Global asymptotic stability in a chemotaxis-growth model for tumor invasion
Global asymptotic stability in a chemotaxis-growth model for tumor invasion
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DOI：
10.3934/dcdss.2020011
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发表时间：
2020
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影响因子：
--
通讯作者：
Kentarou Fujie
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文献类型：
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作者：
Kentarou Fujie
本文给出了趋化-增长方程组解的整体存在性和渐近性态.该方程的解的存在性和渐近性主要体现在方程组的解的整体存在性和渐近性上.0，\w_t=-wz，\qquad x\in\Omega，\t>；0，\z_t=\Delta z-z+u，\qquad x\in\Omega，\t>；0，\end{数组}\右。$\end{Document}在平滑有界的域中\Begin{Document}$\Omega\Subset\mathbb{R}^n$\end{Document}，\Begin{Document}$n\le 3$\end{Document}，其中\Begin{Document}$\end{Document}，\Begin{Document}$\Mu>；0$\end{Document}和\Begin{Document}$\Alpha>；1$\end{Document}。在没有逻辑信号源的情况下，Fujie，Ito，Winkler和Yokota(2016)证明了该系统的稳定性，但特别是关于渐近行为，逻辑信号源直接干扰了该方法的应用。本文介绍了一种走出这一困境的方法，并精确地确定了具有Logistic源的系统解的渐近行为。
This paper presents global existence and asymptotic behavior of solutions to the chemotaxis-growth system \begin{document}$ \left\{ \begin{array}{l} u_t = \Delta u - \nabla \cdot (u\nabla v) + ru -\mu u^\alpha, \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{array} \right. $\end{document} in a smoothly bounded domain \begin{document}$ \Omega \subset \mathbb{R}^n $\end{document} , \begin{document}$ n \le 3 $\end{document} , where \begin{document}$ r>0 $\end{document} , \begin{document}$ \mu>0 $\end{document} and \begin{document}$ \alpha>1 $\end{document} . Without the logistic source \begin{document}$ ru-\mu u^\alpha $\end{document} , the stabilization of this system has been shown by Fujie, Ito, Winkler and Yokota (2016), whereas especially about asymptotic behavior, the logistic source disturbs applying this method directly. In the present paper, a way out of this difficulty is introduced and the asymptotic behavior of solutions to the system with logistic source is precisely determined.