A quasilinear attraction repulsion chemotaxis system of parabolic-elliptic type with logistic source

This study considers the following quasilinear attraction-repulsion chemotaxis system of parabolic elliptic type with logistic source {ut = del. (D(u)del u) - del.(chi u del v) + del. (xi u del w) +f(u), x is an element of Omega, t > 0, 0 = Delta v + alpha u - beta v, x is an element of Omega, t > 0, 0 = Delta w + gamma u - delta w, x is an element of Omega, t > 0, under homogeneous Neumann boundary conditions in a bounded domain Omega subset of R-n (n >= 2) with smooth boundary, where D(u) >= c(D)(u + sigma)(m-1) with m >= 1, sigma >= 0, and c(D) > 0, and f(u) <= a- bu(n) with a >= 0, b > 0, and eta > 1. In the case of non-degenerate diffusion (i.e., sigma > 0), we show that the system admits a unique global bounded classical solution provided that the repulsion prevails over the attraction in the sense that xi gamma- chi alpha > 0, or the logistic dampening is sufficiently strong, or the diffusion is sufficiently strong, while in the case of degenerate diffusion (i.e., sigma = 0), we show that the system admits a global bounded weak solution at least under the same assumptions. Finally, we obtain the large-time behavior of solutions for a specific logistic source. (C) 2016 Elsevier Inc. All rights reserved.