Asymptotic behavior of a three-dimensional haptotactic cross-diffusion system modeling oncolytic virotherapy

This paper deals with an initial-boundary value problem for a doubly haptotactic cross-diffusion system arising from the oncolytic virotherapy \begin{equation*} \left\{ \begin{array}{lll} u_t=\Delta u-\nabla \cdot(u\nabla v)+\mu u(1-u)-uz,\\ v_t=-(u+w)v,\\ w_t=\Delta w-\nabla \cdot(w\nabla v)-w+uz,\\ z_t=D_z\Delta z-z-uz+\beta w, \end{array} \right. \end{equation*} in a smoothly bounded domain $\Omega\subset \mathbb{R}^3$ with $\beta>0$,~$\mu>0$ and $D_z>0$. Based on a self-map argument, it is shown that under the assumption $\beta \max \{1,\|u_0\|_{L^{\infty}(\Omega)}\}<1+ (1+\frac1{\min_{x\in \Omega}u_0(x)})^{-1}$, this problem possesses a uniquely determined global classical solution $(u,v,w,z)$ for certain type of small data $(u_0,v_0,w_0,z_0)$. Moreover, $(u,v,w,z)$ is globally bounded and exponentially stabilizes towards its spatially homogeneous equilibrium %constant equilibrium $(1,0,0,0)$ as $t\rightarrow \infty$.