Comprehensive Researches of Differential Equations

对于实域常微分方程的可解性和解的性质进行了积极的研究，在线性方程的振荡解或周期解的存在性和不存在性，中立型泛函微分方程的振荡解和非振荡解等方面取得了许多显著的成果。对于包含无限延迟的方程，发现了与有限延迟不同的是，等极限有界性并不是极限有界性的结果。在复杂域常微分方程中，对多自变量超几何函数的研究不断蓬勃发展，在单一性理论、同调理论、连接问题等方面取得了重要成果。将painlevel方程推广到偏微分方程，得到了满意的结果。关于含偏微分方程的演化方程的可控性、可观测性、可辨识性等方面的研究结果得到了极大的推广。如上所述，应当注意到常微分方程和偏微分方程之间的交流得到了极大的促进。对于线性偏微分方程，在初值问题、边值问题、混合问题、半椭圆问题、散射问题等方面得到了大量的结果。对无限简并算子的亚椭圆性进行了深入的研究，得到了一个算子是亚椭圆但不是微亚椭圆的例子。建立了含向量值时变量方程的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性与不存在性、无穷远处的渐近性、黏性解的方法等问题持续活跃。对于数学物理或生物学方程，在以下方面取得了许多显著的成果：Navier-Stokes方程解的衰减、热对流方程和可压缩粘性气体方程、包含自由边界的变分问题、两种不同物质之间界面的运动、等离子体中波的薛定谔极限、波的散射理论或弹性方程、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。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