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arXiv 1209.5899math.AP
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具有Hartree型非线性项的分数阶薛定谔方程的Cauchy问题

On the Cauchy problem of fractional Schrödinger equation with Hartree type nonlinearity

Yonggeun Cho, Gyeongha Hwang, Hichem Hajaiej, Tohru Ozawa

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摘要：

研究了分数阶薛定谔方程$$ i\partial_tu = (m^2-Δ)^\frac\alpha2 u + F(u) in \mathbb{R}^{1+n}, $$的Cauchy问题，其中$ n \ge 1$，$m \ge 0$，$1 < α< 2$，$F$代表Hartree型非线性：$$F(u) = λ(\frac{ψ(\cdot)}{|\cdot|^γ} * |u|^2)u$$ with $λ= \pm1, 0 <γ< n$，以及$0 \le ψ\in L^\infty(\mathbb R^n)$ .我们证明了某些$α$，$γ$，$λ$，$ψ$的局部和整体解的存在性和唯一性。我们还注意到当$λ= -1$。

英文摘要：

We study the Cauchy problem for the fractional Schrödinger equation $$ i\partial_tu = (m^2-Δ)^\frac\alpha2 u + F(u) in \mathbb{R}^{1+n}, $$ where $ n \ge 1$, $m \ge 0$, $1 < α< 2$, and $F$ stands for the nonlinearity of Hartree type: $$F(u) = λ(\frac{ψ(\cdot)}{|\cdot|^γ} * |u|^2)u$$ with $λ= \pm1, 0 <γ< n$, and $0 \le ψ\in L^\infty(\mathbb R^n)$. 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 $λ= -1$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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