Boundedness of the solution to a higher-dimensional triply haptotactic cross-diffusion system modeling oncolytic virotherapy

This paper investigates a model of oncolytic virotherapy formulated as a triply haptotactic cross-diffusion system 0.1 \(\begin{aligned} \left\{ \begin{aligned}&u_t=D_u \Delta u-\xi _u\nabla \cdot (u\nabla v)+\mu _uu(1-u)-\rho _u uz,&\quad&x\in \Omega ,t>0,\\&v_t=-(\alpha _u u+\alpha _w h(w))v+\mu _v v(1-v),&\quad&x\in \Omega ,t>0,\\&w_t=D_w \Delta w-\xi _w\nabla \cdot (w\nabla v)-\delta _w w+\rho _w uz,&\quad&x\in \Omega ,t>0,\\&z_t=D_z \Delta z-\xi _z\nabla \cdot (z\nabla v)-\delta _z z-\rho _z uz+\beta w,&\quad&x\in \Omega ,t>0,\\&\qquad \quad (D_u \nabla u-\xi _u u\nabla v)\cdot \nu =(D_w \nabla w-\xi _w w\nabla v)\cdot \nu =(D_z\nabla z-\xi _z z\nabla v)\cdot \nu =0,&\quad&x\in \partial \Omega ,t>0,\\&u(x,0)=u_0(x),\,\,v(x,0)=v_0(x),\,\,w(x,0)=w_0(x),\,\,z(x,0)=z_0(x),&\quad&x\in \Omega , \end{aligned} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cccc}& u_t=D_u\mathrm{\Delta }u-{\xi }_u\mathrm{\nabla }·(u\mathrm{\nabla }v)+{\mu }_{u}u(1-u)-{\rho }_{u}uz,\hfill & \hfill & x\in \mathrm{\Omega },t>0,\hfill \\ \hfill & v_t=-({\alpha }_{u}u+{\alpha }_{w}h(w))v+{\mu }_{v}v(1-v),\hfill & \hfill & x\in \mathrm{\Omega },t>0,\hfill \\ \hfill & w_t=D_w\mathrm{\Delta }w-{\xi }_w\mathrm{\nabla }·(w\mathrm{\nabla }v)-{\delta }_{w}w+{\rho }_{w}uz,\hfill & \hfill & x\in \mathrm{\Omega },t>0,\hfill \\ \hfill & z_t=D_z\mathrm{\Delta }z-{\xi }_z\mathrm{\nabla }·(z\mathrm{\nabla }v)-{\delta }_{z}z-{\rho }_{z}uz+\beta w,\hfill & \hfill & x\in \mathrm{\Omega },t>0,\hfill \\ \hfill & (D_u\mathrm{\nabla }u-{\xi }_{u}u\mathrm{\nabla }v)·\nu =(D_w\mathrm{\nabla }w-{\xi }_{w}w\mathrm{\nabla }v)·\nu =(D_z\mathrm{\nabla }z-{\xi }_{z}z\mathrm{\nabla }v)·\nu =0,\hfill & \hfill & x\in \partial \mathrm{\Omega },t>0,\hfill \\ \hfill & u(x,0)=u_0(x),v(x,0)=v_0(x),w(x,0)=w_0(x),z(x,0)=z_0(x),\hfill & \hfill & x\in \mathrm{\Omega },\hfill \end{array}\right)\end{array}$$ in a bounded domain \(\Omega \subset \mathbb {R}^N\) $\mathrm{\Omega }\subset {\mathbb{R}}^N$ ( \(N\ge 3\) $N\ge 3$ is the dimension of sapce) with smooth boundary, where \(h\in C^{2}([0,\infty ))\) $h\in C^2([0,\infty ))$ satisfies \(0\le h(s)\le K_ws^{k_w}\) $0\le h(s)\le K_w{s}^{k_w}$ for all \(s\ge 0\) $s\ge 0$ with some \(K_w>0\) $K_w>0$ and \(k_w>0\) $k_w>0$ . We demonstrate that whenever \(D_u\mu _u>\frac{N-2}{N}\xi _u\alpha _u\max \{1,\Vert v_0\Vert _{L^\infty (\Omega )}\}\) $D_u{\mu }_u>\frac{N-2}{N}{\xi }_u{\alpha }_{u}max\{1,\Vert v_0{\Vert }_{L^{\infty }(\mathrm{\Omega })}\}$ and \(0<k_w<\frac{2}{N}\) $0<k_w<\frac{2}{N}$ , the system admits a globally bounded solution satisfying that \(\begin{aligned} \Vert u(\cdot ,t)\Vert _{L^\infty (\Omega )}+\Vert v(\cdot ,t)\Vert _{L^\infty (\Omega )}+\Vert w(\cdot ,t)\Vert _{L^\infty (\Omega )}+\Vert z(\cdot ,t)\Vert _{L^\infty (\Omega )}\le C \end{aligned}\) $$\begin{array}{c}\hfill {\Vert u(·,t)\Vert }_{L^{\infty }(\mathrm{\Omega })}+{\Vert v(·,t)\Vert }_{L^{\infty }(\mathrm{\Omega })}+{\Vert w(·,t)\Vert }_{L^{\infty }(\mathrm{\Omega })}+{\Vert z(·,t)\Vert }_{L^{\infty }(\mathrm{\Omega })}\le C\end{array}$$ for all \(t>0\) $t>0$ with some constant \(C>0\) $C>0$ . To our knowledge, this constitutes the most recent advancement in the triply haptotactic cross-diffusion model in three-dimensional or higher-dimensional spaces.