On the cauchy problem of fractional schrödinger equation with hartree type nonlinearity

Yonggeun Cho; Hichem Hajaiej; Gyeongha Hwang; Tohru Ozawa

doi:10.1619/fesi.56.193

On the cauchy problem of fractional schrödinger equation with hartree type nonlinearity

Yonggeun Cho^*
, Hichem Hajaiej
, Gyeongha Hwang
, Tohru Ozawa

^*Corresponding author for this work

Department of Mathematics

King Saud University
Seoul National University
Waseda University

Research output: Contribution to journal › Journal article › peer-review

100 Scopus citations

Abstract

We study the Cauchy problem for the fractional Schrodinger equation iδ_tu = (m² -Δ) ^α/2u + F(u) in R¹⁺ⁿ, where n ≥1, m ≥ 0, 1 < α < 2, and F stands for the nonlinearity of Hartree type F(u)-λ(ψ(.)|•|^-γ*|u|²)u with λ= ±1, 0 < γ < n, and 0 ≤ ψ ∈ L^∞(Rⁿ). We prove the existence and uniqueness of local and global solutions for certain α, γ, λ, ψ. We also remark on finite time blowup of solutions when λ =-1.

Original language	English
Pages (from-to)	193-224
Number of pages	32
Journal	Funkcialaj Ekvacioj
Volume	56
Issue number	2
DOIs	https://doi.org/10.1619/fesi.56.193
State	Published - 2013.08.26

Keywords

Finite time blowup
Fractional schrödinger equation
Hartree type nonlinearity
Strichartz estimates

Quacquarelli Symonds(QS) Subject Topics

Mathematics

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10.1619/fesi.56.193

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@article{1eea8b7c3a094ae29e8cfe7cfa98f603,

title = "On the cauchy problem of fractional schr{\"o}dinger equation with hartree type nonlinearity",

abstract = "We study the Cauchy problem for the fractional Schrodinger equation iδtu = (m2 -Δ) α/2u + F(u) in R1+n, where n ≥1, m ≥ 0, 1 < α < 2, and F stands for the nonlinearity of Hartree type F(u)-λ(ψ(.)|•|-γ*|u|2)u with λ= ±1, 0 < γ < n, and 0 ≤ ψ ∈ L∞(Rn). We prove the existence and uniqueness of local and global solutions for certain α, γ, λ, ψ. We also remark on finite time blowup of solutions when λ =-1.",

keywords = "Finite time blowup, Fractional schr{\"o}dinger equation, Hartree type nonlinearity, Strichartz estimates",

author = "Yonggeun Cho and Hichem Hajaiej and Gyeongha Hwang and Tohru Ozawa",

year = "2013",

month = aug,

day = "26",

doi = "10.1619/fesi.56.193",

language = "English",

volume = "56",

pages = "193--224",

journal = "Funkcialaj Ekvacioj",

issn = "0532-8721",

number = "2",

}

TY - JOUR

T1 - On the cauchy problem of fractional schrödinger equation with hartree type nonlinearity

AU - Cho, Yonggeun

AU - Hajaiej, Hichem

AU - Hwang, Gyeongha

AU - Ozawa, Tohru

PY - 2013/8/26

Y1 - 2013/8/26

N2 - We study the Cauchy problem for the fractional Schrodinger equation iδtu = (m2 -Δ) α/2u + F(u) in R1+n, where n ≥1, m ≥ 0, 1 < α < 2, and F stands for the nonlinearity of Hartree type F(u)-λ(ψ(.)|•|-γ*|u|2)u with λ= ±1, 0 < γ < n, and 0 ≤ ψ ∈ L∞(Rn). We prove the existence and uniqueness of local and global solutions for certain α, γ, λ, ψ. We also remark on finite time blowup of solutions when λ =-1.

AB - We study the Cauchy problem for the fractional Schrodinger equation iδtu = (m2 -Δ) α/2u + F(u) in R1+n, where n ≥1, m ≥ 0, 1 < α < 2, and F stands for the nonlinearity of Hartree type F(u)-λ(ψ(.)|•|-γ*|u|2)u with λ= ±1, 0 < γ < n, and 0 ≤ ψ ∈ L∞(Rn). We prove the existence and uniqueness of local and global solutions for certain α, γ, λ, ψ. We also remark on finite time blowup of solutions when λ =-1.

KW - Finite time blowup

KW - Fractional schrödinger equation

KW - Hartree type nonlinearity

KW - Strichartz estimates

UR - https://www.scopus.com/pages/publications/84883184429

U2 - 10.1619/fesi.56.193

DO - 10.1619/fesi.56.193

M3 - Journal article

AN - SCOPUS:84883184429

SN - 0532-8721

VL - 56

SP - 193

EP - 224

JO - Funkcialaj Ekvacioj

JF - Funkcialaj Ekvacioj

IS - 2

ER -

On the cauchy problem of fractional schrödinger equation with hartree type nonlinearity

Abstract

Keywords

Quacquarelli Symonds(QS) Subject Topics

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