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Ill-Posedness Issue on a Multidimensional Chemotaxis Equations in the Critical Besov Spaces

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Abstract

In this paper, we aim to solving the open question left in [Nie, Yuan: Nonlinear Anal 196 (2020); J. Math. Anal. Appl 505 (2022) and Xiao, Fei: J. Math. Anal. Appl 514 (2022)]. We prove that a multidimensional chemotaxis system is ill-posedness in \(\dot{B}_{2d, r}^{-\frac{3}{2}} \times \big (\dot{B}_{2d, r}^{-\frac{1}{2}}\big )^{d}\) when \(1\le r<d\) due to the lack of continuity of the solution.

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Acknowledgements

The authors would like to express their gratitude to the anonymous referees for valuable suggestions and comments which greatly improved the paper. J. Li is supported by the National Natural Science Foundation of China (11801090 and 12161004) and Jiangxi Provincial Natural Science Foundation (20212BAB211004). Y. Yu is supported by the National Natural Science Foundation of China (12101011). W. Zhu is supported by the National Natural Science Foundation of China (12201118) and Guangdong Basic and Applied Basic Research Foundation (2021A1515111018).

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Authors and Affiliations

  1. School of Mathematics and Computer Sciences, Gannan Normal University, Ganzhou, 341000, China

    Jinlu Li

  2. School of Mathematics and Statistics, Anhui Normal University, Wuhu, 241002, China

    Yanghai Yu

  3. School of Mathematics and Big Data, Foshan University, Foshan, Guangdong, 528000, China

    Weipeng Zhu

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  1. Jinlu Li
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  2. Yanghai Yu
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Correspondence to Yanghai Yu.

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Li, J., Yu, Y. & Zhu, W. Ill-Posedness Issue on a Multidimensional Chemotaxis Equations in the Critical Besov Spaces. J Geom Anal 33, 84 (2023). https://doi.org/10.1007/s12220-022-01140-6

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