Boundedness in a chemotaxis model with oxygen consumption by bacteria

This paper deals with the Keller–Segel model { u t = Δ 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 smooth bounded domain Ω ⊂ R n , n ⩾ 2 , with nonnegative initial data ( u ( ⋅ , 0 ) , v ( ⋅ , 0 ) ) ∈ ( W 1 , r ( Ω ) ) 2 (for some r > n ), ‖ u ( ⋅ , 0 ) ‖ L 1 ( Ω ) > 0 and ‖ v ( ⋅ , 0 ) ‖ L ∞ ( Ω ) > 0 . This model describes bacteria movement toward the concentration gradient of the oxygen that is consumed by the bacteria. It is proved that if 0 < χ ⩽ 1 6 ( n + 1 ) ‖ v ( ⋅ , 0 ) ‖ L ∞ ( Ω ) then the corresponding initial–boundary value problem possesses a unique global solution that is uniformly bounded.