Boundedness of solutions to a parabolic attraction-repulsion chemotaxis system in R2: The attractive dominant case

We discuss the Cauchy problem for a parabolic attraction-repulsion chemotaxis system: {partial derivative(t)u = Delta u - del.(beta(1)u del v(1)) + del.(beta(2)u del v(2)), t > 0, x is an element of R-2, partial derivative(t)v(j) = Delta v(j) - lambda(j)v(j) + u, t > 0, x is an element of R-2 (j = 1,2), u(0, x) = u(0)(x), v(j)(0, x) = v(j0)(x), x is an element of R-2 (j = 1, 2) with positive constants beta(j), lambda(j) > 0 (j = 1, 2) satisfying beta(1) > beta(2). In our companion paper, the authors proved the existence of global-in-time solutions for any initial data with (beta(1) - beta(2)) integral(R2) u(0) dx < 8 pi. In this paper, we prove that every solution stays bounded as t -> infinity provided that (beta(1) - beta(2)) integral(R2) u(0) dx < 4 pi. (C) 2021 Elsevier Ltd. All rights reserved.