Radius of analyticity for the Camassa–Holm equation on the line. (September 2018)

• Record Type: Journal Article
• Authors: Himonas, A. Alexandrou; Petronilho, Gerson
• Is Part Of: Nonlinear analysis. Volume 174(2018)
• Page Start: 1
• Page End: 16
• Publication Date: 2018-09
• Subjects: primary 35Q53 37K10 35C07; Camassa–Holm equation -- Cauchy problem -- Analytic data -- Global in time solutions -- Analytic spaces -- Exponential lower bound on the radius of analyticity; 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.2018.04.007
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: Using estimates in Sobolev spaces we prove that the solution to the Cauchy problem for the Camassa–Holm equation on the line with analytic initial data u 0 ( x ) and satisfying the McKean condition, that is the quantity m 0 ( x ) = ( 1 − ∂ x 2 ) u 0 ( x ) does not change sign, is analytic in the spatial variable for all time. Furthermore, we obtain explicit lower bounds for the radius of spatial analyticity r ( t ) given by r ( t ) ≥ A − 1 ( 1 + C 1 B t ) − 1 exp { − C 0 ‖ u 0 ‖ H 1 t }, where A, B, C 1 and C 0 are suitable positive constants.