A higher dispersion KdV equation on the line

Author(s)

Figueira, Renata ^[1] ; Himonas, A. Alexandrou ^[2] ; Yan, Fangchi ^[2]

Affiliation

^[1] Univ Fed Sao Carlos, Dept Math, BR-13565905 Sao Carlos, SP - Brazil

^[2] Univ Notre Dame, Dept Math, Notre Dame, IN 46556 - USA

• Document type: Journal article
• Source: NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS; v. 199, OCT 2020.
• Web of Science Citations: 1

Abstract

The Cauchy problem for a Korteweg-deVries equation with dispersion of order m = 2j + 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(delta,s), delta > 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 delta(t) >= ct(-alpha), with alpha = 4/3 + epsilon, for any epsilon > 0, when j = 1, and alpha = 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. (C) 2020 Elsevier Ltd. All rights reserved. (AU)

• FAPESP's process: 17/12499-0 - Analytic and Gevrey well-posedness for the "good" Boussinesq equation
• Grantee: Renata de Oliveira Figueira
• Support Opportunities: Scholarships abroad - Research Internship - Doctorate
• FAPESP's process: 15/24109-7 - Analytic and Gevrey well-posedness of the "good" Boussinesq equation
• Grantee: Renata de Oliveira Figueira
• Support Opportunities: Scholarships in Brazil - Doctorate