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

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 compactne...

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Bibliographic Details
Published inAnnales de l'Institut Henri Poincaré. Analyse non linéaire Vol. 35; no. 2; pp. 283 - 326
Main Authors Masaki, Satoshi, Segata, Jun-ichi
Format Journal Article
LanguageEnglish
Published Elsevier Masson SAS 01.03.2018
Subjects
Generalized Korteweg–de Vries equation
Scattering problem
Threshold solution
secondary
Generalized Korteweg–de Vries equation
Threshold solution
Scattering problem
primary
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Summary: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.
ISSN:0294-1449
1873-1430
DOI:10.1016/j.anihpc.2017.04.003