Boundedness in the higher-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), 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<2n−1, the corresponding initial–boundary problem possesses a unique global-in-time classical solution which is uniformly bounded. This paper develops some Lp-estimate techniques and thereby extends boundedness results in n≤3 to arbitrary space dimensions.