Boundedness in a two‐species chemotaxis system

This paper is concerned with the following coupled chemotaxis systemu t = Δu − χ 1 ∇ · ( u ∇ w ) + μ 1 u ( 1 − u − a 1 v ) ,x ∈ Ω , t > 0 ,v t = Δv − χ 2 ∇ · ( v ∇ w ) + μ 2 v ( 1 − a 2 u − v ) ,x ∈ Ω , t > 0 ,τ w t = Δw − λw + b 1 u + b 2 v ,x ∈ Ω , t > 0with homogeneous Neumann boundary conditions in a bounded domain Ω⊂ R n ( n ≥2) with smooth boundary, where λ , χ 1 , χ 2 , μ 1 , μ 2 , a 1 , a 2 , b 1 , and b 2 are supposed to be positive and τ = 0,1. In the case τ = 0, based on some energy estimates for both u and v , it is shown that for any parameters, the system possesses a unique globally bounded solution if n = 2. Moreover, when τ = 1, relying on a comparison principle, for a range of parameters, the existence of a unique global bounded classical solution of problem is established for any n ≥2 if Ω is convex. Copyright © 2015 John Wiley & Sons, Ltd.