复域上的微分方程和偏微分方程的亚纯函数解研究

本项目拟利用Nevanlinna值分布理论研究复域上的Fermat型微分方程、Fermat型偏微分方程以及非线性偏微分方程的亚纯函数解。首先，我们运用单复变函数Nevanlinna值分布理论和复微分方程理论研究Fermat型复微分方程亚纯函数解的具体形式以及存在性、增长性等性质。其次，我们结合利用多复变函数Nevanlinna值分布理论和偏微分方程中的Lagrange-Charpit特征法对Fermat型复偏微分方程进行研究，探究其亚纯函数解的精确形式和存在性。最后，我们以多复变函数Nevanlinna值分布理论为主要工具探讨一类非线性偏微分方程亚纯函数解的存在性、素性或者拟素性。本项目的研究不仅能丰富Nevanlinna值分布理论，又可以推动复域中的微分方程和偏微分方程的发展。

In this project, we study meromorphic solutions of the Fermat-type differential equations, the Fermat-type partial differential equations and the nonlinear partial differential equations in the complex domain by using Nevanlinna’s value distribution theory. First, we explore the exact solutions and properties such as the existence and growth of meromorphic solutions of the Fermat-type complex differential equations by using Nevanlinna’s value distribution theory of one complex variable and complex differential equation theory. Secondly, we investigate the exact solutions and existence of meromorphic solutions of the Fermat-type complex partial differential equations by using Nevanlinna’s value distribution theory in several complex variables and the Lagrange-Charpit characteristic method of partial differential equations. Finally, we study the existence of meromorohic solutions of a class of nonlinear partial differential equations by using Nevanlinna’s value distribution theory in several complex variables, and study the primeness or pseudo-primeness of meromorphic solutions of the partial differential equations. This research will not only enrich Nevanlinna’s value distribution theory, but also promote the development of differential equations and partial differential equations in the complex domain.