Yang Juan, Morgan Jeff, Tang Bao Quoc
School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000 China.
Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, 8010 Graz, Austria.
Nonlinear Differ Equ Appl. 2024;31(5):98. doi: 10.1007/s00030-024-00985-w. Epub 2024 Aug 6.
The global existence and boundedness of solutions to quasi-linear reaction-diffusion systems are investigated. The system arises from compartmental models describing the spread of infectious diseases proposed in Viguerie et al. (Appl Math Lett 111:106617, 2021); Viguerie et al. (Comput Mech 66(5):1131-1152, 2020), where the diffusion rate is assumed to depend on the total population, leading to quasilinear diffusion with possible degeneracy. The mathematical analysis of this model has been addressed recently in Auricchio et al. (Math Methods Appl Sci 46:12529-12548, 2023) where it was essentially assumed that all sub-populations diffuse at the same rate, which yields a positive lower bound of the total population, thus removing the degeneracy. In this work, we remove this assumption completely and show the global existence and boundedness of solutions by exploiting a recently developed -energy method. Our approach is applicable to a larger class of systems and is sufficiently robust to allow model variants and different boundary conditions.
研究了拟线性反应扩散系统解的全局存在性和有界性。该系统源于Viguerie等人（《应用数学快报》111:106617，2021年）；Viguerie等人（《计算力学》66(5):1131 - 1152，2020年）提出的描述传染病传播的 compartmental 模型，其中扩散率假定依赖于总人口，导致可能退化的拟线性扩散。最近在Auricchio等人（《数学方法与应用科学》46:12529 - 12548，2023年）中对该模型进行了数学分析，其中本质上假定所有子种群以相同速率扩散，这产生了总人口的正下界，从而消除了退化。在这项工作中，我们完全去除了这个假设，并通过利用最近发展的 -能量方法证明了解的全局存在性和有界性。我们的方法适用于更大类的系统，并且足够稳健以允许模型变体和不同的边界条件。
