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Blow-up Analysis for the \({\varvec{ab}}\)-Family of Equations

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Abstract

This paper investigates the Cauchy problem for the ab-family of equations with cubic nonlinearities, which contains the integrable modified Camassa–Holm equation (\(a = \frac{1}{3}\), \(b = 2\)) and the Novikov equation (\(a = 0\), \(b = 3\)) as two special cases. When \(3a + b \ne 3\), the ab-family of equations does not possess the \(H^1\)-norm conservation law. We give the local well-posedness results of this Cauchy problem in Besov spaces and Sobolev spaces. Furthermore, we provide a blow-up criterion, the precise blow-up scenario and a sufficient condition on the initial data for the blow-up of strong solutions to the ab-family of equations. Our blow-up analysis does not rely on the use of the conservation laws.

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Acknowledgements

The work of Cheng is partially supported by the National Natural Science Foundation of China (No. 12201417), and the Project funded by China Postdoctoral Science Foundation (No. 2023M733173). The work of Lin is partially supported by the National Natural Science Foundation of China (No. 12375006).

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Authors and Affiliations

  1. Department of Physics, Zhejiang Normal University, Jinhua, 321004, Zhejiang, China

    Wenguang Cheng & Ji Lin

  2. Department of Mathematics, Shaoxing University, Shaoxing, 312000, Zhejiang, China

    Wenguang Cheng

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  1. Wenguang Cheng
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Cheng, W., Lin, J. Blow-up Analysis for the \({\varvec{ab}}\)-Family of Equations. J. Math. Fluid Mech. 26, 22 (2024). https://doi.org/10.1007/s00021-024-00857-4

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