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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 \begin{align*} u_t=&\nabla\cdot(D(u)\nabla u)-χ\nabla\cdot(u\nabla v)-ξ\nabla\cdot(u\nabla w)+μu(1-u-w),\\ v_t=&Δv-v+u,\\ w_t=&-vw\end{align*} under homogeneous Neumann boundary conditions in a bounded smooth domain $Ω\subset\mathbb{R}^n$, $n=2, 3, 4$, where $χ, ξ$ and $μ$ are given nonnegative parameters. The diffusivity $D(u)$ is assumed to satisfy $D(u)\geqδu^{m-1}$ for all $u>0$ with some $δ>0$. It is proved that for sufficiently regular initial data global bounded solutions exist whenever $m>2-\frac{2}{n}$. For the case of non-degenerate diffusion (i.e. $D(0)>0$) the solutions are classical; for the case of possibly degenerate diffusion ($D(0)\geq 0$), the existence of bounded weak solutions is shown.