发表机构
Research Institute of Intelligent Complex Systems, Fudan University; School of Mathematics, South China Normal University（复旦大学智能复杂系统研究院; 华南师范大学数学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了趋化-消耗系统中正非恒定稳态的非线性稳定性，在一般二维和三维有界域上，对于足够大的营养物扩散系数，系统具有唯一稳态且小扰动解指数收敛。
AI 中文摘要
我们证明了具有营养物依赖性运动性和固定边界营养物浓度 $b$ 的趋化-消耗系统中正的非恒定稳态的非线性稳定性。运动性满足由幂律和饱和函数满足的两个单调性条件。对于每个给定的种群质量，足够大的营养物扩散系数 $D$ 产生唯一的正稳态。小的相容扰动产生唯一的全局解，该解以指数速度收敛到具有其守恒质量的稳态。该结果在二维和三维的一般有界域上成立，在区间上有一个较低正则性版本。在 $D^{-1}$ 阶上，营养物亏缺是扭转函数乘以平均种群密度和边界营养物值；种群对比度还取决于 $\kappa_b=b\phi'(b)/\phi(b)$。计算说明了从相容数据出发的松弛以及当初始质量变化时收敛到不同稳态的情况。
英文摘要
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.
Comments32 pages
