This paper studies a model for oncolytic virotherapy given by a triply haptotactic cross-diffusion system 0.1 \(\begin{aligned} \left\{ \begin{aligned}&u_t=D_u \Delta u-\xi _u\nabla \cdot (u\nabla v)+\mu _uu(1-u)-\rho _u uz,&\quad&x\in \Omega ,t>0,\\&v_t=-(\alpha _u u+\alpha _w w)v+\mu _v v(1-v),&\quad&x\in \Omega ,t>0,\\&w_t=D_w \Delta w-\xi _w\nabla \cdot (w\nabla v)-\delta _w w+\rho _w uz,&\quad&x\in \Omega ,t>0,\\&z_t=D_z \Delta z-\xi _z\nabla \cdot (z\nabla v)-\delta _z z-\rho _z uz+\beta w,&\quad&x\in \Omega ,t>0,\\&(D_u \nabla u-\xi _u u\nabla v)\cdot \nu =(D_w \nabla w-\xi _w w\nabla v)\cdot \nu =(D_z\nabla z-\xi _z z\nabla v)\cdot \nu =0,&\quad&x\in \partial \Omega ,t>0,\\&u(x,0)=u_0(x),\,\,v(x,0)=v_0(x),\,\,w(x,0)=w_0(x),\,\,z(x,0)=z_0(x),&\quad&x\in \Omega , \end{aligned} \right. \end{aligned}\) in a bounded domain \(\Omega \subset {\mathbb {R}}^2\) with smooth boundary. We prove that there exists a positive constant K such that if \(\xi _w\alpha _w\le K\) , then the unique solution is globally bounded. This finding represents an improvement over previous work in Li and Wang (J Differ Equ 270:94-113, 2021), Tao and Zhou (J Differ Equ 308:57-76, 2022) and Zheng and Xie (J Differ Equ 340:111-150, 2022), which imposed more restrictive conditions on the system. By contrast, our result relaxes these assumptions and establishes global boundedness under simpler and more general conditions.
