Qualitative behavior of solutions for a chemotaxis-haptotaxis model with flux limitation

In this paper, the following the coupled chemotaxis-haptotatxis system under zero-flux boundary conditions is studied:

$ \begin{align} \begin{cases} u_t = \Delta u-\chi\nabla\cdot\Bigg(\frac{u\nabla v}{(1+|\nabla v|^2)^\alpha}\Bigg)-\xi\nabla\cdot(u\nabla w)+\mu u(1-u-w), \, \, & x\in\Omega, \, \, t>0, \\ 0 = \Delta v-v+u, & x\in\Omega, \, \, t>0, \\ w_t = -vw, & x\in\Omega, \, \, t>0, \end{cases} \end{align} $

in a smooth and bounded domain $ \Omega\subset \mathbb{R}^n $ ($ n\ge2 $) under homogeneous Neumann boundary conditions, where $ \chi, \, \xi, \, and \mu $ are given positive parameters. For sufficiently smooth initial data ($ u_0, w_0 $), it is demonstrated that the problem possesses a unique global bounded classical solution, which is uniformly bounded in time.