Boundedness in the higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion

We consider the chemotaxis-haptotaxis model, 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-1, the corresponding initial-boundary problem possesses a unique global-in-time classical solution which is uniformly bounded. This paper develops some L^p-estimate techniques and thereby extends boundedness results in n≤3 to arbitrary space dimensions.