The comprehensive study of differential equations

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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.