概周期问题的若干研究

In this project, we study the properties of almost periodic type functions and the existence of almost periodic solutions for some nonlinear evolution equations. More specifically, we investigate four problems. The first problem is to study the properties of some almost periodic type functions and the relationship between almost periodic type functions. The second one is to explore the existence of almost periodic solutions to semilinear evolution equations in the case of the Lipschitz constants of nonlinear terms being not sufficiently small. The third one is to investigate the applications of some nonlinear analysis tools involving compactness and variational methods in almost periodic differential equations. The last one is to discuss some problems with different almost periodic type functions for the coefficients and the solutions of some evolution equations.

本项目拟研究概周期型函数的性质和几类非线性发展方程的概周期型解。具体来说，包括四类问题：第一类是研究几类概周期型函数的性质和概周期型函数之间的关系；第二类是在非线性项的李普希兹常数不充分小情形下，探索半线性发展方程概周期型解的存在性；第三类是探讨紧性工具和变分方法在概周期问题中的应用；第四类是讨论发展方程系数和解的概周期函数类型不一致的问题。

本项目对几类概周期函数的性质、几类非线性发展方程（包括抽象半线性发展方程、无穷维随机微分方程、神经网络模型等）的概周期性、变分方法在概周期椭圆方程中的应用等问题展开了系列研究。特别的，对于渐近概周期函数的Kadets型定理和Loomis型定理这两个难点问题，我们取得了突破性进展。本项目的研究成果一定程度上推动了概周期函数理论、泛函分析、随机微分方程等领域理论和方法的发展和交叉融合。本项目共发表或录用学术论文12篇，获省自然科学奖二等奖1项，入选省千人计划1人，培养了3名青年教师、4名博士研究生、13名硕士研究生。