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

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    * Corresponding author

The authors are supported in part by the National Natural Science Foundation of China (No. 11671079, No. 11701290, No. 11601127 and No. 11171063), the Natural Science Foundation of Jiangsu Province (No. BK20170896), the Postgraduate Research and Practice Innovation Program of Jiangsu Province(No.KYCX21_0077)

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  • 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.

    Mathematics Subject Classification: 35K65, 35Q92, 35B45, 92C17.


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