Abstract
In this paper, we investigate the existence of a weak solution and blow-up of strong solutions to a two-component Fornberg–Whitham system. Due to the absence of some useful conservation laws, we establish the existence of a weak solution to the system in lower order Sobolev spaces \(H^{s}\times H^{s-1}\) (\(s\in (1,3/2]\)) via a modified pseudo-parabolic regularization method. And then, a blow-up scenario for strong solutions to this system is shown. By the analysis of Riccati-type inequalities recently, we present some sufficient conditions on the initial data that lead to the blow-up for corresponding strong solutions to the system.
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Acknowledgements
The authors would like to express their gratitude to the reviewers for their thoughtful comments and their help in improving this article. This research was supported by Zhejiang Provincial Natural Science Foundation of China under Grant No. LY21A010008.
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Bai, Z., Wang, Y. & Wei, L. Existence of a weak solution and blow-up of strong solutions for a two-component Fornberg–Whitham system. J. Evol. Equ. 24, 10 (2024). https://doi.org/10.1007/s00028-023-00941-8
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DOI: https://doi.org/10.1007/s00028-023-00941-8