This paper studies the properties of weak solutions to a class of space-time fractional parabolic-elliptic Keller-Segel equations with logistic source terms in \({\mathbb {R}}^{n}\), \(n\ge 2\). The global existence and \(L^{\infty }\)-bound of weak solutions are established. We mainly divide the damping coefficient into two cases: (i) \(b>1-\frac{\alpha }{n}\), for any initial value and birth rate; (ii) \(0<b\le 1-\frac{\alpha }{n}\), for small initial value and small birth rate. The existence result is obtained by verifying the existence of a solution to the constructed regularization equation and incorporate the generalized compactness criterion of time fractional partial differential equation. At the same time, we get the \(L^{\infty }\)-bound of weak solutions by establishing the fractional differential inequality and using the Moser iterative method. Furthermore, we prove the uniqueness of weak solutions by using the hyper-contractive estimates when the damping coefficient is strong.

本文研究了一类时空分数阶抛物线-椭圆 Keller-Segel 方程的弱解的性质，其逻辑源项在 \（{\mathbb {R}}^{n}\）， \（n\ge 2\） 中。弱解的全局存在和 \（L^{\infty }\） 界已建立。我们主要将阻尼系数分为两种情况：（i） \（b>1-\frac{\alpha }{n}\），对于任何初始值和出生率;（ii） \（0<b\le 1-\frac{\alpha }{n}\），初始值小，出生率小。通过验证所构建的正则化方程的解的存在性，并结合时间分数偏微分方程的广义紧凑性准则，得到存在性结果。同时，我们通过建立分数阶微分不等式并使用 Moser 迭代方法得到弱解的 \（L^{\infty }\） 界。此外，当阻尼系数较强时，我们通过使用超收缩估计来证明弱解的唯一性。