We consider the Cauchy problem for an attraction-repulsion chemotaxis system in two-dimensional space. The system consists of three partial differential equations; a drift-diffusion equation incorporating terms for both chemoattraction and chemorepulsion, and two elliptic equations. We denote by β1 the coefficient of the attractant and by β2 that of the repellent. The boundedness of nonnegative solutions to the Cauchy problem was shown in the repulsive dominant case β1 < β2 and the balance case β1 = β2. In this paper, we study the boundedness problem to the Cauchy problem in the attractive dominant case β1 > β2.
