2026/08/05 by Yiming Jiang, Haohang Li, Yawei Wei
Mathematics · #math.AP
arxiv created 2026/08/05 · arxiv updated 2026/08/06
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 ∇⋅(u∇ v[a]). In addition, random environmental fluctuations are incorporated through multiplicative noise u dW(t), which represents stochastic perturbations in population dynamics. For a≥1 and m≥2a+1, we establish the global existence of martingale solutions, uniform a priori estimates, and preservation of non-negativity. The condition m≥2a+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.