Global attractors in a two-species chemotaxis system with two chemicals and variable logistic sources

This paper deals with the higher dimension quasilinear parabolic-parabolic chemotaxis model involving a source term of logistic type (formula presented), subject to the homogeneous Neumann boundary conditions in a Ω ⊂ RN (N ≥ 1) with smooth boundary. It is shown that for the attractionrepulsion case with χ2 ≤ 0, the global boundedness of solutions can be ensured by µ1, µ2 > 0 without any other assumptions, due to the contribution of the logistic sources included in addition to the repulsion mechanism. While for the attraction-attraction case with χ2 > 0, the global boundedness of solutions has to require logistic coefficients µ1, µ2 > 0 such that µ2 properly large.