A higher dispersion KdV equation on the line. (October 2020)

• Record Type: Journal Article
• Authors: Figueira, Renata; Himonas, A. Alexandrou; Yan, Fangchi
• Is Part Of: Nonlinear analysis. Volume 199(2020)
• Publication Date: 2020-10
• Subjects: primary 35Q53; Korteweg–deVries equation -- Higher dispersion -- Initial value problem -- Well-posedness -- Analytic Gevrey spaces -- Uniform radius of spatial analyticity -- Sobolev spaces -- Bilinear estimates -- Bourgain spaces; 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.2020.112055
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: The Cauchy problem for a Korteweg–deVries equation with dispersion of order m = 2 j + 1, where j is a positive integer, (KdVm), is studied with data in Sobolev and analytic spaces. First, optimal bilinear estimates in Bourgain spaces are proved and using them well-posedness in Sobolev spaces H s, s > − j + 1 4, is established. Then, well-posedness in analytic Gevrey spaces G δ, s, δ > 0, is proved by using an analytic version of the bilinear estimates. This implies that the uniform radius of analyticity persist for some time. For the later times a lower bound for the radius of spacial analyticity is derived, which is given by δ ( t ) ≥ c t − α, with α = 4 3 + ε, for any ε > 0, when j = 1, and α = 1 when j ≥ 2 . Finally, it is shown that the regularity of the solution in the time variable is Gevrey of order m, and this is optimal.