Asymptotic behavior of a three-dimensional haptotactic cross-diffusion system modeling oncolytic virotherapy

TL;DR

This paper analyzes a complex three-dimensional cross-diffusion system modeling oncolytic virotherapy, proving global existence, boundedness, and exponential stabilization of solutions under specific small data conditions.

Contribution

It establishes the global well-posedness and stability of solutions for a novel haptotactic cross-diffusion model in three dimensions, which was previously unaddressed.

Findings

Existence of unique global classical solutions under small data conditions.

Solutions are globally bounded and exponentially stabilize to equilibrium.

The model's solutions exhibit asymptotic behavior consistent with biological expectations.

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=\Delta u-\nabla \cdot(u\nabla v)+\mu u(1-u)-uz,\\ v_t=-(u+w)v,\\ w_t=\Delta w-\nabla \cdot(w\nabla v)-w+uz,\\ z_t=D_z\Delta z-z-uz+\beta w, \end{array} \right. \end{equation*} in a smoothly bounded domain $\Omega\subset \mathbb{R}^3$ with $\beta>0$,~$\mu>0$ and $D_z>0$. Based on a self-map argument, it is shown that under the assumption $\beta \max \{1,\|u_0\|_{L^{\infty}(\Omega)}\}< 1+ (1+\frac1{\min_{x\in \Omega}u_0(x)})^{-1}$, 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 $t\rightarrow \infty$.