Large time behavior in a chemotaxis model with logistic growth and indirect signal production

Abstract

This paper is concerned with the following chemotaxis-growth system ut=Δu−∇⋅u∇v+μ(u−uα),x∈Ω,t>0,vt=Δv−v+w,x∈Ω,t>0,wt=Δw−w+u,x∈Ω,t>0,in a smooth bounded domain Ω⊂Rn(n⩾2) with nonnegative initial data and null Neumann boundary condition, where μ>0,α>1. It is stated that if α>n4+12, the solution is globally bounded. Moreover, if μ>0 is sufficiently large, the solution (u,v,w) emanating from nonnegative initial data u0,v0,w0 with u0⁄≡0 is globally bounded and satisfies ‖u⋅,t−1‖L∞Ω+‖v⋅,t−1‖L∞Ω+‖w⋅,t−1‖L∞Ω→0as t→∞.