Global existence and decay estimates for the classical solution of fractional attraction–repulsion chemotaxis system

Abstract

The purpose of this paper is devoted to investigate the generalized Keller–Segel system with a fractional parabolic equation and two classical elliptic equations in Rn with n⩾2. When the chemotaxis sensitivities ξ1 and ξ2 are constants, the global well-posedness of solution to (1.1) is derived if repulsion prevails over attraction in the sense that ξ1μ1−ξ2μ2>0. In the process of proving the above results, we introduce Riesz transform and Kato–Ponce’s commutator inequalities to deal with the challenge arising from fractional diffusion. However, there are few results in the case that ξ1 and ξ2 are functions. We develop a unified framework to deal with the existence, uniqueness and decay estimates in such case. At the same time, the global existence, uniqueness and decay estimates of classical solution to the generalized system (1.2) for small initial data are obtained by selecting an appropriate functional space.