Asymptotic behavior of a doubly haptotactic cross-diffusion model for oncolytic virotherapy

This paper considers a model for oncolytic virotherapy given by the doubly haptotactic cross-diffusion system \begin{equation*} \left\{\begin{array}{ll} u_t=D_u\Delta u-\xi_u\nabla\cdot(u\nabla v)+\mu_u u(1-u)-\rho uz, v_t=- (\alpha_u u+\alpha_w w)v,\\ w_t=D_w\Delta w-\xi_w\nabla\cdot(w\nabla v)- w+\rho uz,\\ z_t=D_z\Delta z-\delta_z z- \rho uz+\beta w, \end{array}\right. \end{equation*} with positive parameters $D_u,D_w,D_z,\xi_u,\xi_w,\delta_z,\rho$, $\alpha_u,\alpha_w,\mu_u,\beta$. When posed under no-flux boundary conditions in a smoothly bounded domain $\Omega\subset {\mathbb{R}}^2$, and along with initial conditions involving suitably regular data, the global existence of classical solution to this system was asserted in Tao and Winkler (2020). Based on the suitable quasi-Lyapunov functional, it is shown that when the virus replication rate $\beta<1$, the global classical solution $(u,v,w,z)$ is uniformly bounded and exponentially stabilizes to the constant equilibrium $(1, 0, 0, 0)$ in the topology $(L^\infty(\Omega))^4$ as $t\rightarrow \infty$.