Boundedness in a parabolic-elliptic chemotaxis-growth system under a critical parameter condition

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• 作者：Bingran Hu^a ; ^{hubingran@qq.com" class="auth_mail" title="E-mail the corresponding author} ; Youshan Tao^b ; ^{taoys@dhu.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Boundedness ; Chemotaxis-growth system ; Critical parameter condition
• 刊名：Applied Mathematics Letters
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：64
• 期：Complete
• 页码：1-7
• 全文大小：588 K

文摘

We consider the parabolic–elliptic chemotaxis-growth system under no-flux boundary conditions in a smoothly bounded domain $\Omega \subset {\mathbb{R}}^N$, 985e9b35aa8249ff094dedbb7789a518" title="Click to view the MathML source">N≥1$N\ge 1$, where χ,μ,m,α$\chi ,\mu ,m,\alpha$ and e56c06a360e25c135adb9f4" title="Click to view the MathML source">γ$\gamma$ are prescribed positive parameters fulfilling m≥1$m\ge 1$ and γ≥1$\gamma \ge 1$.

Recently, it has been proved in Galakhov et al. (2016) that if either α>m+γ−1$\alpha >m+\gamma -1$ or bb76a7793198fd2f2c2259ec" title="Click to view the MathML source">α=m+γ−1$\alpha =m+\gamma -1$ and ba64762">$\mu >\frac{N\alpha -2}{2(m-1)+N\alpha }\chi$, for any given e4f61bcbf79cea95fcf08e94c26471">$u_0\in W^{1,\infty }\left(\Omega \right)$ this system possesses a global and bounded classical solution. The present work further shows that the same conclusion still holds for the critical case bb76a7793198fd2f2c2259ec" title="Click to view the MathML source">α=m+γ−1$\alpha =m+\gamma -1$ and $\mu =\frac{N\alpha -2}{2(m-1)+N\alpha }\chi$.