Global boundedness and asymptotic stabilization in a chemotaxis system with density-suppressed motility and nonlinear signal production

Abstract

In this paper, we study the following chemotaxis model with density-suppressed motility and nonlinear production{ut=Δ(φ(v)u)+ru−μuα,x∈Ω,t>0,vt=Δv−v+wβ,x∈Ω,t>0,wt=Δw−w+uγ,x∈Ω,t>0 under homogeneous Neumann boundary conditions in a bounded domain Ω⊂Rn(n≥2) with smooth boundary, where r∈R,μ,β,γ>0 and α>1. The positive motility function satisfies φ(s)∈C3([0,∞)) and φ′(s)≤0 for all s≥0. It is showed that the system admits a globally bounded and classical solution under some conditions on α,β and γ. Then, under stricter constraints on φ, we obtained that the parameter β has a wider range than before, which is enough to ensure the global boundedness of the solution. Furthermore, if μ is sufficiently large, we proved that the solution converges to ((r+μ)1α−1,(r+μ)βγα−1,(r+μ)γα−1) in L∞(Ω) as t→∞ for all r∈R,μ,β,γ>0 and α>1, where r+:=max⁡{r,0}. Finally, we showed that the convergence is exponential in cases r>0 and r<0.