Boundedness and large time behavior for a chemotaxis system with signal-dependent motility and indirect signal consumption

Abstract

In this paper, we study the following chemotaxis system with signal-dependent motility, indirect signal consumption and logistic source ut=Δ(uγ(v))+ρu−μul,x∈Ω,t>0,vt=Δv−vw,x∈Ω,t>0,wt=−δw+u,x∈Ω,t>0under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂Rn, where the motility function γ(v)∈C3((0,+∞)),γ(v)>0,γ′(v)<0 on (0,+∞), limv→∞γ(v)=0, ρ>0,μ>0,l>1 and δ>0. The purpose of this paper is to prove that if l>max{1,n2}, then the system possesses a global solution. In addition, if l satisfies l≥2,if n≤3,>n2,if n≥4,then the solution (u,v,w) satisfies ‖u(⋅,t)−(ρμ)1l−1‖L∞(Ω)+‖v(⋅,t)‖L∞(Ω)+‖w(⋅,t)−1δ(ρμ)1l−1‖L∞(Ω)→0ast→∞.