Journal of Guangxi Normal University(Natural Science Edition) ›› 2022, Vol. 40 ›› Issue (2): 125-131.doi: 10.16088/j.issn.1001-6600.2021061201

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Optimal Time-decay Rates of the Hyperbolic-parabolic System Modeling Chemotaxis in R3

WANG Han, ZHANG Yinghui*   

  1. School of Mathematics and Statistics, Guangxi Normal University, Guilin Guangxi 541006, China
  • Received:2021-06-12 Revised:2021-07-02 Published:2022-05-31

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Abstract: The large-time behavior of solutions to the Cauchy problem of a 3D hyperbolic-parabolic system modeling chemotaxis is investigated. The optimal time decay rates of the higher-order spatial derivatives of the solutions are obtained. Compared with previous results, the main innovation of this paper is to give the highest order spatial derivative of the solutions which is the same as that of the heat equation. The proof is mainly based on low-frequency and high-frequency decomposition and delicate energy estimates.

Key words: hyperbolic-parabolic system, optimal decay rates, low-frequency and high-frequency decomposition, large-time behavior, chemotaxis

CLC Number: 

  • O29
[1] OTHMER H G, STEVENS A. Aggregation, blowup, and collapse: the ABCs of taxis in reinforced random walks[J]. SIAM Journal on Applied Mathematics, 1997, 57(4): 1044-1081.
[2] LEVINE H A, SLEEMAN B D. A system of reaction diffusion equations arising in the theory of reinforced random walks[J]. SIAM Journal on Applied Mathematics, 1997, 57(3): 683-730.
[3] GUO J, XIAO J X, ZHAO H J, et al. Global solutions to a hyperbolic-parabolic coupled system with large initial data[J]. Acta Mathematica Scientia, 2009, 29(3): 629-641.
[4] XIE W J, ZHANG Y H, XIAO Y D, et al. Global existence and convergence rates for thestrong solutions in H2 to the 3D chemotaxis model[J]. Journal of Applied Mathematics, 2013, 2013(1): 1-9.
[5] ZHANG M, ZHU C J. Global existence of solutions to a hyperbolicparabolic system[J]. Proceedings of the American Mathematical Society, 2007, 135(4): 1017-1027.
[6] 张映辉, 谭忠, 赖柏顺. 一个模拟趋化现象的广义双曲-抛物系统的光滑解的全局分析[J]. 数学年刊A辑, 2012, 33(1): 27-38.
[7] ZHANG Y H, TAN Z, SUN M B. Global existence and asymptotic behavior of smooth solutions to a coupled hyperbolic-parabolic system[J]. Nonlinear Analysis: Real World Applications, 2013, 14(1): 465-482.
[8] ZHANG Y H, DENG H Y, SUN M B. Global analysis of smooth solutions to a hyperbolic-parabolic coupled system[J]. Frontiers of Mathematics in China, 2013, 8(6): 1437-1460.
[9] ZHANG Y H, XIE W J. Global existence and exponential stability for the strong solutions in H2 to the 3-D chemotaxis model[J]. Boundary Value Problems, 2015, 116: 1-13.
[10] LI D, PAN R H, ZHAO K. Quantitative decay of a one-dimensional hybrid chemotaxis model with large data[J]. Nonlinearity, 2015, 28(7): 2181-2210.
[11] 涂馨予, 穆春来, 郑攀. 带有流晕限制的拟线性趋化模型解的整体有界性[J]. 中国科学: 数学, 2021, 51(6): 1003-1012.
[12] 张婕燕, 辛巧, 穆春来. 具有非线性扩散的趋化模型弱解的有界性[J]. 西南大学学报(自然科学版), 2021, 43(5): 88-95.
[13] 王曦, 刘祖汉, 周玲. 具不同分数阶扩散趋化模型的衰减估计[J]. 数学年刊A辑, 2020, 41(2): 175-200.
[14] 张映辉, 谭忠, 孙明保. 一个耦合双曲-抛物系统的全局光滑解[J]. 数学年刊A辑, 2013, 34(1): 29-46.
[15] LI D, PAN R H, ZHAO K. On a hybrid type chemotaxis model on bounded domains with large data[J]. SIAM Journal on Applied Mathematics, 2012, 72(7): 417-443.
[16] HAO C C. Global well-posedness for a multidimensional chemotaxis model in critical Besov spaces[J]. Zeitschrift für Angewandte Mathematik und Physik, 2012, 63: 825-834.
[17] LI D, LI T, ZHAO K. On a hyperbolic-parabolic system modeling chemotaxis[J]. Mathematical Models and Methods in Applied Sciences, 2011, 21(8): 1631-1650.
[18] ZHANG Y H, LI C, XIE W J. Decay of a 3-D hyperbolic parabolic system modeling chemotaxis[J]. Journal of Information and Optimization Sciences, 2018, 39(7): 1505-1525.
[19] NIRENBERG L. On elliptic partial differential equations[J]. Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 1959, 13(2): 115-162.
[20] MAJDA A J, BERTOZZI A L. Vorticity and incompressible flow[M]. Cambridge: Cambridge University Press, 2002.
[21] JU N. Existence and uniqueness of the solution to the dissipative 2D quasi-geostrophic equations in the Sobolev space[J]. Communications in Mathematical Physics, 2004, 251: 365-376.
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