Boundedness in a chemotaxis-haptotaxis model with gradient-dependent flux limitation

Abstract This paper deals with a chemotaxis–haptotaxis system with gradient-dependent flux-limitation u t = Δ u − χ ∇ ⋅ ( u f ( | ∇ v | 2 ) ∇ v ) − ξ ∇ ⋅ ( u ∇ w ) + μ u ( 1 − u − w ) , x ∈ Ω , t > 0 , v t = Δ v − v + u , x ∈ Ω , t > 0 , w t = − v w , x ∈ Ω , t > 0 , under a smooth bounded domain Ω ⊂ R n , n ∈ { 2 , 3 } , where χ , ξ and μ are positive parameters, f ∈ C 2 ( [ 0 , ∞ ) ) satisfies the condition f ( | ∇ v | 2 ) ≤ ( 1 + ( | ∇ v | 2 ) ) p − 2 2 , with 1 p n n − 1 . It is proved that for sufficiently smooth initial data ( u 0 , v 0 , w 0 ) , the corresponding initial–boundary problem possesses a unique classical solution, which is uniformly bounded in time.