带有多孔介质扩散和非线性趋化敏感性的随机Keller-Segel系统
Stochastic Keller Segel System with Porous Medium Diffusion and Nonlinear Chemotactic Sensitivity
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中文总结 AI 辅助
本文研究带多孔介质扩散和非线性趋化敏感性的随机Keller-Segel系统，对a≥1且m≥2a+1的情况建立鞅解全局存在性等，揭示非线性趋化聚集与扩散的平衡，用多种方法克服相关困难。
中文摘要 AI 辅助
本文在有界一维区域上研究带有多孔介质扩散和非线性趋化敏感性的随机Keller-Segel系统。该模型描述复杂环境中的细胞聚集，其中细胞的扩散由依赖密度的扩散项Δu^[m]控制，反映多孔介质与种群拥挤的综合效应；对趋化因子的感知遵循史蒂文斯幂律，形成非线性趋化敏感性项∇·(u∇v^[a])。此外，通过乘性噪声u dW(t)纳入随机环境波动，代表种群动力学中的随机扰动。对于a≥1且m≥2a+1的情况，我们建立了鞅解的全局存在性、一致先验估计以及非负性的保持。条件m≥2a+1揭示了非线性趋化聚集与多孔介质扩散之间的平衡：更强的感知响应需要更强的扩散来防止过度聚集。证明结合了解耦辅助系统、能量估计以及随机Schauder-Tychonoff不动点论证，以克服由退化扩散、非线性漂移和随机扰动带来的困难。
英文摘要
In this paper, we investigate a stochastic Keller--Segel system with porous medium diffusion and nonlinear chemotactic sensitivity on a bounded one-dimensional domain. The model describes cell aggregation in complex environments, where the dispersal of cells is governed by density-dependent diffusion $Δu^{[m]}$, reflecting the combined effects of porous media and population crowding, and the perception of chemoattractants follows Stevens' power law, leading to the nonlinear chemotactic sensitivity $\nabla\cdot(u\nabla v^{[a]})$. In addition, random environmental fluctuations are incorporated through multiplicative noise $u\,dW(t)$, which represents stochastic perturbations in population dynamics. For $a\geq1$ and $m\geq2a+1$, we establish the global existence of martingale solutions, uniform a priori estimates, and preservation of non-negativity. The condition $m\geq2a+1$ reveals a balance between nonlinear chemotactic aggregation and porous-medium diffusion: stronger sensing response requires stronger diffusion to prevent excessive aggregation. The proof combines a decoupled auxiliary system, energy estimates, and a stochastic Schauder--Tychonoff fixed point argument to overcome the difficulties caused by degenerate diffusion, nonlinear drift, and stochastic perturbations.
