Electron. J. Differential Equations, Vol. 2025 (2025), No. 70, pp. 1-23.
Wave-breaking for two-component Fornberg-Whitham systems with dissipation
Xi Zhu, Min Zhu, Ying Wang, Ke Wang
Abstract:
In this article, we study the Cauchy problem for a two-component Fornberg-Whitham (2FW)
system in fluid dynamics, incorporating a dissipation term to account for energy loss.
In the 2FW system, the analysis of blow-up phenomena is complicated due to its
non-integrable structure and the lack of sufficient useful conservation laws.
Adding dissipation term makes the problem even more challenging, since the \(L^2\)
norm of \(u\) grows exponentially in time rather than polynomially. Unlike previous
works that focus on Riccati-type inequalities with polynomial expressions,
we consider a case where the involved term exhibits exponential growth.
This induces an extension of the Riccati-type inequalities to handle exponential
forms, from which we obtain a new blow-up analysis result.
As a consequence, we establish a novel blow-up criterion and obtain three blow-up
results.
Submitted January 16, 2025. Published July 11, 2025.
Math Subject Classifications: 35B44, 35G25, 35Q35.
Key Words: Two-component Fornberg-Whitham system; blow-up; local well-posedness.
DOI: 10.58997/ejde.2025.70
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Xi Zhu
School of Mathematical Sciences
University of Electronic Science and Technology of China
Chengdu 611731, China
email: zhuxi199901@163.com
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Min Zhu
Department of Mathematics
Nanjing Forestry University,
Nanjing 210037, China
email: zhumin@njfu.edu.cn
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Ying Wang
School of Mathematical Sciences
University of Electronic Science and Technology of China
Chengdu 611731, China
email: nadine_1979@163.com
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Ke Wang
Basic Teaching Department of Chengdu Technology University
Yibin Research Institute of Chengdu Technology University
Yibin 644000, China
email: yuwk77@163.com
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