含饱和型密度依赖奇异敏感性的高维斑秃趋化系统全局有界性

Global Boundedness of High-Dimensional Alopecia Chemotaxis Systems with Saturated Density-Dependent Singular Sensitivity

摘要: 本文研究一类描述斑秃发病机制的三组分趋化系统动力学行为，将系统中CD4⁺T细胞的趋化敏感性设定为饱和型密度依赖奇异形式${\chi }_1\left(u\right)=\frac{{\chi }_0}{1+ku}$ ，在带光滑边界的有界凸区域$\Omega \subset R^n$ $\left(n=4,5\right)$ 内。通过构造能量泛函、运用Young’s不等式、Gronwall’s不等式及热半群理论，证明当Logistic阻尼系数${\mu }_1$ ，${\mu }_2$ 充分大时，该修正系统存在全局一致有界的经典解。该结果将常数趋化敏感性下的系统研究拓展至饱和密度依赖奇异场景，为斑秃病灶中免疫细胞聚集的非线性调控机制提供了更贴合生物实际的数学描述。

Abstract: This paper investigates the dynamical behaviors of a three-component chemotaxis system describing the pathogenesis of alopecia areata. CD4⁺T cell chemotactic sensitivity in the system is taken as a saturated density-dependent singular form ${\chi }_1\left(u\right)=\frac{{\chi }_0}{1+ku}$ , within a bounded convex domain $\Omega \subset R^n$ $\left(n=4,5\right)$ with smooth boundaries. By constructing energy functionals and applying Young’s inequality, Gronwall’s inequality and heat semigroup theory, it is proved that there exist globally uniformly bounded classical solutions to the modified system when the Logistic damping coefficient ${\mu }_1$ and ${\mu }_2$ is sufficiently large. This result extends the research on systems with constant chemotactic sensitivity to the setting of saturated density-dependent singular sensitivity, and provides a more biologically realistic mathematical characterization for the nonlinear regulation mechanism of immune cell aggregation in alopecia areata lesions.