On the scattering of subcritical defocusing generalized Korteweg-de Vries equation

Taegyu Kim

doi:10.3934/dcds.2023009

Article Contents

2023, Volume 43, Issue 6: 2241-2269. Doi: 10.3934/dcds.2023009

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On the scattering of subcritical defocusing generalized Korteweg-de Vries equation

Taegyu Kim

Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, Republic of Korea

Received: December 2021

Revised: December 2022

Early access: February 10, 2023

Published online: February 10, 2023

The author is partially supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (NRF-2019R1A5A1028324).

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Abstract

In this paper, we consider the scattering problem of the mass-subcritical defocusing generalized Korteweg-de Vries equation $ \partial_tu+\partial_{xxx}u-\partial_x(|u|^{2\alpha}u) = 0 $ in a scaling critical space $ |\partial_x|^{-\sigma}\hat{M}^{\beta}_{2,\delta} $ which is a Morrey type space. We prove that if the solution $ u $ satisfies a priori bound

$ \bigg\|\big\|u\big\|_{\langle\partial_x\rangle^{-\sigma}\hat{M}^{\beta}_{2,\delta} }\bigg\|_{L_t^\infty }<\infty, $

then $ u $ is global and scatters. To prove this, we follow Dodson's argument in [2]. We combine Tao's monotonicity formula [21] and the existence of minimal blowup solution adopted to $ |\partial_x|^{-\sigma}\hat{M}^{\beta}_{2,\delta} $-framework [18] which is done by Masaki and Segata.

Keywords:

Generalized Korteweg-de Vries equation,
scattering problem.

Mathematics Subject Classification: Primary: 35B40, 35Q53.

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References

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