Boundedness for the chemotaxis system in a flux limitation with indirect signal production

A fully parabolic chemotaxis system with indirect signal production{ut=Δu−∇⋅(uf(|∇v|2)∇v),x∈Ω,t>0,vt=Δv−v+w,x∈Ω,t>0,wt=Δw−w+u,x∈Ω,t>0 is considered in a smooth bounded domain Ω⊂Rn (n≥1) under homogeneous Neumann boundary conditions, where the flux limitation f∈C2([0,∞)) is given suitably regular satisfyingf(ξ)≤Kf(1+ξ)−αforallξ≥0 with some Kf>0 and α∈R. The range of α depending on the dimension n ensures the existence of a global bounded classical solution for the associated initial-boundary value problem, in which sense any collapse is excluded.