Asymptotic behavior of a three-dimensional haptotactic cross-diffusion system modeling oncolytic virotherapy
Yifu Wang, Chi Xu
TL;DR
This work analyzes a three-dimensional haptotactic cross-diffusion system modeling oncolytic virotherapy, proving global existence and exponential stabilization of solutions under a small-data condition on the virus production parameter. The authors employ a transformation $a=ue^{-v}$, $b=we^{-v}$ and a self-map argument to derive sharp a priori estimates, including exponential decay of $v$ and $z$, which lead to global boundedness of $(u,v,w,z)$ and convergence to the equilibrium $(1,0,0,0)$. A bootstrap strategy then yields exponential decay of $u-1$ in $L^p$ norms ($p<6$) and ultimately in $L^{\infty}$, together with decay of $\nabla v$ and sustained regularity. Overall, the paper provides a rigorous 3D analysis of a doubly haptotactic cross-diffusion system in the oncolytic virotherapy context, establishing global solvability and quantitative long-time behavior for small initial perturbations.
Abstract
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 $Ω\subset \mathbb{R}^3$ with $β>0$,~$μ>0$ and $D_z>0$. Based on a self-map argument, it is shown that under the assumption $β\max \{1,\|u_0\|_{L^{\infty}(Ω)}\}< 1+ (1+\frac1{\min_{x\in Ω}u_0(x)})^{-1}$, 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 $t\rightarrow \infty$.