Studies on linear and nonlinear differential equations with time delay

利用Puiseux和Laplace展开式构造基本解，研究了具有时滞的微分-积分方程陈方程。由Mittag-Leffler函数表征的渐近级数详细地确定了方程的长时间性态。陈氏方程最初描述的是球形粒子在流体中的运动，与二组分流体混合物的动力学方程有着密切的联系。莫斯科大学的Radkevich教授在我系呆了四个半月，直到1998年3月。马斯卡特问题是描述油水界面运动的非线性椭圆型偏微分方程组，曾与他共同研究过。联合工作仍在进行中。我们的主要思想是将Muskat问题的弱解刻画为具有与界面宽度对应的小参数的非线性抛物型偏微分方程解的弱极限。这个想法实际上适用于对称域为球的情况。这些结果将发表在期刊《适用分析》上。

The differential-integral equation with time lag, Chen's equation, was studied by constructing the fundamental solutions using Puiseux and Laplace expansions. Also, long-time behavior of the equation was determined in detail by the asymptotic series characterized by the Mittag-Leffler function. Chen's equation originally describes the motion of a spherical particle in fluid have a close connection with the equations of dynamics of 2-species fluids mixture.Professor Radkevich in Moscow University had been staying at our department for four and half months through March 1998. The Muskat problem, nonlinear elliptic partial differential equation describing the interfacial motion between oil and water, has been studied jointly with him. The joint work is still going on. Our main idea is to characterize the weak solution to the Muskat problem as the weak-limit of the smooth solution of the nonlinear parabolic partial differential equation which has the small parameter corresponding the width of the interface. This idea is actually true for the case of the symmetric domain as a ball. These results will appear in the journal, Applicable Analysis.