Acta mathematica scientia,Series A ›› 2019, Vol. 39 ›› Issue (5): 1102-1114.

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

Jie Wu1,*(),Hongxia Lin2   

  1. 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
  • 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)


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

CLC Number: 

  • O175.29