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=Δu-\nabla \cdot(u\nabla v)+μu(1-u)-uz,\ v_t=-(u+w)v,\ w_t=Δw-\nabla \cdot(w\nabla v)-w+uz,\ z_t=D_zΔz-z-uz+βw, \end{array} \right. \end{equation*} in a smoothly bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^3$ with $\beta >0$,~$\mu >0$ and $D_z>0$. Based on a self-map argument, it is shown that under the assumption $\beta \max \{1,\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(\mathrm{\Omega })}\}<1+(1+\frac{1}{{\min }_{x\in \mathrm{\Omega }}{u}_0(x)}{)}^{-1}$, 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 $t\to \mathrm{\infty }$.