Shemtaga Hewan, Shen Wenxian, Sukhtaiev Selim
Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849 USA.
J Evol Equ. 2025;25(1):7. doi: 10.1007/s00028-024-01033-x. Epub 2024 Dec 15.
Chemotaxis phenomena govern the directed movement of microorganisms in response to chemical stimuli. In this paper, we investigate two Keller-Segel systems of reaction-advection-diffusion equations modeling chemotaxis on thin networks. The distinction between two systems is driven by the rate of diffusion of the chemo-attractant. The intermediate rate of diffusion is modeled by a coupled pair of parabolic equations, while the rapid rate is described by a parabolic equation coupled with an elliptic one. Assuming the polynomial rate of growth of the chemotaxis sensitivity coefficient, we prove local well-posedness of both systems on compact metric graphs, and, in particular, prove existence of unique classical solutions. This is achieved by constructing sufficiently regular mild solutions via analytic semigroup methods and combinatorial description of the heat kernel on metric graphs. The regularity of mild solutions is shown by applying abstract semigroup results to semi-linear parabolic equations on compact graphs. In addition, for logistic-type Keller-Segel systems we prove global well-posedness and, in some special cases, global uniform boundedness of solutions.
趋化现象控制着微生物响应化学刺激的定向运动。在本文中，我们研究了两个反应 - 对流 - 扩散方程的凯勒 - 塞格尔系统，它们对细网络上的趋化作用进行建模。两个系统之间的区别由化学引诱剂的扩散速率驱动。中间扩散速率由一对耦合的抛物型方程建模，而快速扩散速率由一个抛物型方程与一个椭圆型方程耦合描述。假设趋化敏感性系数的多项式增长率，我们证明了这两个系统在紧致度量图上的局部适定性，特别是证明了唯一经典解的存在性。这是通过解析半群方法构造足够正则的温和解以及度量图上热核的组合描述来实现的。通过将抽象半群结果应用于紧致图上的半线性抛物型方程，展示了温和解的正则性。此外，对于逻辑斯谛型凯勒 - 塞格尔系统，我们证明了全局适定性，并在某些特殊情况下证明了解的全局一致有界性。
