On the Cauchy problem for the integrable modified Camassa-Holm equation with cubic nonlinearity

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:255

Article

Fu, Ying¹  Gui, Guilong¹  Liu, Yue^2,3  Qu, Changzheng³ 

[1] Northwest Univ, Dept Math, Xian 710069, Peoples R China

[2] Univ Texas Arlington, Dept Math, Arlington, TX 76019 USA

[3] Ningbo Univ, Dept Math, Ningbo 315211, Zhejiang, Peoples R China

关键词: Modified Camassa-Holm equation;    Besov space;    Local well-posedness;    Blow-up;    Traveling waves;   

DOI  :  10.1016/j.jde.2013.05.024

来源: Elsevier

【 摘 要 】

Considered in this paper is the modified Camassa-Holm equation with cubic nonlinearity, which is integrable and admits the single peaked solitons and multi-peakon solutions. The short-wave limit of this equation is known as the short-pulse equation. The main investigation is the Cauchy problem of the modified Camassa-Holm equation with qualitative properties of its solutions. It is firstly shown that the equation is locally well-posed in a range of the Besov spaces. The blow-up scenario and the lower bound of the maximal time of existence are then determined. A blow-up mechanism for solutions with certain initial profiles is described in detail and nonexistence of the smooth traveling wave solutions is also demonstrated. (c) 2013 Elsevier Inc. All rights reserved.

【 授权许可】

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