On the fractional Keller–Segel chemotaxis model

Abstract

This paper studies a modified Keller–Segel chemotaxis model that uses fractional time derivatives (of order $\alpha$ and $\beta$, where $0<\alpha$, $\beta <1$) to incorporate memory effects. This approach better captures the subdiffusive behavior observed in biological systems. We perform our analysis in weak-Herz spaces by using a fixed-point argument and the decay properties of the Mittag–Leffler operators. We prove the local existence and uniqueness of mild solutions. Moreover, we show that the solutions depend continuously on the initial data, and we give a detailed description of the asymptotic behavior of solutions as $t\to 0^{+}$. We establish conditions for either extending solutions or encountering blow-up (chemotactic collapse).