This paper is concerned with the higher-dimensional haptotactic system modeling oncolytic virotherapy, which was initially proposed by Alzahrani–Eftimie–Trucu [Multiscale modelling of cancer response to oncolytic viral therapy, Math. Biosci. 310 (2019) 76–95] (see also the survey Bellomo–Outada et al. [Chemotaxis and cross-diffusion models in complex environments: Models and analytic problems toward a multiscale vision, Math. Models Methods Appl. Sci. 32 (2022) 713–792]) to model the process of oncolytic viral therapy. We consider this problem in a bounded domain $\mathrm{\Omega }\subseteq {ℝ}^N(N=2,3)$ with zero-flux boundary conditions. Although the $L^{\infty }$-norm of the extracellular matrix density $v$ is easily obtainable, the remodeling process still causes difficulty due to the deficiency of regularity for $v$. Relying on some $L^p$-estimate techniques, in this paper, under the mild condition on parameters, we finally established the existence of global-in-time classical solution, which is bounded uniformly. Moreover, the large time behavior of solutions to the problem is also investigated. Specially speaking, when ${\kappa }_u=0$, the corresponding solution of the system decays to $(0,0,0)$ algebraically. To the best of our knowledge, these are the first results on boundedness and asymptotic behavior of the system in three-dimensional space.

本文关注的是溶瘤病毒治疗的高维触触系统建模，该系统最初由 Alzahrani-Eftimie-Trucu 提出[癌症对溶瘤病毒治疗反应的多尺度建模，数学。生物科学。 310 (2019) 76–95]（另请参阅 Bellomo-Outada等人的调查[复杂环境中的趋化性和交叉扩散模型：面向多尺度视觉的模型和分析问题，Math . ModelsMethodsAppl.Sci.32 （2022 年） ）713–792]）来模拟溶瘤病毒治疗的过程。我们在有界域中考虑这个问题$\mathrm{\Omega }\subseteq {ℝ}^{氮}（氮=2,3）$具有零通量边界条件。虽然$L^{\mathrm{无穷大}}$-细胞外基质密度的范数$v$虽然很容易获得，但由于缺乏规律性，重构过程仍然存在困难。$v$。依靠一些$L^p$-估计技术，本文在参数温和的条件下，最终建立了一致有界的全局时间经典解的存在性。此外，还研究了问题解决方案的长时间行为。特别来说，当${\kappa }_{你}=0$，系统相应的解衰减为$（0,0,0）$代数上。据我们所知，这些是三维空间中系统有界性和渐近行为的第一个结果。