The Cauchy problem and blow-up phenomena for a new integrable two-component peakon system with cubic nonlinearities

Xiuting Li; Lei Zhang

doi:10.3934/dcds.2017140

Article Contents

2017, Volume 37, Issue 6: 3301-3325. Doi: 10.3934/dcds.2017140

This issue Previous Article Local well-posedness of the Camassa-Holm equation on the real line Next Article On the positive vector solutions for nonlinear fractional Laplacian systems with linear coupling

The Cauchy problem and blow-up phenomena for a new integrable two-component peakon system with cubic nonlinearities

Xiuting Li^1, and
Lei Zhang^2, ,

1.
School of Automation, Huazhong University of Science and Technology, Wuhan 430074, Hubei, China
2.
School of Mathematics and Statistics, Huazhong University of Science, and Technology, Wuhan 430074, Hubei, China

* Corresponding author: Lei Zhang
* Corresponding author: Lei Zhang

Received: October 31, 2016

Revised: December 31, 2016

Published: May 2017

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Abstract

This paper studies the local well-posedness and blow-up phenomena for a new integrable two-component peakon system in the Besov space. Firstly, by utilizing the Littlewood-Paley theory, the logarithmic interpolation inequality and the Osgood's Lemma, we investigate the existence and uniqueness of the solution to the system in the critical Besov space $B_{2, 1}^{\frac{1}{2}}(\mathbb{R})× B_{2, 1}^{\frac{1}{2}}(\mathbb{R})$, and show that the data-to-solution mapping is Hölder continuous. Secondly, we derive a blow-up criteria for the Cauchy problem in the critical Besov space. Finally, with suitable conditions on the initial data, a new blow-up criteria for the system is obtained by virtue of the global conservative property of the potential densities $m$ and $n$ along the characteristics and the blow-up criteria at hand.

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Mathematics Subject Classification: Primary: 35R10, 35G20, 35D35, 35B44.

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