LAUSR

Abstract In this study, we consider the quasilinear parabolic–parabolic chemotaxis model: { u t = ∇ ⋅ ( D ( u ) ∇ u ) − ∇ ⋅ ( S ( u ) ∇ v ) , x ∈ Ω , t > 0 , v t = Δ v − u v , x ∈ Ω , t > 0 , subject to homogeneous Neumann boundary conditions, where Ω is a convex bounded domain of R n ( n ≥ 2 ) with smooth boundary. The diffusivity sensitivity D ( s ) and chemotactic sensitivity S ( s ) satisfy K 1 e − β − s ≤ D ( s ) ≤ K 2 e − β + s and S ( s ) / D ( s ) ≤ K 3 e α s for s ≥ 0 with constants K i > 0 ( i = 1 , 2 , 3 ) , β − ≥ β + > 0 , and α ∈ [ 0 , β + / ( n + 1 ) ) . If the initial data are u 0 ∈ C 0 ( Ω ¯ ) and v 0 ∈ W 1 , ∞ ( Ω ) , then the classical solutions to this model are globally bounded.