A chemotaxis-haptotaxis system with haptoattractant remodeling: Boundedness enforced by mild saturation of signal production

We consider the chemotaxis-haptotaxis system

$ \begin{eqnarray*} \left\{ \begin{array}{l} u_t = \Delta u - \chi \nabla \cdot (u\nabla v) - \xi \nabla \cdot (u\nabla w) + \mu u(1-u-w), \\ v_t = \Delta v-v+f(u), \\ w_t = -vw+\eta w(1-u-w), \end{array} \right. \end{eqnarray*} $

in a bounded convex domain $ \Omega\subset \mathbb{R} ^n $ with smooth boundary, where $ \chi, \xi, \mu $ and $ \eta $ are positive constants, and where $ f \in C^1([0,\infty)) $ is a given function fulfilling $ f(0) \ge 0 $ and

$ \begin{eqnarray*} f(s) \le K_f (s+1)^\alpha \qquad \mbox{for all } s\ge 0 \end{eqnarray*} $

with some $ K_f >0 $ and $ \alpha>0 $.

It is asserted that whenever

$ \begin{eqnarray*} \alpha < \left\{ \begin{array}{ll} \frac{3}{2} \qquad & \mbox{if } n = 1, \\ \frac{n+6}{2(n+2)} \qquad & \mbox{if } n\ge 2, \end{array} \right. \end{eqnarray*} $

the Neumann boundary problem with suitably regular initial data possesses a unique global and bounded classical solution.