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

Abstract

The quasilinear chemotaxis–haptotaxis system{ut=∇⋅(D(u)∇u)−χ∇⋅(u∇v)−ξ∇⋅(u∇w)ut=+μu(1−u−w),x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0,wt=−vw,x∈Ω,t>0, is considered under homogeneous Neumann boundary conditions in a bounded and smooth domain Ω⊂R3. Here χ>0, ξ>0 and μ>0, D(u)≥cDum−1 for all u>0 with some cD>0 and D(u)>0 for all u≥0. It is shown that if the ratio χμ is sufficiently small, then the system possesses a unique global classical solution that is uniformly bounded. Our result is independent of m.