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, $\left\{ \begin{array}{l}{u_t} = \Delta {u^m} - \chi \nabla \cdot \left( {u \cdot \nabla v} \right) - \xi \nabla \cdot \left( {u \cdot \nabla w} \right) + \mu u\left( {1 - u - w} \right),{\rm{in}}\;\Omega \times {{\mathbb{R}}^ + },\\{v_t} - \nabla v + v = u,\;{\rm{in}}\;\Omega \times {{\mathbb{R}}^ + },\\{w_t} = - vw,\;\;{\rm{in}}\;\Omega \times {{\mathbb{R}}^ + },\end{array} \right.$ where $Ω\subset\mathbb R^N$ ( $N≥ 3$ ) is a bounded domain. Under zero-flux boundary conditions, we showed that for any $m>0$ , the problem admits a global bounded weak solution for any large initial datum if $\frac{χ}{μ}$ is appropriately small. The slow diffusion case ( $m>1$ ) 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 $m>\frac{2N}{N+2}$ , but the cases $m≤ \frac{2N}{N+2}$ remain open.