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This paper investigates Cauchy problems for nonlinear fractional time-spacegeneralized Keller-Segel equation$^c_0D_t^\beta\rho+(-\triangle)^{\frac{\alpha}{2}}\rho+\nabla\cdot(\rhoB(\rho))=0$, where Caputo derivative $^c_0D_t^\beta\rho$ models memory effectsin time, fractional Laplacian $(-\triangle)^{\frac{\alpha}{2}}\rho$ representsL\'evy diffusion and$B(\rho)=-s_{n,\gamma}\int_{R^n}\frac{x-y}{|x-y|^{n-\gamma+2}}\rho(y)dy $ isthe general potential with a singular kernel which takes into account the longrang interaction. We first establish $L^r-L^q$ estimates and weighted estimatesof the fundamental solutions $(P(x,t), Y(x,t))$ (or equivalently, the solutionoperators $(S_\alpha^\beta(t), T_\alpha^\beta(t))$). Then, we prove theexistence and uniqueness of the mild solutions when initial data are in $L^p$spaces, or the weighted spaces. Similar to Keller-Segel equations, if theinitial data are small in critical space $L^{p_c}(\mathbb{R}^n)$($p_c=\frac{n}{\alpha+\gamma-2}$), we construct the global existence.Furthermore, we prove the $L^1$ integrability and integral preservation whenthe initial data are in $L^1(\mathbb{R}^n)\cap L^p(\mathbb{R}^n)$ or$L^1(\mathbb{R}^n)\cap L^{p_c}(\mathbb{R}^n)$. Finally, some importantproperties of the mild solutions including the nonnegativity preservation, massconservation and blowup behaviors are established.

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