Boundedness in a chemotaxis system with consumed chemoattractant and produced chemorepellent

We study this zero-flux attraction-repulsion chemotaxis model, with linear and superlinear production g for the chemorepellent and sublinear rate f for the chemoattractant: {u(t) = Delta u - chi del . (u del v) + xi del . (u del w) in Omega x (0, T-max), v(t) = Delta v - f(u)v in Omega x (0, T-max), (lozenge) 0 = Delta w - delta w + g(u) in Omega x (0, T-max), In this problem, Omega is a bounded and smooth domain of R-n, for n >= 1, chi,xi, delta > 0, f(u) and g(u) reasonably regular functions generalizing the prototypes f(u) = Ku(alpha) and g(u) = gamma u(l), with K, gamma > 0 and proper alpha, l > 0. Once it is indicated that any sufficiently smooth u(x, 0) = u(0)(x) >= 0 and v(x, 0) = v(0)(x) >= 0 produce a unique classical and nonnegative solution (u, v, w) to (lozenge), which is defined in Omega x (0, T-max), we establish that for any such (u(0), v(0)), the life span T-max = infinity and u, v and w are uniformly bounded in Omega x (0,infinity), (i) for l = 1, n is an element of {1, 2}, alpha is an element of (0, 1/2 + 1/n)boolean AND(0, 1) and any xi > 0, (ii) for l = 1, n >= 3, alpha is an element of (0, 1/2+ 1/n) and xi larger than a quantity depending on chi parallel to v(0)parallel to L-infinity(Omega), (iii) for l > 1, alpha is an element of (0, 1/2 + 1/n) boolean AND (0, 1), any xi > 0, and in any dimensional settings. Finally, an illustrative analysis about the effect by logistic and repulsive actions on chemotactic phenomena is proposed by comparing the results herein derived for the linear production case with those in Lankeit and Wang (2017). (C) 2021 Elsevier Ltd. All rights reserved.