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: (Formula presented.) under the smallness condition on the initial data, where 0 < β < 1, 1 < α ≤ 2 and n ≥ 2, u and v denote the cell density and the concentration of the chemoattractant, respectively, and (Formula presented.) denotes the Caputo fractional derivative of order β with respect to time t. The nonlocal operator (-Δ)^α/2, defined with respect to the space variable x, is known as the Laplacian of order (Formula presented.). We establish the existence of weak solution to the above system by fixed-point arguments under suitable conditions on u<inf>0</inf> and v<inf>0</inf>.