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: \{ {align*}{}₀ᶜDₜβ u+(-Δ)α/2u+∇·(u ∇ v)&=0,&&xⁿ,\ t>0,\\{}₀ᶜDβₜ v+(-Δ)α/2v-u&=0,&&xⁿ,\ t>0,\\ u(x,0)=u₀(x), v(x,0)&=v₀(x),&&xⁿ,{align*}. 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 {}₀ᶜDβₜ 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 α/2 . We establish the existence of weak solution to the above system by fixed-point arguments under suitable conditions on u₀ and v₀ .