Boundedness of classical solutions for a chemotaxis model with consumption of chemoattractant

Abstract

In this paper, we study the chemotaxis system:{ut=∇⋅(ξ∇u−χu∇v),x∈Ω,t>0,vt=Δv−uv,x∈Ω,t>0, under homogeneous Neumann boundary conditions in a bounded domain Ω⊂Rn,n≥1, with smooth boundary. Here, ξ and χ are some positive constants.We prove that the classical solutions to the above system are uniformly in-time-bounded provided that:‖v0‖L∞(Ω)<{1χξ2(n+1)[π+2arctan⁡((1−ξ)22(n+1)ξ)],if0<ξ<1,πχ2(n+1),ifξ=1,1χξ2(n+1)[π−2arctan⁡((ξ−1)22(n+1)ξ)],ifξ>1. In the case of ξ=1, the recent results show that the classical solutions are global and bounded provided that 0<‖v0‖L∞(Ω)≤16(n+1)χ. Because of 16(n+1)χ<πχ2(n+1), or more precisely, limn→∞⁡πχ2(n+1)16(n+1)χ=+∞, our results extend the recent results.