Comprehensive Researches of Differential Equations

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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