# THE CAUCHY PROBLEM FOR AN INTEGRABLE GENERALIZED CAMASSA-HOLM EQUATION WITH CUBIC NONLINEARITY

• Liu, Bin (School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology) ;
• Zhang, Lei (School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology)
• Accepted : 2017.05.23
• Published : 2018.01.31

#### Abstract

This paper studies the Cauchy problem and blow-up phenomena for a new generalized Camassa-Holm equation with cubic nonlinearity in the nonhomogeneous Besov spaces. First, by means of the Littlewood-Paley decomposition theory, we investigate the local well-posedness of the equation in $B^s_{p,r}$ with s > $max\{{\frac{1}{p}},\;{\frac{1}{2}},\;1-{\frac{1}{p}}\},\;p,\;r{\in}[0,{\infty}]$. Second, we prove that the equation is locally well-posed in $B^s_{2,r}$ with the critical index $s={\frac{1}{2}}$ by virtue of the logarithmic interpolation inequality and the Osgood's Lemma, and it is shown that the data-to-solution mapping is $H{\ddot{o}}lder$ continuous. Finally, we derive two kinds of blow-up criteria for the strong solution by using induction and the conservative property of m along the characteristics.

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