Stabilization in three-dimensional chemotaxis-growth model with indirect attractant production

Abstract

This paper deals with the chemotaxis-growth system: ut=Δu−∇⋅(u∇v)+μu(1−u), vt=Δv−v+w, τwt+δw=u in a smooth bounded domain Ω⊂R3 with zero-flux boundary conditions, where μ, δ, and τ are given positive parameters. It is shown that the solution (u,v,w) exponentially stabilizes to the constant stationary solution (1,1δ,1δ) in the norm of L∞(Ω) as t→∞ provided that μ>0 and any given nonnegative and suitably smooth initial data (u0,v0,w0) fulfills u0≢0, which extends the condition μ>18δ2 in [8].