Global existence and convergence to steady states for an attraction–repulsion chemotaxis system

Abstract

This paper deals with a Neumann initial–boundary value problem in the two-dimensional space for a coupled chemotaxis model {ut=Δu−∇⋅(χu∇v)+∇⋅(ξu∇w),x∈Ω,t>0,vt=Δv+αu−βv,x∈Ω,t>0,wt=Δw+γu−δw,x∈Ω,t>0, which describes the competition between attractive and repulsive signals produced by the same species. The parameters χ,ξ,α,β,γ and δ are positive.Relying on a new entropy-type inequality, we obtain the global existence and boundedness of solutions to this model if the initial data u0 satisfies ‖u0‖L1(Ω)<1kχα, where k is a constant depending only on Ω. The convergence to steady states is also given through this paper.