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

We show the existence of a global solution to time-space fractional fully parabolic Keller–Segel system: \left\{ \begin{align*}{}_{0}^{c}D_{t}^{\beta} u+(-\Delta)^{\alpha/2}u+\nabla\cdot(u \nabla v)&=0,&&x\in\mathbb{R}^{n},\ t>0,\\{}_{0}^{c}D^{\beta}_{t} v+(-\Delta)^{\alpha/2}v-u&=0,&&x\in\mathbb{R}^{n},\ t>0,\\ u(x,0)=u_{0}(x),\quad v(x,0)&=v_{0}(x),&&x\in\mathbb{R}^{n},\end{align*}\right. under the smallness condition on the initial data, where 0<\beta<1 , 1<\alpha\leq 2 and n\geq 2 , u and v denote the cell density and the concentration of the chemoattractant, respectively, and {}_{0}^{c}D^{\beta}_{t} 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} .