# American Institute of Mathematical Sciences

May  2014, 13(3): 1283-1304. doi: 10.3934/cpaa.2014.13.1283

## On the Cauchy problem for a generalized Camassa-Holm equation with both quadratic and cubic nonlinearity

 1 Department of mathematics, China University of Mining and Technology, Xuzhou, Jiangsu 221116, China 2 Department of Mathematics, University of Texas-Pan American, Edinburg, Texas 78541, United States 3 Department of Mathematics, Zhongshan University, 510275 Guangzhou, China

Received  June 2013 Revised  October 2013 Published  December 2013

In this paper, we study the Cauchy problem for a generalized integrable Camassa-Holm equation with both quadratic and cubic nonlinearity. By overcoming the difficulties caused by the complicated mixed nonlinear structure, we firstly establish the local well-posedness result in Besov spaces, and then present a precise blow-up scenario for strong solutions. Furthermore, we show the existence of single peakon by the method of analysis.
Citation: Xingxing Liu, Zhijun Qiao, Zhaoyang Yin. On the Cauchy problem for a generalized Camassa-Holm equation with both quadratic and cubic nonlinearity. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1283-1304. doi: 10.3934/cpaa.2014.13.1283
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