The comprehensive study of differential equations

目前的研究项目包括常微分方程和偏微分方程。由于本课题课题组成员的积极、热情和积极的研究，在微分方程的各个分支中取得了许多有意义和重要的成果，如下：(1)实域非线性常微分方程的定性理论；(2)复域常微分方程和偏微分方程的解析理论；(3)双曲型、椭圆型和抛物型线性偏微分方程的基本存在唯一性理论；(4)伪微分算子理论；(5)波动和弹性波动方程的散射理论；(6)非线性椭圆型偏微分方程在无界区域上正解的存在性和定性行为问题；(7)抛物型和双曲型非线性偏微分方程（或系统）解的整体存在性和渐近性；(8)流体动力学和其他应用科学中出现的非线性偏微分方程的各种基本问题，如Navier-Stokes方程和Boltzmann方程。特别提到了首席研究员关于(1)中性型高阶泛函微分方程振荡理论的新结果；（ii）推广规定平均曲率方程的二阶拟线性椭圆方程的各种正完整解的存在性；(3)一类任意阶半线性椭圆方程正全解的分类。

The present research project covers both ordinary differential equations and partial differential equations. As a consequence of active, enthusiastic and energetic investigations by the members of the group of researchers, organized for this project, a great number of significant and important results have been obtained in various branches of differential equations as listed below:(1) qualitative theory of nonlinear ordinary differential equations in the real domain;(2) analytic theory of ordinary and partial differential equations in the complex domain;(3) fundamential existence-uniqueness theory for linear partial differential equations of hyperbolic, elliptic and parabolic types;(4) theory of pseudo-differential operators;(5) scattering theory for wave and elastic wave equations;(6) the problem of existence and qualitative behavior of positive solutions of nonlinear elliptic partial differential equations in unbounded domains;(7) global existence and asymptotic behavior of solutions for nonlinear partial differential equations (or systems) of parabolic and hyperbolic types;(8) various fundamental problems for nonlinear partial differential equations arising in fulid dynamics and other applied sciences, such as the Navier-Stokes equation and the Boltzmann equation.Praticular mention is made of the new results obtained by the head investigator regarding(i) oscillation theory of higher order functional differential equations of neutral type;(ii) existence of various types of positive entire solutions for second order quasilinear elliptic equations generalizing the equation of prescribed mean curvature; and(iii) classification of positive entire solutions of a class of semilinear elliptic equations of arbitrary order.