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The Cauchy problem for an integrable generalized Camassa-Holm equation with cubic nonlinearity

Lei Zhang|Bin Liu
대한수학회 (The Korean Mathematical Society)|2018.01|대한수학회보 (Bulletin of the KMS)|제55권 제1호|pp. 267-296

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참고문헌

키워드

Cauchy problemgeneralized Camassa-Holm equationnonhomogeneous Besov spacecubic nonlinearityblow-up phenomena
주제분류수학

초록

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.

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.

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