A nonlinear attraction-repulsion Keller–Segel model with double sublinear absorptions: criteria toward boundedness

This paper deals with the zero-flux attraction-repulsion chemo-taxis model {u(t) = del center dot ((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) + h(u) (lozenge) v(t) = Delta v - f (u)v in Omega x (0, T-max), w(t) = Delta w - g(u)w in Omega x (0, T-max), in the unknown (u, v, w)= (u(x, t), v(x, t), w(x, t)). Here, x is an element of Omega, a bounded and smooth domain of R-n(n >= 1), t, chi, xi > 0, m(1), m(2), m(3) is an element of R, and f (u), g(u) and h(u) sufficiently regular functions generalizing the prototypes f(u) = K(1)u(alpha), g(u) = K2u(gamma) and h(u) = ku - mu u(beta), with K-1, K-2, mu > 0, k is an element of R, beta > 1 and suitable alpha, gamma > 0. Besides, further regular initial data u(x, 0) = u(0)(x), v(x, 0) = v(0)(x), w(x, 0) = w(0)(x) >= 0 are given, whereas T-max is an element of (0, infinity] stands for the maximal instant of time up to which solutions to the system exist. We will derive relations between the parameters involved in (>) capable to warrant that u, v, w are global and uniformly bounded in time. The article generalizes and extends to the case of nonlinear effects and logistic perturbations some results recently developed in [3] where, for the linear counterpart and in the absence of logistics, criteria towards boundedness are established.