Boundedness and global solvability to a chemotaxis-haptotaxis model with slow and fast diffusion

In this paper, we deal with the following coupled chemotaxis-haptotaxis system modeling cancer invasionwith nonlinear diffusion, \begin{document}$\left\{ \begin{array}{l}{u_t} = \Delta {u^m} - \chi \nabla \cdot \left( {u \cdot \nabla v} \right) - \xi \nabla \cdot \left( {u \cdot \nabla w} \right) + \mu u\left( {1 - u - w} \right),{\rm{in}}\;\Omega \times {{\mathbb{R}}^ + },\\{v_t} - \nabla v + v = u,\;{\rm{in}}\;\Omega \times {{\mathbb{R}}^ + },\\{w_t} = - vw,\;\;{\rm{in}}\;\Omega \times {{\mathbb{R}}^ + },\end{array} \right.$ \end{document} where \begin{document} $Ω\subset\mathbb R^N$ \end{document} ( \begin{document} $N≥ 3$ \end{document} ) is a bounded domain. Under zero-flux boundary conditions, we showed that for any \begin{document} $m>0$ \end{document} , the problem admits a global bounded weak solution for any large initial datum if \begin{document} $\frac{χ}{μ}$ \end{document} is appropriately small. The slow diffusion case ( \begin{document} $m>1$ \end{document} ) of this problem have been studied by many authors [ 14 , 7 , 19 , 23 ], in which, the boundedness and the global in time solution are established for \begin{document} $m>\frac{2N}{N+2}$ \end{document} , but the cases \begin{document} $m≤ \frac{2N}{N+2}$ \end{document} remain open.