Blow-up phenomena and local well-posedness for a generalized Camassa–Holm equation in the critical Besov space. (November 2015)

• Record Type: Journal Article
• Authors: Tu, Xi; Yin, Zhaoyang
• Is Part Of: Nonlinear analysis. Volume 128(2015)
• Page Start: 1
• Page End: 19
• Publication Date: 2015-11
• Subjects: 35Q53 -- 35A01 -- 35B44 -- 35B65; A generalized Camassa–Holm equation -- Local well-posedness -- The critical Besov space -- Blow-up; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2015.07.017
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper we mainly study the Cauchy problem for a generalized Camassa–Holm equation in a critical Besov space. First, by using the Littlewood–Paley decomposition, transport equations theory, logarithmic interpolation inequalities and Osgood's lemma, we establish the local well-posedness for the Cauchy problem of the equation in the critical Besov space B 2, 1 1 2 . Then we give a new blow-up criterion for the Cauchy problem of the equation. Finally, we present a new blow-up result and the exact blow-up rate of strong solutions to the equation by making use of the conservation law and the obtained blow-up criterion.