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http://dx.doi.org/10.4134/BKMS.b161006

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)
Publication Information
Bulletin of the Korean Mathematical Society / v.55, no.1, 2018 , pp. 267-296 More about this Journal
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.
Keywords
Cauchy problem; generalized Camassa-Holm equation; nonhomogeneous Besov space; cubic nonlinearity; blow-up phenomena;
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