部分退化反应扩散系统的非平面波

In recent years, a lot of partially degenerate reaction-diffusion systems arise from epidemiology, chemistry, population dynamics and other fields, which attract many scholars' great interests, such as academician Volpert et al.. Since the solution maps associated with such systems are not compact with respect to the compact open topology, which makes the study on the related problems very difficult, hence it is a significant project to study such problems. We will study nonplanar traveling waves of partially degenerate reaction-diffusion systems by means of dynamical system, nonlinear analysis and partial differential equations. Our main concerns are the following: the study of the time periodic V-shaped traveling waves in the two dimensional space, pyramidal traveling waves and conical traveling waves in the multidimensional space, the classification of traveling waves in the two dimensional space of time periodic partially degenerate reaction-diffusion systems and the impact of time delay on nonplanar traveling waves. The purpose is to make a comprehensive understanding of the nonlinear dynamical behavior of the equations from the viewpoint of the classification of traveling waves and their complex mode of transmission. From the viewpoint of dynamical systems, the study on nonplanar traveling waves can not only help us to determine the structure of the maximum invariant set (global attractor), but also help us to make a good understanding of transient dynamics of the systems.

近年来，在传染病学、化学、人口动力学等领域的研究中导出了大量部分退化反应扩散系统，并引起了许多学者如Volpert院士等的极大兴趣。由于在紧开拓扑意义下系统的解半流紧性缺失，从而使得相关问题的研究变得十分困难，因此对其研究是具有重要意义的课题。本项目将借助动力系统、非线性分析以及偏微分方程等工具，研究部分退化反应扩散系统的非平面波。主要内容包括时间周期部分退化反应扩散系统的V形波、高维空间中的棱锥形波和圆锥形波、二维空间中行波解的分类问题以及时滞对非平面波的影响。我们试图从行波解的分类和复杂传播方式的角度全面认识系统的非线性动力学行为。从动力系统的观点看，对非平面波的研究不仅能够帮助我们确定最大不变集（全局吸引子）的结构，而且可以很好地理解系统的瞬态动力学。

本项目系统研究了时间周期反应扩散方程和反应扩散系统的非平面行波解及其定性性质。主要研究内容包括：1.双稳型反应扩散系统的非平面行波解及高维稳定性问题。获得了反应扩散系统非平面行波解的存在性。借助半群理论，证明了平面行波解的代数稳定性。2.时间周期双稳型反应扩散方程的非平面行波解及高维稳定性问题。给出了时间周期非平面行波解的存在性和渐近稳定性。得到了空间衰减扰动下平面行波解和V形行波解的高维稳定性。3.时间周期反应对流扩散方程的整体解和稳定性问题。研究了柱体上时间周期反应对流扩散方程整体解的存在性和定性性质。证明了柱体上时间周期反应对流扩散方程行波解是全局指数稳定性的。我们试图从行波解的分类和复杂传播方式角度全面认识方程的非线性动力学行为。.受本项目的资助，2015年至2017年共发表SCI科研论文9篇。