Global existence of solutions to the Cauchy problem for an attraction-repulsion chemotaxis system in R2 in the attractive dominant case

We consider the Cauchy problem for an attraction-repulsion chemotaxis system in R2 with the chemotactic coefficient of the attractant β1 and that of the repellent β2. It is known that in the repulsive dominant case β1<β2 or the balance case β1=β2, the nonnegative solutions to the Cauchy problem exist globally in time, whereas in the attractive dominant case β1>β2, there are blowing-up solutions in finite time under the assumption (β1−β2)∫R2u0dx>8π on the nonnegative initial data u0. In this paper, we show the global existence of nonnegative solutions to the Cauchy problem under the assumption (β1−β2)∫R2u0dx<8π in the attractive dominant case.