Neumann problem for the Korteweg-de Vries equation

Nakao Hayashi, Elena I. Kaikina

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)

Abstract

We consider Neumann initial-boundary value problem for the Korteweg-de Vries equation on a half-line{A formula is presented} We prove that if the initial data u0 ∈ H1 0, frac(21,4) ∩ H21, frac(7, 2) and the norm {norm of matrix} u0 {norm of matrix}H10, frac(21, 4) + {norm of matrix} u0 {norm of matrix}H21, frac(7, 2) {less-than or slanted equal to} ε, where ε > 0 is small enough {Mathematical expression}, 〈 x 〉 = sqrt(1 + x2) and λ ∫0 x u0 ( x ) d x = λ θ < 0. Then there exists a unique solution u ∈ C ( [ 0, ∞ ), H21, frac(7, 2) ) ∩ L2 ( 0, ∞ ; H22, 3 ) of the initial-boundary value problem (0.1). Moreover there exists a constant C such that the solution has the following asymptotics{A formula is presented} for t → ∞ uniformly with respect to x > 0, where η = - 9 θ λ ∫0 A i′ 2 ( z ) d z and A i ( q ) is the Airy function{A formula is presented}.

Original languageEnglish
Pages (from-to)168-201
Number of pages34
JournalJournal of Differential Equations
Volume225
Issue number1
DOIs
Publication statusPublished - 2006 Jun 1
Externally publishedYes

Keywords

  • Half-line
  • Korteweg-de Vries equation
  • Large time asymptotics
  • Nonlinear evolution equation

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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