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 $D_u,D_w,D_z,{\xi }_u,{\xi }_w,{\delta }_z,\rho$, ${\alpha }_u,{\alpha }_w,{\mu }_u,\beta$. 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 $t\to \infty$.

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