Comprehensive Researches of Differential Equations

关于真实的域上的常微分方程的可解性和解的性质的研究也很活跃，关于线性方程的振动解或周期解的存在性与不存在性，中立型泛函微分方程的振动解与非振动解的存在性与非存在性，都有一些显著的结果，对于含有无限时滞的方程，人们发现，等终极有界性不是最终有界性的结果，这与有限时滞的情况不同。关于复域上的常微分方程，多个自变量的超几何函数的研究一直很活跃，在单值理论、同调理论、联络问题等方面都取得了重要的成果，并尝试将Painleve方程推广到偏微分方程，得到了满意的结果.对于已知时滞泛函微分方程的控制理论， 关于我们 将能控性、能观性、能识性等结果推广到含偏微分方程的发展方程。正如注意到从什么是上述应该指出的是，交流之间的常微分方程和偏微分方程是大大促进。对于线性偏微分方程的初值问题、边值问题、混合问题、亚椭圆性、散射问题等，已有大量的结果。深入研究了无限退化算子的亚椭圆性，并给出了一个亚椭圆而非微亚椭圆算子的例子。建立了一类含向量值时间变量方程的Cauchy-Kovalevskaja型定理。关于非线性椭圆型方程整体解或振动解的存在性与不存在性、无穷远点的渐近性态、粘性解的方法等问题一直很活跃。在数学物理或生物学方程中，Navier-Stokes方程解的衰减、热对流和可压缩粘性气体方程、含自由边界的变分问题、两种不同物质之间界面的运动、等离子体中波的Schrodinger极限、波动或弹性方程的散射理论、行波解的稳定性，拟线性抽象微分方程的全局可解性及其在种群动力学中的应用，等等。

As for ordinary differential equations in real domains solvability and properties of solutions were studied actively, and there were a number of remarkable results concerning the existence and non-existence of oscillatory or periodic solutions of Linear equations and of oscillatory and non-oscillatory solutions of neutral functional differential equations, etc. For equations containing an infinite delay it was discovered that equi-ultimate boundedness is not a consequence of ultimate boundedness unlike the case of a finite delay. Concerning ordinary differential equations in complex domains researches on hypergeometric functions of several independent variables were vigorous continuously, and significant results were established for the monodromy theory, homology theory, connection problems, etc. An extension of Painleve equations to partial differential equations was attempted with a satisfactory outcome. As for the control theory of retarded functional differential equations known re … More sults on controllability, observability, identifiability, etc. were greatly extended for evolution equations containing partial differential equations. As is noticed from what is stated above it should be remarked that the intercourse between ordinary differential equations and partial differential equations was greatly promoted. As for linear partial differential equations numerous results were obtained for initial value problems, boundary value problems, mixed problems, hypoellipticity, scattering problems, etc. The hypoellipticity of infinitely degenerate operators was deeply studied, and an example of an operator which is hypoelliptic but not microhypoelliptic was obtained. A Cauchy-Kovalevskaja type theorem for an equation with a vector valued time variable was established. Concerning nonlinear elliptic equations the existence and non-existence of global solutions or oscillatory solutions, asymptotic behavior at infinity, the method of viscosity solutions, etc. were continuously active. As for equations of mathematical physics or biology a great number of remarkable results were obtained for the decay of solutions of Navier-Stokes equations, equations of thermal convection and compressible viscous gases, variational problems containing a free boundary, the motion of an interface between two different kinds of substances, the Schrodinger limit of waves in plasma, scattering theory of wave or elasticity equations, the stability of traveling wave solutions, global solvabilty of quasi-linear abstract differential equations with applications to population dynamics, and so on. Less