LAUSR

Abstract In this paper, we study the chemotaxis system: { u t = ∇ ⋅ ( ξ ∇ u − χ u ∇ v ) , x ∈ Ω , t > 0 , v t = Δ v − u v , x ∈ Ω , t > 0 , under homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ R n , n ≥ 1 , with smooth boundary. Here, ξ and χ are some positive constants. We prove that the classical solutions to the above system are uniformly in-time-bounded provided that: ‖ v 0 ‖ L ∞ ( Ω ) { 1 χ ξ 2 ( n + 1 ) [ π + 2 arctan ⁡ ( ( 1 − ξ ) 2 2 ( n + 1 ) ξ ) ] , if 0 ξ 1 , π χ 2 ( n + 1 ) , if ξ = 1 , 1 χ ξ 2 ( n + 1 ) [ π − 2 arctan ⁡ ( ( ξ − 1 ) 2 2 ( n + 1 ) ξ ) ] , if ξ > 1 . In the case of ξ = 1 , the recent results show that the classical solutions are global and bounded provided that 0 ‖ v 0 ‖ L ∞ ( Ω ) ≤ 1 6 ( n + 1 ) χ . Because of 1 6 ( n + 1 ) χ π χ 2 ( n + 1 ) , or more precisely, lim n → ∞ ⁡ π χ 2 ( n + 1 ) 1 6 ( n + 1 ) χ = + ∞ , our results extend the recent results.