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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2210.00484 |
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Table of Contents:
- 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 $t\rightarrow \infty$.