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{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} $

where $ \Omega\subset \mathbb{R}^3 $ is a bounded domain with smooth boundary and the parameters $ \chi, \xi, d_{v}, d_{m},\gamma_{1}>0 $. Under homogeneous boundary conditions of Neumann type for $ u $, $ v $, $ m $ and $ w $, it is proved that, for suitable smooth initial data $ (u_0, v_0, m_0, w_0) $, the corresponding Neumann initial-boundary value problem possesses a global generalized solution.