Stochastic Keller Segel System with Porous Medium Diffusion and Nonlinear Chemotactic Sensitivity
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 $\Delta u^{[m]}$, 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.