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하트리유형의 비선형 분수슈뢰딩거 방정식에 관하여

Title
하트리유형의 비선형 분수슈뢰딩거 방정식에 관하여
Authors
황경하
Date Issued
2012
Publisher
포항공과대학교
Abstract
In this dissertation we consider for the fractional Schrödinger equation iut = (-Δ)^α/2 u + F(u) in R^1+n, n ≥ 1 with the Lévy index 1 < α < 2 and the nonlinearity F(u) = λ(jxj^-ν* juj^2)u
0 < ν < n. In Chapter 1 we study the Cauchy problem for the fractional Schröodinger equation. We prove the existence and uniqueness of local and global solutions for certain α and ν. We also remark on finite time blowup of solutions when λ = -1. In Chapter 2 we develop a profile decomposition of fractional Schröodinger equation with Lévvy index 1 < α < 2. One the main difficulty is the non-locality of fractional operator which causes the lack of Galilean invariance. The second one is the regularity loss of Strichartz estimate stemming from the low index α < 2. To overcome these difficulties we assume radial symmetry and use the recently developed Strichartz estimates. We will apply the profile decomposition to the blowup profile of fractional Hartree equations.
URI
http://postech.dcollection.net/jsp/common/DcLoOrgPer.jsp?sItemId=000001396260
http://oasis.postech.ac.kr/handle/2014.oak/1698
Article Type
Thesis
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