This paper discusses an initial-boundary value problem for a doubly haptotactic cross-diffusion system arising from the oncolytic virotherapy \(\begin{aligned} {\left\{ \begin{array}{ll} u_t=\Delta u^m-\nabla \cdot (u\nabla v) +\mu u(1-u)-uz,\;\;& \;x\in \Omega ,~t>0, \\ v_t=-(u+w)v,\;\;& \;x\in \Omega ,~t>0, \\ w_t=\Delta w-\nabla \cdot (w\nabla v)-w+uz,\;\;& \;x\in \Omega ,~t>0, \\ z_t=D\Delta z-z-uz+\beta w,\;\;& \;x\in \Omega ,~t>0, \\ \frac{\partial u^m}{\partial \nu }-u\frac{\partial v}{\partial \nu }=\frac{\partial w}{\partial \nu }-w\frac{\partial v}{\partial \nu }=\frac{\partial z}{\partial \nu }=0,\;\;& \;x\in \partial \Omega ,~t>0, \\ u(x,0)=u_{0},~v(x,0)=v_{0},~w(x,0)=w_{0}, \\ z(x,0)=z_{0},\;\;& \;x\in \Omega , \end{array}\right. } \end{aligned}\) in a smooth bounded domain \( \Omega \subset {\mathbb {R}}^{N}(N=1,2) \) with \( m>1, \beta>0, \mu >0 \) , and \( D>0\) . We prove that for any large initial datum, the problem admits a global ‘very’ weak solution for any \(m>1\) .
