Boundedness in a multi-dimensional chemotaxis–haptotaxis model with nonlinear diffusion

We consider the chemotaxis–haptotaxis model {ut=∇⋅(D(u)∇u)−χ∇⋅(u∇v)−ξ∇⋅(u∇w)+μu(1−u−w),x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0,wt=−vw,x∈Ω,t>0 in a bounded smooth domain Ω⊂Rn(n≥2)Ω⊂Rn(n≥2), where χ,ξχ,ξ and μμ are positive parameters, and the diffusivity D(u)D(u) is assumed to generalize the prototype D(u)=δ(u+1)−αD(u)=δ(u+1)−α with α∈Rα∈R. Under zero-flux boundary conditions, it is shown that for sufficiently smooth initial data (u0,v0,w0)(u0,v0,w0) and α<2−nn+2, the corresponding initial–boundary problem possesses a unique global-in-time classical solution which is uniformly bounded, which improves the previous results.