空间非局部时滞微分系统的渐近行为

In view of problems of non-monotone and non-compactness on delay differential systems with spatial non-locality, we plan to systematically study asymptotic behavior of delay differential systems with spatial non-locality by using the theory for semi-groups of linear operators, the theory of dissipative systems, the theory of monotone dynamical systems, perturbation theory, the method of super-sub solutions, the fluctuation method and compact open topology method, etc. The research emphases are as follows. 1) We first build up the spectral theory on a class of differential operators with spatial non-locality. And then by appropriately improving the existing fluctuation method and the maximum principle, and by using the relaxation variable method, we try to develop a new method to deal with a non-monotone system transferring a quasi-monotone system. 2) By setting up new appropriate comparison principle, and applying the theory of dissipative systems, the theory for semi-groups of linear operators, the theory of monotone dynamical systems and compact open topology method, we explore a new approach on studying a non-compact solution semi-flow and a unbounded domain. 3) We mean to establish a theory of asymptotic behavior of delay differential systems with spatial non-locality. 4) We expect to find out the relationships between asymptotic behavior of part classical delay biological models with spatial non-locality and their parameters. This project is looking forward to enriching the content of qualitative theory of functional differential equations, improving the stability theory of delay differential systems with spatial non-locality, and providing theoretical guidance for further study to the problems of non-local space in other applied fields, such as, the population ecology, disease transmission and the aerodynamics.

针对空间非局部时滞微分系统的非单调与缺紧性问题，本项目拟运用算子半群理论、耗散系统理论、单调动力系统理论、扰动理论、上下解方法、波动方法及紧开拓扑方法等理论方法，对空间非局部时滞微分系统的渐近行为展开深入研究。其研究重点：1）建立一类非局部微分算子的谱理论，适当改进现有波动方法和最大值原理，结合运用放松变量方法，形成将非单调系统化为拟单调增加系统的理论方法；2）建立新的适当比较原则，结合运用耗散系统理论、算子半群理论、单调动力系统理论及紧开拓扑方法，建立处理非紧解半流与无界区域的理论方法；3）建立非单调缺紧性空间非局部时滞微分系统的渐近行为理论；4）建立部分经典空间非局部时滞生物模型渐近行为与其参数的关系。本项目的完成将有助于丰富泛函微分方程定性理论的内容，有助于完善空间非局部时滞微分系统的稳定性理论，并将为种群生态学、疾病传播学和空气动力学等应用学科中空间非局部问题的研究提供理论指导。

针对空间非局部时滞微分系统的非单调与缺紧性问题，本项目运用算子半群理论和扰动理论，对非局部微分算子的特征值问题进行了深入分析，建立了该类算子的谱理论；通过改进最大值原理，找到了一种将混拟单调系统化为拟单调系统的方法；通过改进波动方法，获得了一种将非单调系统化为混拟单调系统的方法，进而获得了一类非局部时滞微分系统的平衡态的存在唯一性、全局吸引性和稳定性等渐近性质；通过运用上下解方法和对积分核的精细分析以及非局部微分算子的谱理论，获得了非单调非局部时滞微分方程平衡态的存在唯一性和全局渐近稳定性；通过推广Sattinger关于非线性椭圆与抛物边值问题中的单调方法，找到了一种研究空间非局部时滞微分系统非常数平衡态稳定性的方法；通过运用算子半群理论、耗散系统理论、上下解方法、紧开拓扑方法、单调动力系统理论以及改进的波动方法，获得了空间非局部时滞微分系统的平衡态的存在唯一性、全局吸引性和稳定性等渐近性质，进而形成了一套处理非单调性问题和缺紧性问题的方法，初步建立了空间非局部时滞微分系统的渐近行为理论；通过编程自行设计模型求解软件和数值模拟等方法，建立了部分经典空间非局部时滞生物模型渐近行为与其参数的关系。本项目研究结果将丰富泛函微分方程定性理论的内容，有助于完善空间非局部时滞微分系统的稳定性理论，并对种群生态学、疾病传播学和空气动力学等应用学科中空间非局部问题具有重要的理论与实践意义。