Boundedness in a chemotaxis–haptotaxis model with nonlinear diffusion

This article deals with an initial-boundary value problem for the coupled chemotaxis-haptotaxis system with nonlinear diffusion {align*} u_t=&∇·(D(u)∇ u)-χ∇·(u∇ v)-ξ∇·(u∇ w)+μ u(1-u-w),\\ v_t=&Δ v-v+u,\\ w_t=&-vw{align*} under homogeneous Neumann boundary conditions in a bounded smooth domain Ωⁿ, $n=2, 3, 4$, where χ, ξ and μ are given nonnegative parameters. The diffusivity $D(u)$ is assumed to satisfy D(u)≥δ uᵐ⁻¹ for all $u>0$ with some δ>0. It is proved that for sufficiently regular initial data global bounded solutions exist whenever m>2-2/n. For the case of non-degenerate diffusion (i.e. $D(0)>0$) the solutions are classical; for the case of possibly degenerate diffusion (D(0)≥ 0), the existence of bounded weak solutions is shown.