Study on the nonlinear phenomena with moving boundaries

本研究项目主要关注以下两个课题，这两个课题是涉及移动边界的非线性现象的热门研究课题。1：相分离的研究。2 .数学金融学中练习边界的研究。作者一直在分析T.Eguchi教授提出的模型方程，该模型方程用两个未知变量来描述相分离。一个是局部浓度，另一个是局部有序度。该模型由两个耦合的演化方程组成；一个是4^< >阶，因此分析比较困难。我们主要研究一维情况，得到了很多结果。第一个是解的存在性和稳态解的结构。接着分析奇异摄动问题和分岔问题。我们把这些论文发表在各种杂志上，这些杂志被匿名审稿人估计得相当准确。本文给出了takaoka提出的模型方程的一组精确解，扩展了著名的Black-Scholes分析。在此基础上，对考虑交易费用影响的偏微分方程进行了求解。论文已经提交了。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，目前工作正在进行中。

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.