几类微分方程的行波解性态研究

演化方程的动力学行为与其特殊解存在性及其性态紧密相关，而行波解是其一类特殊解，在演化方程的定性理论研究中占有重要的地位。在一定条件下，行波解通常可作为方程的稳态解决定初值问题解的长时间行为。刻画行波解的性态对研究演化方程的动力学行为，以及理解实际问题中发生的传播现象具有重要的意义。本项目拟研究三类具有应用背景的演化方程或系统的传播动力学问题，如：分布时滞扩散方程、时间周期扩散方程、格动力系统的行波解的单调性，唯一性和稳定性问题。我们试图发展一些新的方法和技巧建立这些微分方程的行波解的单调性，唯一性和稳定性结果，以丰富微分方程定性理论和行波解理论，并为理解实际问题中的传播现象提供理论依据。

The dynamic behavior of evolution equation is closely related to the existence and behavior of its special solution. The traveling wave solution is a kind of special solution, which plays an important role in the qualitative theoretical research of evolution equation. Under certain conditions, the traveling wave solution can usually serve as the steady-state solution to determine the long-term behavior of the solution of the initial value problem. It is of great significance to characterize the behavior of traveling wave solutions for studying the dynamic behaviors of evolution equations and understanding the propagation phenomena in practical problems. This project is devoted to the study of the propagation dynamics of three kinds of evolution equations or systems with application background, such as the monotonicity, uniqueness and stability of traveling wave solutions of distributed delay diffusion equation, time periodic diffusion equation and lattice dynamical system. We try to develop some new methods and techniques to establish the monotonicity, uniqueness and stability results of the traveling wave solutions of these differential equations, so as to enrich the qualitative theory and the traveling wave solution theory of the differential equations, and provide a theoretical basis for understanding the propagation phenomenon in practical problems.