Boundedness in a chemotaxis model with exponentially decaying diffusivity and consumption of chemoattractant

Abstract This paper is devoted to the following chemotaxis model u t = ∇ ⋅ ( D ( u ) ∇ u ) − ∇ ⋅ ( S ( u ) ∇ v ) , x ∈ Ω , t > 0 , v t = Δ v − u v , x ∈ Ω , t > 0 , under homogeneous Neumann boundary conditions in a c o n v e x smooth bounded domain Ω ⊂ R n ( n ≥ 2 ). It is proved that if D and S are sufficiently smooth nonnegative functions on [ 0 , ∞ ) satisfying K 0 e − β − s ≤ D ( s ) ≤ K 1 e − β + s for all s ≥ 0 with some K 0 > 0 , K 1 > 0 , β − ≥ β + and β + > 0 , then under the condition that S ( s ) D ( s ) ≤ K 2 s α for all s ≥ 0 with some K 2 > 0 and α ≥ 0 , for the initial data ( u 0 , v 0 ) are sufficiently regular satisfying u 0 ≥ 0 and v 0 > 0 , the classical solutions to the system are uniformly-in-time bounded.