Finite-time blow-up for the attractive dominant case of an attraction-repulsion chemotaxis system in the whole space

We consider the Cauchy problem derived from an attraction-repulsion chemotaxis system in $\mathbb R^n$. This system consists of the partial differential equations of the parabolic-elliptic-elliptic type, and possesses two components representing the chemoattractant and chemorepellent. In this paper, we show that the system for the three or four-dimensional case admits the solution which blows up in finite time in the attractive dominant case by use of the moment method.