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Asymptotic stability of spatial homogeneity in a haptotaxis model for oncolytic virotherapy
Proceedings of the Royal Society of Edinburgh Section A: Mathematics ( IF 1.3 ) Pub Date : 2021-01-08 , DOI: 10.1017/prm.2020.97 Youshan Tao 1 , Michael Winkler 2
Proceedings of the Royal Society of Edinburgh Section A: Mathematics ( IF 1.3 ) Pub Date : 2021-01-08 , DOI: 10.1017/prm.2020.97 Youshan Tao 1 , Michael Winkler 2
Affiliation
This study considers a model for oncolytic virotherapy, as given by the reaction–diffusion–taxis system \[\begin{eqnarray*} \left\{ \begin{array}{l} u_t = \Delta u - \nabla (u\nabla v)-\rho uz, \\ v_t = - (u+w)v, \\ w_t = D_w \Delta w - w + uz, \\ z_t = D_z \Delta z - z - uz + \beta w, \end{array} \right. \end{eqnarray*}\] in a smoothly bounded domain Ω ⊂ ℝ2 , with parameters D w > 0, D z > 0, β > 0 and ρ ⩾ 0.Previous analysis has asserted that for all reasonably regular initial data, an associated no-flux type initial-boundary value problem admits a global classical solution, and that this solution is bounded if β < 1, whereas whenever β > 1 and $({1}/{|\Omega |})\int _\Omega u(\cdot ,0) > 1/(\beta -1)$ , infinite-time blow-up occurs at least in the particular case when ρ = 0.In order to provide an appropriate complement to this, the current study reveals that for any ρ ⩾ 0 and arbitrary β > 0, at each prescribed level γ ∈ (0, 1/(β − 1)+ ) one can identify an L ∞ -neighbourhood of the homogeneous distribution (u , v , w , z ) ≡ (γ, 0, 0, 0) within which all initial data lead to globally bounded solutions that stabilize towards the constant equilibrium (u ∞ , 0, 0, 0) with some u ∞ > 0.
中文翻译:
溶瘤病毒疗法的趋向性模型中空间同质性的渐近稳定性
本研究考虑了一种溶瘤病毒疗法模型,由反应-扩散-出租车系统给出\[\begin{eqnarray*} \left\{ \begin{array}{l} u_t = \Delta u - \nabla (u\nabla v)-\rho uz, \\ v_t = - (u+w)v , \\ w_t = D_w \Delta w - w + uz, \\ z_t = D_z \Delta z - z - uz + \beta w, \end{array} \right。\end{eqnarray*}\] 在平滑有界域 Ω ⊂ ℝ2 , 带参数D w > 0,D z > 0,β > 0 和ρ ⩾ 0.先前的分析断言,对于所有合理规则的初始数据,相关的无通量类型初始边界值问题承认全局经典解决方案,并且该解决方案是有界的,如果β < 1,而无论何时β > 1 和$({1}/{|\Omega |})\int _\Omega u(\cdot ,0) > 1/(\beta -1)$ , 至少在特定情况下发生无限时间爆炸ρ = 0。为了对此提供适当的补充,目前的研究表明,对于任何ρ ⩾ 0 和任意β > 0, 在每个规定的水平 γ ∈ (0, 1/(β - 1)+ ) 可以识别一个大号 ∞ -均匀分布的邻域(你 ,v ,w ,z ) ≡ (γ, 0, 0, 0) 在其中所有初始数据导致全局有界解稳定在恒定平衡 (你 ∞ , 0, 0, 0) 与一些你 ∞ > 0。
更新日期:2021-01-08
中文翻译:
溶瘤病毒疗法的趋向性模型中空间同质性的渐近稳定性
本研究考虑了一种溶瘤病毒疗法模型,由反应-扩散-出租车系统给出