部分退化的时滞反应扩散系统的主特征值问题

在微分方程的动力学研究中，特征值是一个非常重要的工具，比如用于研究稳定性问题和分支问题。本项目拟对部分扩散系数为零（部分退化）的合作的时滞反应扩散系统的主特征值问题进行研究，探索主特征值存在与否的本质原因。这一问题来源于benthic-drift种群模型动力学分析的过程。系统部分退化导致解映射缺乏紧性。紧性的缺失为其动力学行为的分析带来了本质性的困难，而研究主特征值问题是探索系统动力学行为的首要步骤。我们将从以下两个方面进行讨论：尝试给出该系统主特征值存在的充分条件；将零扩散系数扰动为正的，使得扰动系统所有扩散系数均大于零，我们将研究当扰动的扩散系数趋于零时，主特征值与主特征函数的极限行为。值得注意的是，这种极限行为可以反映部分退化系统主特征值存在与否的本质原因。

Eigenvalues are critical tools to study the dynamical behavior of differential equations, e.g., to analyze the stability and bifurcation. This project aims to investigate the principal eigenvalue problem for a reaction-diffusion cooperative system with some zero diffusion coefficients (partially degenerate) and time-delay, and to explore the mechanism for the existence or non-existence of the principal eigenvalue. The problem arises from the process of the dynamical analysis of benthic-drift population model. Since some diffusion coefficients are zero, the corresponding solution map is non-compact, which brings an essential difficulty to investigate the dynamical behavior of benthic-drift population model. The vital problem is to study the associated eigenvalue problem. We will discuss it from two perspectives. The first one is to show a sufficient condition for the existence of the principal eigenvalue. The other one is to investigate a perturbation problem. By changing the sign of zero diffusion coefficients to be positive, we introduce a family of perturbed reaction-diffusion systems whose diffusion coefficients are greater than zero. We will study the limit behavior of the principal eigenvalue and eigenfunction as perturbed diffusion coefficients go to zero. It is worth pointing out that the limit behavior essentially reflects the mechanism of the existence or non-existence of the principal eigenvalue for the degenerate system.