Cauchy problems for Keller-Segel type time-space fractional diffusion equation

This paper investigates Cauchy problems for nonlinear fractional time-space generalized Keller-Segel equation Dtβ0cρ+(−△)α2ρ+∇⋅(ρB(ρ))=0, where Caputo derivative Dtβ0cρ models memory effects in time, fractional Laplacian (−△)α2ρ represents Lévy diffusion and B(ρ)=−sn,γ∫Rnx−y|x−y|n−γ+2ρ(y)dy is the Riesz potential with a singular kernel which takes into account the long rang interaction. We first establish Lr−Lq estimates and weighted estimates of the fundamental solutions (P(x,t),Y(x,t)) (or equivalently, the solution operators (Sαβ(t),Tαβ(t))). Then, we prove the existence and uniqueness of the mild solutions when initial data are in Lp spaces, or the weighted spaces. Similar to Keller-Segel equations, if the initial data are small in critical space Lpc(Rn) (pc=nα+γ−2), we construct the global existence. Furthermore, we prove the L1 integrability and integral preservation when the initial data are in L1(Rn)∩Lp(Rn) or L1(Rn)∩Lpc(Rn). Finally, some important properties of the mild solutions including the nonnegativity preservation, mass conservation and blowup behaviors are established.