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

Abstract This paper deals with the chemotaxis-growth system: u t = Δ u − ∇ ⋅ ( u ∇ v ) + μ u ( 1 − u ) , v t = Δ v − v + w , τ w t + δ w = u in a smooth bounded domain Ω ⊂ R 3 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 , 1 δ , 1 δ ) in the norm of L ∞ ( Ω ) as t → ∞ provided that μ > 0 and any given nonnegative and suitably smooth initial data ( u 0 , v 0 , w 0 ) fulfills u 0 ≢ 0 , which extends the condition μ > 1 8 δ 2 in [8] .