Comprehensive Researches of Differential Equations

对于实区域上的常微分方程解的可解性和性质进行了积极的研究，关于线性方程的振动解或周期解的存在与不存在，中立型泛函微分方程解的振动与非振动解的存在与不存在等结果，等等。对于含有无限时滞的方程，我们发现与有限时滞不同，等终有界性并不是最终有界性的结果。关于复域上的常微分方程组，关于多个自变量的超几何函数的研究一直很活跃，在单调理论、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组，得到了满意的结果。关于已知Re…的时滞泛函微分方程的控制理论对于含有偏微分方程解的发展方程，在能控性、能观测性、可辨识性等方面的结果得到了很大的推广。如上所述，应该指出的是，常微分方程组和偏微分方程组之间的相互作用得到了极大的促进。对于线性偏微分方程，在初值问题、边值问题、混合问题、亚椭圆性、散射问题等方面都得到了大量的结果，深入研究了无穷退化算子的亚椭圆性，得到了一个亚椭圆型而不是微亚椭圆型算子的例子。建立了具有向量值时间变量的方程的Cauchy-Kovalevskaja型定理。对于非线性椭圆型方程，整体解或振荡解的存在与不存在，无穷远处的渐近行为，粘性解的方法等一直活跃着。对于数学物理或生物方程，在Navier-Stokes方程、热对流和可压缩粘性气体方程的解的衰减、含有自由边界的变分问题、两种不同物质之间的界面运动、等离子体中波的薛定谔极限、波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组的整体可解性及其在种群动力学中的应用等方面都取得了许多显著的结果。较少

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