非线性微分方程的动力学研究

The nonlinear differential equations derive from a number of application areas and it has important theoretical significance and scientific value to study their dynamical behaviors. This project is devoted to studying two kinds of problems in the dynamics of nonlinear differential equations. One is the boundary blow up problems of infinity Laplace equation, Monge-Ampere equation and fractional Laplace equation. By using methods such as the perturbation method, the upper and lower solutions method and the continuity technique, this project intends to prove the existence of the solutions to the above three kinds of equations. Based on Karamata regular variation theory, by using the comparison principle, this project intends to study the boundary asymptotic behavior and the uniqueness of the solutions. The other is the Lagrange stability of nonlinear differential equations. Based on the existing results, this project intends to further study the Lagrange stability of the conservative systems with singularity, the inversional systems with singularity and fractional Hamilton systems. The study of this project not only has important significance to the development of the theory of differential equations, but also provides theoretical support and methodological guidance for the development and application of celestial mechanics, image processing and optimal transportation problems, etc.

非线性微分方程源于多个应用领域，对其动力学行为的研究具有重要的理论意义和科学价值。本项目致力于研究非线性微分方程动力学中的两类问题。其一是无穷拉普拉斯方程、Monge-Ampere方程和分数阶拉普拉斯方程的边界爆破问题。本项目拟借助摄动、上下解和连续性等方法证明这三类方程边界爆破解的存在性，并基于Karamata正规变化理论，利用比较原理研究爆破解的唯一性和边界渐近行为。其二是非线性微分方程的Lagrange稳定性。本项目拟在已有的研究基础上进一步研究具有奇点的保守系统、具有奇点的反转系统以及分数阶Hamilton系统的Lagrange稳定性。本项目的深入研究不仅对微分方程和动力系统的理论发展具有重要意义，而且还将为天体力学、图像处理和最优运输问题等的发展应用提供理论支持和方法指导。

微分方程源于各种实际问题，对其动力学行为的定性研究具有重要的理论意义和实践价值。本项目重点研究了几类非线性微分方程边值问题解的存在性、解的性质和一些其它的相关问题，主要研究成果包括：(1).研究了一类无穷拉普拉斯方程边界爆破问题解的精确渐近行为;(2).研究了一类Hessian方程边界爆破问题解的渐近行为;(3).研究了一类p-Laplacian椭圆方程边界爆破问题的解在区域边界附近的二阶展式;(4).研究了几类非线性微分方程解的稳定性理论,并将该理论应用到非线性系统的同步控制中;(5).利用Karamata正规变化理论引进一新的Morrey空间并研究了调和分析中大多数单边积分算子在此空间中的有界性质;(6).研究了其它一些与本项目相关的内容:非线性泛函分析中一类广义超梯度隐迭代算法和隐式外梯度迭代算法的性质及其应用以及群体决策中模糊判断矩阵加性一致性修正问题等。项目执行期间，发表了高水平学术论文12篇，其中SCI收录11篇。