Boundedness in a three-dimensional chemotaxis–haptotaxis model

Abstract

This paper studies the chemotaxis–haptotaxis system

$$\left\{\begin{array}{lll} u_t = \Delta u - \chi\nabla \cdot (u\nabla v) - \xi\nabla \cdot (u\nabla w) + \mu u(1 - u - w), &\quad(x, t)\in \Omega \times (0, T),\\ v_t = \Delta v - v + u, &\quad(x, t) \in \Omega \times (0, T),\\ w_t= - vw, &\quad(x, t)\in \Omega \times (0,T)\end{array} \right.\quad\quad(\star)$$

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.