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The dependences on initial data for the b-family equation in critical Besov space

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Abstract

This paper is concerned with the periodic boundary problem for the b-family equation. At first, we use a different method to prove the local well-posedness in the critical Besov space \(B_{2,1}^{3/2}\). Then we show that if a weaker \(B_{p,r}^q\)-topology is used, the solution map becomes Hölder continuous. Moreover, we show that the dependence on initial data is optimal in \(B_{2,1}^{3/2}\) in the sense that the solution map is continuous but not uniformly continuous. Finally, we obtain the periodic peaked solutions to the b-family equation and apply them to obtain the ill-posedness in \(B_{2,\infty }^{3/2}\).

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Acknowledgments

The authors would like to express their great gratitude to the referees for their valuable suggestions, which have led to a meaningful improvement of the paper. The authors would also like to express their sincere gratitude to Professor Yongsheng Li for his helpful suggestions.

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Authors and Affiliations

  1. Department of Mathematics, South China University of Technology, Guangzhou, Guangdong, 510640, People’s Republic of China

    Hao Tang, Shijie Shi & Zhengrong Liu

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  1. Hao Tang
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  2. Shijie Shi
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  3. Zhengrong Liu
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Corresponding author

Correspondence to Zhengrong Liu.

Additional information

Communicated by J. Escher.

This work is supported by the National Natural Science Foundation of China (No. 11171115).

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Tang, H., Shi, S. & Liu, Z. The dependences on initial data for the b-family equation in critical Besov space. Monatsh Math 177, 471–492 (2015). https://doi.org/10.1007/s00605-014-0673-8

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  • DOI: https://doi.org/10.1007/s00605-014-0673-8

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