Decay estimates for the classical solution of Keller–Segel system with fractional Laplacian in higher dimensions

In this paper, we investigate the generalized Keller–Segel system with two fractional parabolic equations and a classical elliptic equation in with . We develop a framework for a unified treatment of the existence, uniqueness and decay estimates. With the help of selecting appropriate functional space, the existence, uniqueness and decay estimates of the global classical solution are shown at the same time under the assumption that the initial data are small enough.