Global existence of solutions to the Cauchy problem for an attraction–repulsion chemotaxis system in [formula omitted] in the attractive dominant case

Abstract

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.