关于抛物-抛物Keller-Segel类模型的全局解和渐近性

数学物理学报 ›› 2019, Vol. 39 ›› Issue (5): 1102-1114.

关于抛物-抛物Keller-Segel类模型的全局解和渐近性

吴杰^1,*(),林红霞²

¹ 电子科技大学数学科学学院成都 611731
² 成都理工大学数学地质四川省重点实验室成都 610059

收稿日期:2018-11-14 出版日期:2019-10-26 发布日期:2019-11-08
通讯作者: 吴杰 E-mail:dxtxwj@126.com
基金资助:
国家自然科学基金(11571243);国家自然科学基金(11701049);中国博士后科学基金(2017M622989);数学地质四川省重点实验室开放基金(scsxdz201707)

The Global Solution and Asymptotic Behavior of Parabolic-Parabolic Keller-Segel Type Model

Jie Wu^1,*(),Hongxia Lin²

¹ School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731
² Geomathematics Key Laboratory of Sichuan Province, Chengdu University of Technology, Chengdu 610059

Received:2018-11-14 Online:2019-10-26 Published:2019-11-08
Contact: Jie Wu E-mail:dxtxwj@126.com
Supported by:
the NSFC(11571243);the NSFC(11701049);the China Postdoctoral Science Foundation(2017M622989);the Opening Fund of Geomathematics Key Laboratory of Sichuan Province(scsxdz201707)

摘要/Abstract

摘要：

该文考虑了抛物-抛物型Keller-Segel型模型,通过Neumann热半群获得了关于$\rho$依赖于$\chi$的梯度不等式,同时得到了$\|\rho(\cdot,t)\|_{L^1(\Omega)},\, \|n(\cdot,t)\|_{C^\theta(\bar{\Omega})}$和$\|c(\cdot,t)\|_{L^\infty(\Omega)}$关于时间$t$的收敛性.它揭示了精子的质量将收敛到初始精卵细胞的差,卵子全部受精,化学物质的浓度最终耗尽.同时化学物质的耗尽也伴随着卵细胞全部受精.它表明了化学物质的浓度在珊瑚礁受精过程中起着非常重要的作用.

关键词: 趋化流体, Keller-Segel, 珊瑚礁, 全局存在性, 渐近行为

Abstract:

This paper concerns the parabolic-parabolic Keller-Segel type model. By making use of the Neumann heat semigroup, the asymptotic inequality on the gradient of $\rho$ depending on the chemotaxis signal $\chi$ is derived. Meanwhile, the convergence results on $\|\rho(\cdot,t)\|_{L^1(\Omega)}, \|n(\cdot,t)\|_{C^{\theta }(\bar{\Omega})}$ and $\|c(\cdot,t)\|_{L^\infty(\Omega)}$ have also been obtained. It reveals that mass of sperm will tend to the initial difference between sperm and eggs mass in the process of evolution, eggs are all fertilized and the concentration of the chemical substance will be also exhausted eventually. At the same time, the depletion of chemical substance is accompanied by complete fertilization of eggs. It illustrates that concentration of the chemical substance plays a relevant role in the fertilization process of corals.

Key words: Chemotaxis fluid, Keller-Segel, Corals, Global existence, Asymptotic behavior

中图分类号:

O175.29

吴杰,林红霞. 关于抛物-抛物Keller-Segel类模型的全局解和渐近性[J]. 数学物理学报, 2019, 39(5): 1102-1114.

Jie Wu,Hongxia Lin. The Global Solution and Asymptotic Behavior of Parabolic-Parabolic Keller-Segel Type Model[J]. Acta mathematica scientia,Series A, 2019, 39(5): 1102-1114.

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