Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 70, pp. 1-23.

Title: Wave-breaking for two-component Fornberg-Whitham systems with dissipation

Authors: Xi Zhu (Univ. of Electronic Science and Tech., Chengdu, China)
         Min Zhu (Nanjing Forestry Univ., Nanjing, China)
         Ying Wang (Univ. of Electronic Science and Tech., Chengdu, China)
         Ke Wang (Chengdu Technology Univ., Yibin, China)

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.