Stabilization in a chemotaxis model for tumor invasion
Stabilization in a chemotaxis model for tumor invasion
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DOI：
10.3934/dcds.2016.36.151
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发表时间：
2015-06
影响因子：
1.1
通讯作者：
Kentarou Fujie;A. Ito;M. Winkler;T. Yokota
中科院分区：
文献类型：
--
作者：
Kentarou Fujie;A. Ito;M. Winkler;T. Yokota
本文涉及趋化系统 \[ \begin{cases} u_t=\Delta u - \nabla \cdot (u\nabla v), \qquad x\in \Omega, \ t>0, \\ v_t=\Delta v + wz, \qquad x\in \Omega, \ t>0, \\ w_t=-wz, \qquad x\in \Omega, \ t>0, \\ z_t=\Delta z - z + u, \qquad x\in \Omega, \ t>0, \end{cases} \] 在平滑有界域 $\Omega \subset \mathbb{R}^n$, $n \le 3$，最近被提议作为肿瘤侵袭的模型，其中考虑了活性细胞外基质的作用。结果表明，对于任何非负且适当规则的初始数据$(u_0,v_0,w_0,z_0)$，相应的诺依曼型初始边值问题都具有有界的全局解。此外，证明只要$u_0\not\equiv 0$，这些解就接近某种空间均匀平衡，即$t\to\infty$、$u(x,t)\to \overline{u_0}$、$v(x,t) \to \overline{v_0} + \overline{w_0}$， $w(x,t) \to 0$ 和 $z(x,t) \to \overline{u_0}$，统一相对于 $x\in\Omega$，其中 $\overline{u_0}:=\frac{1}{|\Omega|} \int_{\Omega} u_0$， $\overline{v_0}:=\frac{1}{|\Omega|} \int_{\Omega} v_0$ $\overline{w_0}:=\frac{1}{|\Omega|} \int_{\Omega} w_0$。
This paper deals with the chemotaxis system \[ \begin{cases} u_t=\Delta u - \nabla \cdot (u\nabla v), \qquad x\in \Omega, \ t>0, \\ v_t=\Delta v + wz, \qquad x\in \Omega, \ t>0, \\ w_t=-wz, \qquad x\in \Omega, \ t>0, \\ z_t=\Delta z - z + u, \qquad x\in \Omega, \ t>0, \end{cases} \] in a smoothly bounded domain $\Omega \subset \mathbb{R}^n$, $n \le 3$, that has recently been proposed as a model for tumor invasion in which the role of an active extracellular matrix is accounted for. It is shown that for any choice of nonnegative and suitably regular initial data $(u_0,v_0,w_0,z_0)$, a corresponding initial-boundary value problem of Neumann type possesses a global solution which is bounded. Moreover, it is proved that whenever $u_0\not\equiv 0$, these solutions approach a certain spatially homogeneous equilibrium in the sense that as $t\to\infty$, $u(x,t)\to \overline{u_0}$ , $v(x,t) \to \overline{v_0} + \overline{w_0}$, $w(x,t) \to 0$ and $z(x,t) \to \overline{u_0}$, uniformly with respect to $x\in\Omega$, where $\overline{u_0}:=\frac{1}{|\Omega|} \int_{\Omega} u_0$, $\overline{v_0}:=\frac{1}{|\Omega|} \int_{\Omega} v_0$ $\overline{w_0}:=\frac{1}{|\Omega|} \int_{\Omega} w_0$.