The singular perturbation problem for a diffusion-advection equation

本研究项目的目的是利用数值分析或泛函分析来研究扩散-平流方程解的行为。我们首先通过数值方法考虑了其行为的轮廓。在千岁科技大学Kawni教授的有益建议下，我们进行了数值方面的研究。即障碍物后存在尾迹，这是用积分方程研究的方法推测出来的。这个事实还没有得到数学上的证明，但应该得到证明。每位研究者都考虑了自己的课题，并做出了有益的贡献。另一方面，我们研究了半线性抛物方程的解的性质，这是扩散-平流方程的一般情况。我们得到了几个结果。例如，给出了描述不连续现象的脉冲条件与半线性抛物方程解的稳定性之间的关系。我们看到，适当的脉冲条件使爆破溶液稳定，并使溶液在期望的时间爆破。我们应该借助这些结果来研究一个原始问题。

The aim of this research project is to investigate the behavior of the solution of a diffusion-advection equation with the aid of the numerical analysis or the functional analysis. We considered at first the outline of the behavior through the numerical method. We had the numerical aspect under the useful suggestion of Professor Kawni (Chitose Science and technology University). Namely, there exists the wake after an obstacle and this was conjectured by the method of the investigation of an integral equation. This fact is not yet proved mathematically and it should be done. Each investigator considered his subject and made useful contribution.On the other hand, we investigated the behavior of the solution of the semilinear parabolic equation which is general case of a diffusion-advection equation. And we had several results. For example, we have the relation between the impulsive condition which describes the discontinuous phenomena and the stebility of the solution of the semilinear parabolic equation. And we see that the suitable impulsive condition makes a blowing-up solution stable and blows-up the solution at a desired time. We should investigate an original problem with the aid of these results.