广东工业大学学报 ›› 2022, Vol. 39 ›› Issue (01): 93-98.doi: 10.12052/gdutxb.200148

• 综合研究 • 上一篇    下一篇

三维趋化系统全局弱解的存在性和渐近稳定性

唐浩怡, 彭红云   

  1. 广东工业大学 数学与统计学院， 广东 广州 510520
  • 收稿日期:2020-11-03 发布日期:2022-01-20
  • 通信作者: 彭红云(1984-),男,副教授,主要研究方向为非线性偏微分方程,E-mail:hypengmath@gdut.edu.cn
  • 作者简介:唐浩怡(1995-),女,硕士研究生,主要研究方向为非线性偏微分方程
  • 基金资助:
    国家自然科学基金资助项目(11901115)；广东省自然科学基金资助项目(2019A1515010706)；广东工业大学青年百人启动项目(220413228)

Global Weak Solutions and Asymptotics of a 3D Chemotaxis System

Tang Hao-yi, Peng Hong-yun   

  1. School of Mathematics and Statistics, Guangdong University of Technology, Guangzhou 510520, China
  • Received:2020-11-03 Published:2022-01-20

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摘要： 主要研究了一类抛物−双曲型系统在不连续初值下全局弱解的适定性和大时间行为。该系统是利用Cole-Hopf变换从描述肿瘤血管生成的三维PDE-ODE (Partial Differential Equations-Ordinary Differential Equations )趋化模型转换而来的。本文在不连续初值条件下, 证明了当时间趋于无穷时抛物−双曲型系统的解收敛于常数稳态解。

关键词: 趋化性, 渐近稳定性, 不连续初值, 有效粘性通量

Abstract: The well-posedness and large-time behavior for a parabolic-hyperbolic system with discontinuous initial value is mainly considered. The system is transformed from a three dimensional PDE-ODE hybrid chemotaxis system describing the initiation of tumor angiogenesis by a Cole-Hopf type transformation. With discontinuous data, it is proved that the solution of the transformed system converges to a constant equilibrium as time tends to infinity. Lastly, the results for the pre-transformed PDE-ODE hybrid system are obtained by reversing the Cole-Hopf transformation. Compared with the relevant results on continuous initial datum, the asymptotic stability of discontinuous initial values with large oscillations is proved. In the proof, the so-called “effective viscous flux” is used for the analysis to obtain the desired energy estimates and regularity. The "effective viscous flux" technique is rarely used in the literature of chemotaxis model. But it has proved to be a useful tool for studying chemotaxis systems with initial datum having low regularity.

Key words: chemotaxis, asymptotic stability, discontinuous initial data, effective viscous flux

中图分类号: 

  • O175
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