Global generalized solutions of a haptotaxis model describing cancer cells invasion and metastatic spread

In this paper, we consider the following haptotaxis model describing cancer cells invasion and metastatic spread\begin{document}$\begin{array}{*{20}{c}}{\left\{ {\begin{array}{*{20}{l}}{{u_t} = \Delta u - \chi \nabla \cdot (u\nabla w),}&{x \in \Omega ,\;t > 0,}\\{{v_t} = {d_v}\Delta v - \xi \nabla \cdot (v\nabla w),}&{x \in \Omega ,\;t > 0,}\\{{m_t} = {d_m}\Delta m + u - m,}&{x \in \Omega ,\;t > 0,}\\{{w_t} = - \left( {{\gamma _1}u + m} \right)w,}&{x \in \Omega ,\;t > 0,}\end{array}} \right.}&{(0.1)}\end{array}$\end{document}where \begin{document}$ \Omega\subset \mathbb{R}^3 $\end{document} is a bounded domain with smooth boundary and the parameters \begin{document}$ \chi, \xi, d_{v}, d_{m},\gamma_{1}>0 $\end{document} . Under homogeneous boundary conditions of Neumann type for \begin{document}$ u $\end{document} , \begin{document}$ v $\end{document} , \begin{document}$ m $\end{document} and \begin{document}$ w $\end{document} , it is proved that, for suitable smooth initial data \begin{document}$ (u_0, v_0, m_0, w_0) $\end{document} , the corresponding Neumann initial-boundary value problem possesses a global generalized solution.