Persistence of mass in a chemotaxis system with logistic聽source

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• 作者：Youshan Tao^a ; ^{taoys@dhu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Michael Winkler^b ; ^{michael.winkler@math.uni-paderborn.de" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35B40 ; 35K57 ; 35Q92 ; 92C17
• 刊名：Journal of Differential Equations
• 出版年：2015
• 出版时间：5 December 2015
• 年：2015
• 卷：259
• 期：11
• 页码：6142-6161
• 全文大小：318 K

文摘

This paper studies the dynamical properties of the chemotaxis system under homogeneous Neumann boundary conditions in bounded convex domains 惟⊂Rⁿ$\mathrm{惟}\subset {\mathbb{R}}^n$, n≥1$n\ge 1$, with positive constants 蠂, r and 渭.

Numerical simulations but also some rigorous evidence have shown that depending on the relative size of r, 渭   and |惟|$|\mathrm{惟}|$, in comparison to the well-understood case when 蠂=0$蠂=0$, this problem may exhibit quite a complex solution behavior, including unexpected effects such as asymptotic decay of the quantity u within large subdomains of 惟.

The present work indicates that any such extinction phenomenon, if occurring at all, necessarily must be of spatially local nature, whereas the population as a whole always persists. More precisely, it is shown that for any nonnegative global classical solution (u,v)$(u,v)$ of (鈰? with u鈮?$u鈮?/mo><mn>0</mn>$ one can find m_鈰?/sub>>0$m_{鈰?/mo>}>0$ such that

The proof is based on an, in this context, apparently novel analysis of the functional ∫_惟ln鈦${\int }_{\mathrm{惟}}\ln 鈦?/mo><mi>u</mi>$, deriving a lower bound for this quantity along a suitable sequence of times by appropriately exploiting a differential inequality for a suitable linear combination of ∫_惟ln鈦${\int }_{\mathrm{惟}}\ln 鈦?/mo><mi>u</mi>$, ∫_惟u${\int }_{\mathrm{惟}}u$ and ∫_惟v²${\int }_{\mathrm{惟}}{v}^2$.