Global existence and boundedness of classical solutions for a chemotaxis model with consumption of chemoattractant and logistic source

This paper deals with the following chemotaxis system:u t = ∇ · ( δ ∇ u − χ u ∇ v ) + f ( u ) , 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, δ and χ are some positive constants and f is a smooth function that satisfies f ( 0 ) ⩾ 0andf ( s ) ⩽ a s − b s γ , s > 0with some constants a ⩾0, b  > 0, and γ  > 1. We prove that the classical solutions to the preceding system are global and bounded provided that ∥ v 0∥L ∞ ( Ω ) <1 χδ 2 ( n + 1 )π + 2 arctan( 1 − δ ) 22 ( n + 1 ) δ, if 0 < δ < 1 ,π χ 2 ( n + 1 ), if δ = 1 ,1 χδ 2 ( n + 1 )π − 2 arctan( δ − 1 ) 22 ( n + 1 ) δ, if δ > 1 .Copyright © 2016 John Wiley & Sons, Ltd.