The Cauchy problem for an integrable generalized Camassa-Holm equation with cubic nonlinearity
Bull. Korean Math. Soc. 2018 Vol. 55, No. 1, 267-296
https://doi.org/10.4134/BKMS.b161006
Published online January 31, 2018
Bin Liu, Lei Zhang
Huazhong University of Science and Technology, Huazhong University of Science and Technology
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 $B_{p,r}^s$ 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 $B_{2,r}^s$ 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\"{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.
Keywords : Cauchy problem, generalized Camassa-Holm equation, nonhomogeneous Besov space, cubic nonlinearity, blow-up phenomena
MSC numbers : 37L05, 35G25, 35Q35, 35A10
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