个未知变量描述相分离的模型方程进行了分析。一是局部集中度，二是局部有序度。该模型由两个耦合的演化方程组成，其中一个是4阶的，因此分析比较困难。我们集中于一维情形，得到了很多结果。第一个问题是解的存在性和定态解的结构。然后分析了一个奇异摄动问题和一个分支问题。我们在不同的期刊上发表了这些论文，这些论文得到了匿名参考的相当好的评价。2.作者给出了K.Takaoka提出的模型方程的一组精确解，推广了著名的Black-Scholes分析。然后尝试求解考虑交易费用影响的偏微分方程组。本论文现已提交，对相分离的研究仍在继续。关于动力系统的方向似乎很有希望，这项工作现在正在进行中。

This research project is mainly concerned with the next two subjects, among the popular research topics of nonlinear phenomena involving the moving boundaries. 1 : Study on the phase separation. 2 : Study on the exercise boundary in mathematical finances.1.The author has been tackling the analysis of a model equation proposed by Professor T.Eguchi, which describes the phase separation in terms of two unknown variables. One is the local concentration and the other is the local degree of order. The model consists of coupled two evolution equations ; one is 4^<th> order and hence the analysis is tough. We concentrate on the one-dimensional case and we obtain fairy a lot of results. The first one is one the existence of solutions and the structure of steady state solutions. Analysis of a singular perturbation problem and a bifurcation problem then follows. We publish these papers in various journal, which are rather well estimated by anonymous referees.2.The author gives a set of exact solutions for the model equation proposed by K.Takaoka, which extends the famous Black-Scholes analysis. Then the author tries to solve the partial differential equations with the effect of transaction costs. The paper is submitted now.The research of the phase separation still continues. The direction with respect to the dynamical system seems promising and the work is now in progress.