Global solutions for time-space fractional fully parabolic Keller–Segel system

Abstract

We show the existence of a global solution to time-space fractional fully parabolic Keller–Segel system:

under the smallness condition on the initial data, where $0<\beta <1$, $1<\alpha \le 2$ and $n\ge 2$, $u$ and $v$ denote the cell density and the concentration of the chemoattractant, respectively, and ${}_0^c{D}_t^{\beta }$ denotes the Caputo fractional derivative of order $\beta$ with respect to time $t$. The nonlocal operator $(-\Delta {)}^{\alpha /2}$, defined with respect to the space variable $x$, is known as the Laplacian of order $\frac{\alpha }{2}$. We establish the existence of weak solution to the above system by fixed-point arguments under suitable conditions on $u_0$ and $v_0$.