发表机构
School of Mathematics and Statistics, Lanzhou University; School of Mathematical Science, CMA-Shanghai, Shanghai Jiao Tong University（兰州大学数学与统计学院; 上海交通大学数学科学学院，上海数学中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文扩展主特征值性质至非自治抛物算子的主Floquet丛，研究其指数渐近行为，并用于定义SIS模型的临界数以分析疾病灭绝与持续。
AI 中文摘要
Floquet丛理论是椭圆型和周期抛物型算子特征值理论向非自治、非周期抛物型算子谱理论的自然延伸，为更一般的非自治系统中阈值动力学的研究提供了有用的谱工具。然而，与椭圆型或周期抛物型问题的特征值理论相比，目前对归一化主Floquet丛的理解和应用仍然相当有限。在本工作中，我们首先将主特征值的若干基本性质推广到非自治、非周期抛物型算子的归一化主Floquet丛，并获得一系列平行的结果。此外，我们关注在齐次Neumann边界条件下，主Floquet指数关于广义频率和扩散速率在各类典型参数极限（包括独立和耦合参数机制）下的渐近行为。主要工具是本工作中为长时间极限过程发展的抛物型比较原理，该原理通过适当构造的下解和上解来推导主Floquet指数的界，并辅以Harnack不等式和长时间平均技术。最后，我们利用归一化主Floquet丛为非自治空间扩散SIS流行病模型定义两个临界数（作为基本再生数的推广），研究疾病的灭绝和弱持续，并考察基本临界数关于广义频率和扩散速率的极限行为。
英文摘要
Floquet bundle theory serves as a natural extension of the eigenvalue theory for elliptic and periodic parabolic operators to the spectral theory of non-autonomous, non-periodic parabolic operators, which provides a useful spectral tool for studying threshold dynamics in more general non-autonomous systems. However, current understanding and applications of the normalized principal Floquet bundle remain rather limited compared with the eigenvalue theory for elliptic or periodic parabolic problems. In this work, we first extend several fundamental properties of the principal eigenvalue to the normalized principal Floquet bundle of non-autonomous, non-periodic parabolic operators, and obtain a series of parallel results. Furthermore, we focus on the asymptotic behavior of the principal Floquet exponent with respect to the generalized frequency and diffusion rate under various typical parameter limits (including both independent and coupled parameter regimes) for the operator under homogeneous Neumann boundary conditions. The main tools are a parabolic comparison principle developed in this work for long-time limit processes, which is used to derive bounds for the principal Floquet exponent via suitably constructed sub- and super-solutions, supplemented by Harnack's inequality and long-time averaging techniques. Finally, we employ the normalized principal Floquet bundle to define two critical numbers (as extensions of the basic reproduction number) for a non-autonomous spatially diffusive SIS epidemic model, examine the extinction and weak persistence of the disease, and investigate the limiting behavior of the basic critical numbers with respect to the generalized frequency and diffusion rate.
