Boundedness in a two‐dimensional attraction–repulsion system with nonlinear diffusion

This paper is devoted to the attraction–repulsion chemotaxis system with nonlinear diffusion:u t = ∇ · ( D ( u ) ∇ u ) − χ ∇ · ( u ∇ v ) + ζ ∇ · ( u ∇ w ) + uf ( u ) , x ∈ Ω , t > 0 ,v t = Δv − α 1 v + β 1 u , x ∈ Ω , t > 0 ,w t = Δw − α 2 w + β 2 u , x ∈ Ω , t > 0 ,where χ > 0, ζ > 0, α i >0, β i >0 ( i = 1,2) and f ( s )≤ κ − μ s τ . In two‐space dimension, we prove the global existence and uniform boundedness of the classical solution to this model for any μ > 0. Copyright © 2015 John Wiley & Sons, Ltd.