A new criterion to a two-chemical substances chemotaxis system with critical dimension
A new criterion to a two-chemical substances chemotaxis system with critical dimension
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DOI：
10.3934/dcdsb.2018074
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发表时间：
2017-11
影响因子：
1.2
通讯作者：
Xueli Bai;Suying Liu
中科院分区：
文献类型：
--
作者：
Xueli Bai;Suying Liu
我们主要研究如下系统解的全局有界性：\begin{document}$\begin{align*}\begin{cases}u_t=Δu-χ\nabla·(u\nabla v)&\Text{in}Ω×\mathbb R^+，\\v_t=Δv-v+w&\Text{in}Ω×\mathbb R^+，\\w_t=Δw-w+u&\Text{in}Ω×\mathbb R^+，在具有临界空间维度$n=4$的光滑有界域$Ω\子集\mathbb{R}^n$中，具有非负光滑初值的齐次Neumann边界条件下，结束{CASE}\END{ALIGN*}$\END{DOCUMENT}这个问题已被K.Fujie和T.Senba在[5]中考虑过。证明了在对称情形下，当条件为$\int_\omega{u_03]时，我们给出了解整体有界性的一个新判据。作为副产品，我们得到了文[5]中一个主要结果的简化证明。
We mainly investigate the global boundedness of the solution to the following system, \begin{document}$\begin{align*}\begin{cases}u_t = Δ u-χ\nabla·(u\nabla v) &\text{ in }Ω×\mathbb R^+,\\v_t = Δ v-v+w &\text{ in }Ω×\mathbb R^+,\\w_t = Δ w-w+u &\text{ in }Ω×\mathbb R^+,\end{cases}\end{align*}$ \end{document} under homogeneous Neumann boundary conditions with nonnegative smooth initial data in a smooth bounded domain $Ω\subset \mathbb{R}^n$ with critical space dimension $n = 4$. This problem has been considered by K. Fujie and T. Senba in [ 5 ]. They proved that for the symmetric case the condition $\int_\Omega {u_0 3 ], we give a new criterion for global boundedness of the solution. As a byproduct, we obtain a simplified proof for one of the main results in [ 5 ].