Boundedness in a nonlinear attraction-repulsion Keller-Segel system with production and consumption

This paper is focused on the zero-flux attraction-repulsion chemotaxis model {u(t) = del . ((u + 1)(m1-1)del u - chi(u(u+1)m2-1)del v in Omega x (0, T-max), +xi u(u+1)(m3-1)del w) (lozenge) v(t) - Delta v - f(u)v in Omega x (0, T-max), 0 = Delta w - delta w + g(u) in Omega x (0, T-max), defined in Omega, which is a bounded and smooth domain of R-n, for n >= 2, with chi, xi, delta > 0, m(1), m(2), m(3) is an element of R, and 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 appropriate alpha, l > 0. Moreover T-max is finite or infinite and (0, T-max) stands for the maximal temporal interval where solutions to the related initial problem exist. Our main interest is to identify constellations of the impacts m(1), m(2) and m(3)of the diffusion and drift terms, as well as of the growth lof the production gfor the chemorepellent (i.e., w) and the rate alpha of the consumption ffor the chemoattractant (i.e., v), which ensure boundedness of cell densities (i.e., u). Precisely, for any fixed alpha is an element of (0, 1/2 + 1/n) and l >= 1, we prove that whenever m(1) > min {2m(2) + 1 - (m(3) + l), max {2m(2), n - 2/n}}, any sufficiently smooth initial data u(x, 0) = u(0)(x) = 0 and v(x, 0) = v(0)(x) >= 0 produce a unique classical solution (u, v, w) to problem (lozenge) such that its life span T-max = 8 and, moreover, u, v and w are uniformly bounded in Omega x (0, infinity). (c) 2021 Elsevier Inc. All rights reserved.