Global existence, uniqueness and \(L^{\infty }\)-bound of weak solutions of fractional time-space Keller-Segel system

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 \({\mathbb {R}}^{n}\) ${\mathbb{R}}^n$ , \(n\ge 2\) $n\ge 2$ . The global existence and \(L^{\infty }\) $L^{\infty }$ -bound of weak solutions are established. We mainly divide the damping coefficient into two cases: (i) \(b>1-\frac{\alpha }{n}\) $b>1-\frac{\alpha }{n}$ , for any initial value and birth rate; (ii) \(0<b\le 1-\frac{\alpha }{n}\) $0<b\le 1-\frac{\alpha }{n}$ , 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 \(L^{\infty }\) $L^{\infty }$ -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.