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Wave-breaking and weak instability for the stochastic modified two-component Camassa–Holm equations

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Abstract

In this paper, we consider the stochastic modified two-component Camassa–Holm equations. For the periodic boundary value problem for this SPDE, we first study the local existence, uniqueness and blow-up criterion of a solution in Sobolev spaces \(H^s\) with \(s>5/2\). Particularly, for the linear non-autonomous noise case, we study the wave-breaking phenomenon. When wave-breaking occurs, we estimate the corresponding probability and the breaking rate of the solutions. Finally, we study the noise effect on the dependence on initial data. It is shown that the noise cannot improve the stability of exiting times and the continuity of solution map at the same time.

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Acknowledgements

The authors would like to express their great gratitude to the anonymous referee for his/her important suggestions, which hav e led to a significant improvement of this paper. Y. Zhao is supported by Basic and Applied Basic Research Project of Guangzhou under the Grant Numbers 202102020283 and 202102021152. Y. Li is supported by NSFC key project under the Grant Number 11831003, and NSFC under the Grant Number 11971356. F. Chen is supported by the NSFC under Grant Number 12101345 and Natural Science Foundation of Shandong Province of China with Grant Number ZR2021QA017.

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Authors and Affiliations

  1. Department of Basic Courses, Guangzhou Maritime University, Guangzhou, 510725, Guangdong, China

    Yongye Zhao

  2. School of Mathematics, South China University of Technology, Guangzhou, 510640, Guangdong, China

    Yongsheng Li

  3. School of Mathematics and Statistics, Qingdao University, Qingdao, 266071, Shandong, China

    Fei Chen

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  1. Yongye Zhao
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Correspondence to Yongye Zhao.

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Zhao, Y., Li, Y. & Chen, F. Wave-breaking and weak instability for the stochastic modified two-component Camassa–Holm equations. Z. Angew. Math. Phys. 74, 159 (2023). https://doi.org/10.1007/s00033-023-02030-9

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  • DOI: https://doi.org/10.1007/s00033-023-02030-9

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