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

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