An attraction-repulsion chemotaxis system with nonlinear productions

This paper studies the semilinear attraction-repulsion chemotaxis system with nonlinear productions and logistic source: u(t) = Delta u-chi del.(u del v)+xi del.(u del w)+ f (u), 0 = Delta v+alpha u(k) - beta v, 0 = Delta w+gamma u(l)-delta w, in bounded domain Omega subset of R-n, n >= 1, subject to the non-flux boundary conditions, where the nonlinear productions for the attraction and repulsion chemicals are described via alpha u(k) and gamma u(l) respectively, and the logistic source f is an element of C-2[0, infinity) satisfying f (u) <= u(a - bu(s)) with s > 0, f (0) >= 0. It is proved that if one of the random diffusion, logistic source and repulsion mechanisms dominates the attraction with max{l, s, 2/n} > k, the solutions would be globally bounded. Furthermore, under the three balance situations, namely, k = s > l, k = l > s or k = s = l, the boundedness of solutions depends on the sizes of the coefficients involved. This extends the global boundedness criteria established by Zhang and Li (2016) [20] for the attraction-repulsion chemotaxis system with linear productions and logistic source. (C) 2019 Elsevier Inc. All rights reserved.