The Cauchy problem for the energy-critical inhomogeneous nonlinear Schrödinger equation

Yoonjung Lee; Ihyeok Seo

doi:10.1007/s00013-021-01632-x

The Cauchy problem for the energy-critical inhomogeneous nonlinear Schrödinger equation

Yoonjung Lee
, Ihyeok Seo

Department of Mathematics

Sungkyunkwan University

Research output: Contribution to journal › Article › peer-review

13 Scopus citations

Abstract

In this paper, we study the Cauchy problem for the energy-critical inhomogeneous nonlinear Schrödinger equation i∂_tu+ Δ u= λ| x| ^-^α| u| ^βu in H¹. The well-posedness theory in H¹ has been intensively studied in recent years, but the currently known approaches do not work for the critical case β= (4 - 2 α) / (n- 2). It is still an open problem. The main contribution of this paper is to develop the theory in this case.

Original language	English
Pages (from-to)	441-453
Number of pages	13
Journal	Archiv der Mathematik
Volume	117
Issue number	4
DOIs	https://doi.org/10.1007/s00013-021-01632-x
State	Published - Oct 2021

Keywords

Nonlinear Schrödinger equations
Weighted estimates
Well-posedness

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10.1007/s00013-021-01632-x

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title = "The Cauchy problem for the energy-critical inhomogeneous nonlinear Schr{\"o}dinger equation",

abstract = "In this paper, we study the Cauchy problem for the energy-critical inhomogeneous nonlinear Schr{\"o}dinger equation i∂tu+ Δ u= λ| x| -α| u| βu in H1. The well-posedness theory in H1 has been intensively studied in recent years, but the currently known approaches do not work for the critical case β= (4 - 2 α) / (n- 2). It is still an open problem. The main contribution of this paper is to develop the theory in this case.",

keywords = "Nonlinear Schr{\"o}dinger equations, Weighted estimates, Well-posedness",

author = "Yoonjung Lee and Ihyeok Seo",

note = "Publisher Copyright: {\textcopyright} 2021, Springer Nature Switzerland AG.",

year = "2021",

month = oct,

doi = "10.1007/s00013-021-01632-x",

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T1 - The Cauchy problem for the energy-critical inhomogeneous nonlinear Schrödinger equation

AU - Lee, Yoonjung

AU - Seo, Ihyeok

PY - 2021/10

Y1 - 2021/10

N2 - In this paper, we study the Cauchy problem for the energy-critical inhomogeneous nonlinear Schrödinger equation i∂tu+ Δ u= λ| x| -α| u| βu in H1. The well-posedness theory in H1 has been intensively studied in recent years, but the currently known approaches do not work for the critical case β= (4 - 2 α) / (n- 2). It is still an open problem. The main contribution of this paper is to develop the theory in this case.

AB - In this paper, we study the Cauchy problem for the energy-critical inhomogeneous nonlinear Schrödinger equation i∂tu+ Δ u= λ| x| -α| u| βu in H1. The well-posedness theory in H1 has been intensively studied in recent years, but the currently known approaches do not work for the critical case β= (4 - 2 α) / (n- 2). It is still an open problem. The main contribution of this paper is to develop the theory in this case.

KW - Nonlinear Schrödinger equations

KW - Weighted estimates

KW - Well-posedness

UR - https://www.scopus.com/pages/publications/85108822807

U2 - 10.1007/s00013-021-01632-x

DO - 10.1007/s00013-021-01632-x

M3 - Article

AN - SCOPUS:85108822807

SN - 0003-889X

VL - 117

SP - 441

EP - 453

JO - Archiv der Mathematik

JF - Archiv der Mathematik

IS - 4

ER -

The Cauchy problem for the energy-critical inhomogeneous nonlinear Schrödinger equation

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