The Cauchy problem for a family of generalized Camassa–Holm equations. Issue 6 (2nd June 2016)

• Record Type: Journal Article
• Authors: Tu, Xi; Yin, Zhaoyang
• Is Part Of: Applicable analysis. Volume 95:Issue 6(2016)
• Page Start: 1184
• Page End: 1213
• Publication Date: 2016-06-02
• Subjects: a family of generalized Camassa–Holm equations -- local well-posedness -- global existence -- blow-up; 35Q53 -- 35A01 -- 35B44 -- 35B65; Mathematical analysis -- Periodicals; 515
• DOI: 10.1080/00036811.2015.1055730
• Languages: English
• ISSNs: 0003-6811

Abstract

Abstract : In this paper, we first use the energy method to establish the local well-posedness for a family of generalized Camassa–Holm equations in Sobolev spaces. Next, we give a precise blow-up criterion for the Cauchy problem on the family of equations. Then, we show the global existence of strong solutions to the family of equations under some certain sign conditions. Finally, we prove that the family of equations possesses some wave-breaking phenomena.