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Optimal Time-Decay Estimates in the Critical Framework for a Chemotaxis Model

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Abstract

In this paper, we investigate the Cauchy problem of a multidimensional hyperbolic–parabolic system which derived from the Keller–Segel model describing chemotaxis in the critical Besov spaces with initial data close to a constant equilibrium state. The time-decay estimates of global strong solutions are established in more general \(L^{p}\) framework comparing with Xu et al. (J Math Phys 60:091509, 2019) and Xu and Li (Commun Contemp Math, 2021. https://doi.org/10.1142/S0219199721500784), and a new regularity assumption is proposed that the low frequencies have more freedom. The proof mainly depends on tricky and non-classical Besov product estimates about various Sobolev embeddings.

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Notes

  1. Note that for technical reasons, we need a small overlap between low and high frequencies.

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Acknowledgements

The first author is supported by the National Natural Science Foundation of China (12101263) and the Fundamental Research Funds for the Central Universities (JUSRP121047). The third author is supported by the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (19KJD100001). Last but not least, he is very grateful to Professor J. Xu for the suggestion on this question. There are no conflicts of interest to this work.

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Authors and Affiliations

  1. School of Science, Jiangnan University, Lihu, Wuxi, 214122, Jiangsu, People’s Republic of China

    Weixuan Shi

  2. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Jiangjun, Nanjing, 211106, Jiangsu, People’s Republic of China

    Jianzhong Zhang & Mingfeng Xie

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  1. Weixuan Shi
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  2. Jianzhong Zhang
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  3. Mingfeng Xie
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Correspondence to Jianzhong Zhang.

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Communicated by Yong Zhou.

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Shi, W., Zhang, J. & Xie, M. Optimal Time-Decay Estimates in the Critical Framework for a Chemotaxis Model. Bull. Malays. Math. Sci. Soc. 45, 1003–1026 (2022). https://doi.org/10.1007/s40840-021-01240-6

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  • DOI: https://doi.org/10.1007/s40840-021-01240-6

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