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 916302385&_mathId=si2.gif&_user=111111111&_pii=S0893965916302385&_rdoc=1&_issn=08939659&md5=6825181586dd13caeceed51341da455f">916302385-si2.gif">$\Omega \subset {\mathbb{R}}^N$, 916302385&_mathId=si3.gif&_user=111111111&_pii=S0893965916302385&_rdoc=1&_issn=08939659&md5=985e9b35aa8249ff094dedbb7789a518" title="Click to view the MathML source">N≥1$N\ge 1$, where 916302385&_mathId=si4.gif&_user=111111111&_pii=S0893965916302385&_rdoc=1&_issn=08939659&md5=32ac875ec44b7cc72222094ce620ed19" title="Click to view the MathML source">χ,μ,m,α$\chi ,\mu ,m,\alpha$ and 916302385&_mathId=si5.gif&_user=111111111&_pii=S0893965916302385&_rdoc=1&_issn=08939659&md5=de2f026ace56c06a360e25c135adb9f4" title="Click to view the MathML source">γ$\gamma$ are prescribed positive parameters fulfilling 916302385&_mathId=si6.gif&_user=111111111&_pii=S0893965916302385&_rdoc=1&_issn=08939659&md5=74be938fea97002af04ae0ef3531f560" title="Click to view the MathML source">m≥1$m\ge 1$ and 916302385&_mathId=si7.gif&_user=111111111&_pii=S0893965916302385&_rdoc=1&_issn=08939659&md5=9c48b970c57462a0e93d8a58aeb596a6" title="Click to view the MathML source">γ≥1$\gamma \ge 1$.

Recently, it has been proved in Galakhov et al. (2016) that if either 916302385&_mathId=si8.gif&_user=111111111&_pii=S0893965916302385&_rdoc=1&_issn=08939659&md5=e7aa479d7a232068d80f33203d2f5888" title="Click to view the MathML source">α>m+γ−1$\alpha >m+\gamma -1$ or 916302385&_mathId=si9.gif&_user=111111111&_pii=S0893965916302385&_rdoc=1&_issn=08939659&md5=b017c7a3bb76a7793198fd2f2c2259ec" title="Click to view the MathML source">α=m+γ−1$\alpha =m+\gamma -1$ and 916302385&_mathId=si10.gif&_user=111111111&_pii=S0893965916302385&_rdoc=1&_issn=08939659&md5=b42541b1d46e87c31b53dced1ba64762">916302385-si10.gif">$\mu >\frac{N\alpha -2}{2(m-1)+N\alpha }\chi$, for any given 916302385&_mathId=si11.gif&_user=111111111&_pii=S0893965916302385&_rdoc=1&_issn=08939659&md5=18e4f61bcbf79cea95fcf08e94c26471">916302385-si11.gif">$u_0\in W^{1,\infty }\left(\Omega \right)$ this system possesses a global and bounded classical solution. The present work further shows that the same conclusion still holds for the critical case 916302385&_mathId=si9.gif&_user=111111111&_pii=S0893965916302385&_rdoc=1&_issn=08939659&md5=b017c7a3bb76a7793198fd2f2c2259ec" title="Click to view the MathML source">α=m+γ−1$\alpha =m+\gamma -1$ and 916302385&_mathId=si13.gif&_user=111111111&_pii=S0893965916302385&_rdoc=1&_issn=08939659&md5=4e3d3157cbd32d0d7a6c1c04e7e9db5a">916302385-si13.gif">$\mu =\frac{N\alpha -2}{2(m-1)+N\alpha }\chi$.