하트리유형의 비선형 분수슈뢰딩거 방정식에 관하여
- 하트리유형의 비선형 분수슈뢰딩거 방정식에 관하여
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- 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*
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.
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