On the Cauchy Problem of Fractional Schrödinger Equation with Hartree Type Nonlinearity
On the Cauchy Problem of Fractional Schrödinger Equation with Hartree Type Nonlinearity
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DOI：
10.1619/fesi.56.193
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发表时间：
2012-09
期刊：
影响因子：
3.7
通讯作者：
Yonggeun Cho;Gyeongha Hwang;H. Hajaiej;T. Ozawa
中科院分区：
文献类型：
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作者：
Yonggeun Cho;Gyeongha Hwang;H. Hajaiej;T. Ozawa
研究分数阶Schr dinger方程i\partial_tu =（m^2-\Delta）^\frac\alpha2 u + F（u）在\mathbb{R}^{1+n}，$$中的Cauchy问题，其中$ n \ge 1$，$m \ge 0$，$1 < \alpha < 2$，$F$表示Hartree型非线性：$$F（u）= \lambda（\frac{\psi（\cdot）}{|\cdot| ^\gamma} *| u| ^2）u$$，其中$\lambda = \pm1，0 <\gamma < n$，并且$0 \le \psi \in L^\infty（\mathbb R^n）$。我们证明了某些$\alpha$，$\gamma$，$\lambda$，$\psi$的局部解和整体解的存在唯一性。我们还注意到当$\lambda =-1$时解的有限时间爆破。
We study the Cauchy problem for the fractional Schr\"{o}dinger equation $$ i\partial_tu = (m^2-\Delta)^\frac\alpha2 u + F(u) in \mathbb{R}^{1+n}, $$ where $ n \ge 1$, $m \ge 0$, $1 < \alpha < 2$, and $F$ stands for the nonlinearity of Hartree type: $$F(u) = \lambda (\frac{\psi(\cdot)}{|\cdot|^\gamma} * |u|^2)u$$ with $\lambda = \pm1, 0 <\gamma < n$, and $0 \le \psi \in L^\infty(\mathbb R^n)$. We prove the existence and uniqueness of local and global solutions for certain $\alpha$, $\gamma$, $\lambda$, $\psi$. We also remark on finite time blowup of solutions when $\lambda = -1$.