Existence of a minimal non-scattering solution to the mass-subcritical generalized Korteweg–de Vries equation

Satoshi Masaki, Junichi Segata

Research output: Contribution to journalArticle

4 Citations (Scopus)

Abstract

In this article, we prove the existence of a non-scattering solution, which is minimal in some sense, to the mass-subcritical generalized Korteweg–de Vries (gKdV) equation in the scale critical Lˆr space where Lˆr={f∈S(R)|‖f‖r =‖fˆ‖Lr <∞}. We construct this solution by a concentration compactness argument. Then, key ingredients are a linear profile decomposition result adopted to Lˆr-framework and approximation of solutions to the gKdV equation which involves rapid linear oscillation by means of solutions to the nonlinear Schrödinger equation.

Original languageEnglish
Pages (from-to)283-326
Number of pages44
JournalAnnales de l'Institut Henri Poincare (C) Analyse Non Lineaire
Volume35
Issue number2
DOIs
Publication statusPublished - Mar 1 2018
Externally publishedYes

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Korteweg-de Vries Equation
Generalized Equation
Nonlinear equations
Concentration-compactness
Decomposition
Nonlinear Equations
Oscillation
Decompose
Approximation
Framework
Profile

All Science Journal Classification (ASJC) codes

  • Analysis
  • Mathematical Physics

Cite this

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AB - In this article, we prove the existence of a non-scattering solution, which is minimal in some sense, to the mass-subcritical generalized Korteweg–de Vries (gKdV) equation in the scale critical Lˆr space where Lˆr={f∈S′(R)|‖f‖Lˆr =‖fˆ‖Lr′ <∞}. We construct this solution by a concentration compactness argument. Then, key ingredients are a linear profile decomposition result adopted to Lˆr-framework and approximation of solutions to the gKdV equation which involves rapid linear oscillation by means of solutions to the nonlinear Schrödinger equation.

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