We study the time of existence of the solutions of the following nonlinear Schrödinger equation (NLS)$$\begin{aligned} \hbox {i}u_t =(-\Delta +m)^su - |u|^2u \end{aligned}$$on the finite x-interval \([0,\pi ]\) with Dirichlet boundary conditions$$\begin{aligned} u(t,0)=0=u(t,\pi ),\qquad -\infty< t<+\infty , \end{aligned}$$where \((-\Delta +m)^s\) stands for the spectrally defined fractional Laplacian with \(0<s<1/2\). We prove an almost global existence result for the above fractional Schrödinger equation, which generalizes the result in Bambusi and Sire (Dyn PDE 10(2):171–176, 2013) from \(s>1/2\) to \(0<s<1/2\).

中文翻译：

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分数阶Schrödinger方程的几乎整体存在性

我们研究以下非线性Schrödinger方程（NLS）$$ \ begin {aligned} \ hbox {i} u_t =（-\ Delta + m）^ su-| u | ^ 2u \ end {对齐} $$在有限X -interval \（[0，\ PI] \）与狄利克雷边界条件$$ \ {开始对准} U（T，0）= 0 = U（T，\ PI），\ qquad -\ infty <t <+ \ infty，\ end {aligned} $$其中\（（-\ Delta + m）^ s \）代表光谱定义的分数拉普拉斯算子，其中\（0 <s <1/2 \）。我们证明了上面的分数薛定ding方程的一个几乎全局存在的结果，该结果将Bambusi和Sire（Dyn PDE 10（2）：171–176，2013）中的结果从\（s> 1/2 \）推广到\（0 <s <1/2 \）。