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 D-c(0)t(beta) rho + (-Delta)(alpha/2) +del. (rho B(rho)) = 0, where Caputo derivative D-c(0)t(beta)rho models memory effects in time, fractional Laplacian (-Delta)(alpha/2) rho represents Levy diffusion and B(rho) = - s(n),(gamma )integral(Rn) vertical bar x-y/vertical bar(n-gamma+2)rho(y)dy is the Riesz potential with a singular kernel which takes into account the long rang interaction. We first establish L-r- L-q estimates and weighted estimates of the fundamental solutions (P(x, t), Y (x, t)) (or equivalently, the solution operators (S-alpha(beta)(t),T-alpha(beta)( 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 L-Pc (R-n ) (p(c )= n/alpha+gamma-2 ) , we construct the global existence. Furthermore, we prove the L-1 integrability and integral preservation when the initial data are in L-1 (R-n) boolean AND L-P(R-n ) or L-1(R-n) boolean AND L-Pc(R-n). Finally, some important properties of the mild solutions including the nonnegativity preservation, mass conservation and blowup behaviors are established. (C) 2018 Elsevier Inc. All rights reserved.