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 Dw > 0, Dz > 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：

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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Dw> 0,Dz> 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。