A quasilinear fully parabolic chemotaxis system with indirect signal production and logistic source

In this paper we study the quasilinear fully parabolic chemotaxis system with indirect signal production and logistic source: u(t) = del . (D(u)del u - S(u)del v) + f (u), v(t) = Delta v - a(1)v + b(1)w, w(t) = Delta w - a(2)w+b(2)u, under homogeneous Neumann boundary conditions in a bounded and smooth domain Omega subset of R-n (n >= 1), where a(i), b(i) > 0 (i = 1, 2), D, S is an element of C-2 ([0, infinity)) and f : R -> R is a smooth function generalizing the logistic source f (s) = b - mu s(r) for all s >= 0 with b >= 0, mu > 0 and r >= 1. We obtain the global boundedness of solutions in four cases: (i) the self-diffusion dominates the cross-diffusion; (ii) the logistic source suppresses the cross-diffusion; (iii) the logistic dampening balances the cross-diffusion with mu > 0 suitably large; (iv) the self-diffusion and the logistic source both balance the cross-diffusion to some extent with mu > 0 arbitrary. As corollaries, we also consider the global boundedness of solutions for the quasilinear attraction-repulsion chemotaxis model with logistic source: (u) over tilde (t) = del . (D((u) over tilde)del(u) over tilde) - chi del . ((u) over tilde del z) + xi del . ((u) over tilde del(w) over tilde) + f((u) over tilde), z(t) = Delta z - rho z + eta(u) over tilde, (w) over tilde (t) = Delta(w) over tilde) - delta(w) over tilde + gamma(u) over tilde, where chi, eta, xi, gamma, rho, delta > 0. (C) 2019 Elsevier Inc. All rights reserved.