Boundedness of solutions to a parabolic attraction–repulsion chemotaxis system in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e22" altimg="si11.svg"><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>: The attractive dominant case

We discuss the Cauchy problem for a parabolic attraction–repulsion chemotaxis system: ∂tu=Δu−∇⋅(β1u∇v1)+∇⋅(β2u∇v2),t>0,x∈R2,∂tvj=Δvj−λjvj+u,t>0,x∈R2(j=1,2),u(0,x)=u0(x),vj(0,x)=vj0(x),x∈R2(j=1,2)with positive constants βj,λj>0 (j=1,2) satisfying β1>β2. In our companion paper, authors proved existence of global-in-time solutions any initial data with (β1−β2)∫R2u0dx<8π. this we prove that every solution stays bounded as t→∞ provided (β1−β2)∫R2u0dx<4π.