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arXiv 2107.00795math.AP
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$H^{s} (\mathbb R^{n} )$中非齐次非线性薛定谔方程Cauchy问题的连续依赖性

Continuous dependence of the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation in $H^{s} (\mathbb R^{n} )$

JinMyong An, JinMyong Kim, KyuSong Chae

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31 pages
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摘要：

考虑非齐次非线性薛定谔方程\[iu_{t} +Δu=|x|^{-b} f(u),\;u(0)\in H^{s} (\mathbb R^{n} ),\]的Cauchy问题，其中$n\in \mathbb N$，$0<s<\min \{ n,\; 1+n/2\} $，$0<b<\min \{ 2,\;n-s,\;1+\frac{n-2s}{2} \} $和$f(u)$是一个非线性函数，其行为类似于$λ\left|u\right|^{σ} u$，具有$σ>0$和$λ\in \mathbb C$ .最近，An-Kim \cite{AK21}用$0\le s<\min \{ n,\; 1+n/2\}$证明了$H^{s}(\mathbb R^{n} )$解的局部存在性。然而，即使该解决方案是由一个固定点技术，连续的$H^{s}(\mathbb R^{n} )$与$0< s<\min \{ n,\; 1+n/2\}$的标准意义上的依赖性不遵循压缩映射参数。在本文中，我们表明，该解决方案连续依赖于初始数据在$H^{s}(\mathbb R^{n} )$的标准意义下，即在这个意义上，当地的解决方案流是连续的$H^{s}(\mathbb R^{n} )\to H^{s}(\mathbb R^{n} )$，如果$σ$满足一定的假设。

英文摘要：

We consider the Cauchy problem for the inhomogeneous nonlinear Schrödinger (INLS) equation \[iu_{t} +Δu=|x|^{-b} f(u),\;u(0)\in H^{s} (\mathbb R^{n} ),\] where $n\in \mathbb N$, $0<s<\min \{ n,\; 1+n/2\} $, $0<b<\min \{ 2,\;n-s,\;1+\frac{n-2s}{2} \} $ and $f(u)$ is a nonlinear function that behaves like $λ\left|u\right|^{σ} u$ with $σ>0$ and $λ\in \mathbb C$. Recently, An--Kim \cite{AK21} proved the local existence of solutions in $H^{s}(\mathbb R^{n} )$ with $0\le s<\min \{ n,\; 1+n/2\}$. However even though the solution is constructed by a fixed point technique, continuous dependence in the standard sense in $H^{s}(\mathbb R^{n} )$ with $0< s<\min \{ n,\; 1+n/2\}$ doesn't follow from the contraction mapping argument. In this paper, we show that the solution depends continuously on the initial data in the standard sense in $H^{s}(\mathbb R^{n} )$, i.e. in the sense that the local solution flow is continuous $H^{s}(\mathbb R^{n} )\to H^{s}(\mathbb R^{n} )$, if $σ$ satisfies certain assumptions.

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