Boundedness and decay behavior in a higher-dimensional quasilinear chemotaxis system with nonlinear logistic source

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• 作者：Jiashan Zheng^a ; ^{zhengjiashan2008@163.com" class="auth_mail" title="E-mail the corresponding author} ; Yifu Wang^b
• 刊名：Computers & Mathematics with Applications
• 出版年：2016
• 出版时间：November 2016
• 年：2016
• 卷：72
• 期：10
• 页码：2604-2619
• 全文大小：491 K

文摘

This paper is concerned with a class of quasilinear chemotaxis systems generalizing the prototype in a smooth bounded domain $\Omega \subset {\mathbb{R}}^N(N\ge 2)$ with parameters m,r≥1$m,r\ge 1$ and μ≥0$\mu \ge 0$. The PDE system in (0.1) is used in mathematical biology to model the mechanism of chemotaxis, that is, the movement of cells in response to the presence of a chemical signal substance which is in homogeneously distributed in space. It is shown that if and the nonnegative initial data $(u_0,v_0)\in C^{\iota }\left(\bar{\Omega }\right)\times W^{1,\infty }\left(\Omega \right)(\iota >0)$, then (0.1) possesses at least one global bounded weak solution. Apart from this, it is proved that if μ=0$\mu =0$ then both u(⋅,t)$u(\cdot ,t)$ and v(⋅,t)$v(\cdot ,t)$ decay to zero with respect to the norm in $L^{\infty }\left(\Omega \right)$ as t→∞$t\to \infty$.