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

This paper is concerned with the following chemotaxis-growth system u t = Δ u − ∇ ⋅ u ∇ v + μ ( u − u α ) , x ∈ Ω , t > 0 , v t = Δ v − v + w , x ∈ Ω , t > 0 , w t = Δ w − w + u , x ∈ Ω , t > 0 , in a smooth bounded domain Ω ⊂ R n ( n ⩾ 2 ) with nonnegative initial data and null Neumann boundary condition, where μ > 0 , α > 1 . It is stated that if α > n 4 + 1 2 , the solution is globally bounded. Moreover, if μ > 0 is sufficiently large, the solution ( u , v , w ) emanating from nonnegative initial data u 0 , v 0 , w 0 with u 0 ⁄ ≡ 0 is globally bounded and satisfies ‖ u ⋅ , t − 1 ‖ L ∞ Ω + ‖ v ⋅ , t − 1 ‖ L ∞ Ω + ‖ w ⋅ , t − 1 ‖ L ∞ Ω → 0 as t → ∞ .