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

Abstract

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_uΔu-ξ_u\nabla\cdot(u\nabla v)+μ_u u(1-u)-ρuz, v_t=- (α_u u+α_w w)v,\\ w_t=D_wΔw-ξ_w\nabla\cdot(w\nabla v)- w+ρuz,\\ z_t=D_zΔz-δ_z z- ρuz+βw, \end{array}\right. \end{equation*} with positive parameters $D_u,D_w,D_z,ξ_u,ξ_w,δ_z,ρ$, $α_u,α_w,μ_u,β$. When posed under no-flux boundary conditions in a smoothly bounded domain $Ω\subset {\mathbb{R}}^2$, 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 $β<1$, 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(Ω))^4 $ as $t\rightarrow \infty$.