Influence of nonlinear production on the global solvability of an attraction-repulsion chemotaxis system

This paper is dedicated to the attraction-repulsion chemotaxis-system {u(t) = Delta u - chi del.(u del v) + xi del.(u del w) in Omega X (0, T-max), 0 = Delta v + f(u) - beta v in Omega X (0, T-max), (lozenge) 0 = Delta w + g(u) - delta w in Omega X (0, T-max), defined in Omega, a smooth and bounded domain of R-n, with n >= 2. Moreover, beta, delta, chi, xi > 0 and f g are suitably regular functions generalizing, for u >= 0 and alpha, gamma > 0 the prototypes f (u) = alpha u(s), s > 0, and g(u) = gamma u(r), r >= 1. We focus our analysis on the value T-max is an element of (0, infinity], establishing the temporal interval of existence of solutions (u, v, w) to problem (lozenge). When zero-flux boundary conditions are fixed, we prove the following results, all excluding chemotactic collapse scenarios under certain correlations between the attraction and repulsive effects describing the model. To be precise, for every alpha, beta, gamma, delta, chi > 0, and r > s >= 1 (resp. s > r >= 1), there exists xi* > 0 (resp. xi* > 0) such that if xi > (resp. xi >= xi*) any sufficiently regular initial datum u(0) (x) >= 0 (resp. u(0) (x) >= 0 enjoying some smallness assumptions) produces a unique classical solution (u, v, w) to problem (lozenge) which is global, i.e. T-max = infinity, and such that u, v and w are uniformly bounded. Conversely, the same conclusion holds true for every alpha, beta, gamma, delta, chi, xi > 0, 0 < s < 1, r = 1 and any sufficiently regular u(0)(x) >= 0. Further, in a remark of the manuscript, we also address an open question posed in [21].