Kinetic and Related Models (KRM)

On a three-Component Camassa-Holm equation with peakons

Pages: 305 - 339, Volume 7, Issue 2, June 2014      doi:10.3934/krm.2014.7.305

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Yongsheng Mi - College of Mathematics and and Statistics, Chongqing University, Chongqing, 401331, China (email)
Chunlai Mu - College of Mathematics and and Statistics, Chongqing University, Chongqing 401331, China (email)

Abstract: In this paper, we are concerned with three-Component Camassa-Holm equation with peakons. First, We establish the local well-posedness in a range of the Besov spaces $B^{s}_{p,r},p,r\in [1,\infty],s>\mathrm{ max}\{\frac{3}{2},1+\frac{1}{p}\}$ (which generalize the Sobolev spaces $H^{s}$) by using Littlewood-Paley decomposition and transport equation theory. Second, the local well-posedness in critical case (with $s=\frac{3}{2}, p=2,r=1$) is considered. Then, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time. Finally, we consider the initial boundary value problem, our approach is based on sharp extension results for functions on the half-line and several symmetry preserving properties of the equations under discussion.

Keywords:  Besov spaces, Camassa-Holm type equation, local well-posedness.
Mathematics Subject Classification:  35B30, 35G25, 35A10, 35Q53.

Received: March 2013;      Revised: January 2014;      Available Online: March 2014.