Nonlinear Stability of Nonconstant Steady States in Chemotaxis--Consumption Systems with General Motility
Year: 2026
Type: preprint
Abstract: We prove nonlinear stability of positive nonconstant steady states for a chemotaxis--consumption system with nutrient-dependent motility and a fixed boundary nutrient concentration $b$. The motility satisfies two monotonicity conditions met by power laws and saturating functions. For each prescribed population mass, sufficiently large nutrient diffusivity $D$ yields a unique positive steady state. Small compatible perturbations generate unique global solutions converging exponentially to the steady state with their conserved mass. The result holds on general bounded domains in two and three dimensions, with a lower-regularity version on an interval. At order $D^{-1}$, the nutrient deficit is the torsion function multiplied by the mean population density and boundary nutrient value; the population contrast also depends on $κ_b=bϕ'(b)/ϕ(b)$. Computations illustrate relaxation from compatible data and convergence to different steady states when the initial mass changes.
Source: arXiv (Cornell University)
Authors: Cheng, Xinyu, Lai, Xuewen, Song, Xu
Institutions: Fudan University, South China Normal University
Language: English
Subfield: Modeling and Simulation
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G2929595097, G6558892230, G7647130531
OpenAlex ID: W7214713822