Global boundedness in quasilinear attraction-repulsion chemotaxis system with logistic source

This paper studies the quasilinear attraction repulsion chemotaxis system with logistic source ut = del center dot (D(u)del u) -x del center dot (phi(u)del v) +xi del center dot (psi(u) del w) f (u), TVt = Delta u + alpha u - beta v, T epsilon {0, 1}, 0 = Delta w+gamma u-delta w, in bounded domain Omega subset of R-N, N >= 1, subject to the homogeneous Neumann boundary conditions, D, Phi, Psi epsilon C-2[0, +infinity) nonnegative, with D(s) >= (s + 1)(P) for s >= 0, Phi(s) <= xs(q), xi s(r) <= Psi (s) <= zeta s(r) for s > 1, and f smooth satisfying f (s) <= mu s(1-s(k)) for s > 0, f (0) >= 0. It is proved that if the attraction is dominated by one of the other three mechanisms with max{r, k, p + 2/N} > q, then the solutions are globally bounded. Under more interesting balance situations, the behavior of solutions depends on the coefficients involved, i.e., the upper bound coefficient xi for the attraction, the lower bound coefficient for the repulsion, the logistic source coefficient mu as well as the constants alpha and gamma describing the capacity of the cells u to produce chemoattractant and chemorepellent respectively. Three balance situations (attraction repulsion balance, attraction logistic source balance, and attraction repulsion logistic source balance) are considered to establish the boundedness of solutions for the parabolic elliptic elliptic case (with T = 0) and the parabolic parabolic elliptic case (with T = 1) respectively. (C) 2015 Elsevier Ltd. All rights reserved.