杜宛娟.一类具有奇异灵敏度的logistic趋化系统解的渐近行为[J].数学研究及应用,2021,41(5):473~480
一类具有奇异灵敏度的logistic趋化系统解的渐近行为
Asymptotic Behavior of Solutions to a Logistic Chemotaxis System with Singular Sensitivity
投稿时间:2020-09-07  修订日期:2021-01-03
DOI:10.3770/j.issn:2095-2651.2021.05.004
中文关键词:  渐近行为  趋化  奇异灵敏度  logistic源
英文关键词:asymptotic behavior  chemotaxis  singular sensitivity  logistic source
基金项目:西华师范大学创新团队项目(Grant No.CXTD2020-5),西华师范大学英才项目(Grant No.17YC372).
作者单位
杜宛娟 西北大学科学史高等研究院, 陕西 西安 710217
西华师范大学公共数学学院, 四川 南充 637009 
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中文摘要:
      本文在无边界流的光滑有界区域$\Omega\subset\mathbb{R}^n~(n>2)$上研究了具有奇异灵敏度及logistic源的抛物-椭圆趋化系统$$\left\{\begin{array}{ll}u_t=\Delta u-\chi\nabla\cdot(\frac{u}{v}\nabla v)+r u-\mu u^k,&x\in\Omega,\,t>0,\\ 0=\Delta v-v+u,&x\in\Omega,\,t>0\end{array}\right.$$ 其中$\chi$, $r$, $\mu>0$, $k\geq2$. 证明了若当$r$适当大, 则当$t\rightarrow\infty$时该趋化系统全局有界解呈指数收敛于$((\frac{r}{\mu})^{\frac{1}{k-1}}, (\frac{r}{\mu})^{\frac{1}{k-1}})$.
英文摘要:
      In this paper, we study the asymptotic behavior of solutions to the parabolic-elliptic chemotaxis system with singular sensitivity and logistic source $$\left\{\begin{array}{ll}u_t=\Delta u-\chi\nabla\cdot(\frac{u}{v}\nabla v)+r u-\mu u^k,&x\in\Omega,\,t>0,\\ 0=\Delta v-v+u,&x\in\Omega,\,t>0\end{array}\right.$$ in a smooth bounded domain $\Omega\subset\mathbb{R}^n~(n>2)$ with the non-flux boundary, where $\chi$, $r$, $\mu>0$, $k\geq2$. It is proved that the global bounded classical solution will exponentially converge to $((\frac{r}{\mu})^{\frac{1}{k-1}}, (\frac{r}{\mu})^{\frac{1}{k-1}})$ as $t\rightarrow\infty$ if $r$ is suitably large.
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