The periodic Cauchy problem for a two-component non-isospectral cubic Camassa-Holm system

Authors

Lei Zhang
Zhijun Qiao, The University of Texas Rio Grande ValleyFollow

Document Type

Article

Publication Date

1-15-2020

Abstract

In this paper, we study the periodic Cauchy problem for a two-component non-isospectral cubic Camassa-Holm system which includes the Fokas-Olver-Rosenau-Qiao (FORQ) or modified Camassa-Holm (MCH) equation and the two-component MCH system as two special cases. The system is integrable in the sense of possessing a non-isospectral Lax pair with the spectrum depending on time t, and admits multi-peakon solutions in an explicit form. Furthermore, we establish the local well-posedness for the system in the Besov space Bs2,r(T) with s>3/2, 1≤r≤∞, where the key ingredients include the Friedrichs regularization method, the Littlewood-Paley decomposition theory, and the transport theory in Besov spaces. Then we derive a precise blow-up criteria, which is dependent of the parameters α(t) and γ(t). Moreover, by the intrinsic structure of the system, we obtain a new blow-up result for strong solutions with sufficient conditions on the initial data and parameters. The entire proof procedure relies upon a newly derived transport equation which is involved in nonlocal velocity term along the characteristic curves.

Comments

https://doi.org/10.1016/j.jde.2019.08.043

Recommended Citation

Zhang, Lei, and Zhijun Qiao. "The periodic Cauchy problem for a two-component non-isospectral cubic Camassa-Holm system." Journal of Differential Equations 268, no. 3 (2020): 1270-1305. https://doi.org/10.1016/j.jde.2019.08.043

Publication Title

Journal of Differential Equations

DOI

10.1016/j.jde.2019.08.043