Boundedness in a three-dimensional chemotaxis-haptotaxis model

Abstract: This paper studies the chemotaxis-haptotaxis system \begin{equation}\nonumber \left{ \begin{array}{llc} u_t=\Delta u-\chi\nabla\cdot(u\nabla v)-\xi\nabla\cdot(u\nabla w)+\mu u(1-u-w), &(x,t)\in \Omega\times (0,T),\ v_t=\Delta v-v+u, &(x,t)\in\Omega\times (0,T),\ w_t=-vw,&(x,t)\in \Omega\times (0,T) \end{array} \right.\quad\quad(\star) \end{equation} under Neumann boundary conditions. Here $\Omega\subset\mathbb{R}^3$ is a bounded domain with smooth boundary and the parameters $\xi,\chi,\mu>0$. We prove that for nonnegative and suitably smooth initial data $(u_0,v_0,w_0)$, if $\chi/\mu$ is sufficiently small, ($\star$) possesses a global classical solution which is bounded in $\Omega\times(0,\infty)$. We underline that the result fully parallels the corresponding parabolic-elliptic-ODE system.