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

Abstract

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. Finally, we also propose a blow-up criterion for weak solutions, that is, if a weak solution blows up in finite time, then for all $h>q$, the $L^h$-norms of the weak solution blow up at the same time.

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