Local well-posedness for the Cauchy problem of the quadratic Schrödinger equation with nonlinearity $\bar u^2$
Local well-posedness for the Cauchy problem of the quadratic Schrödinger equation with nonlinearity $\bar u^2$
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非线性二次薛定谔方程柯西问题的局部适定性 $ar u^2$
DOI：
10.3934/cpaa.2008.7.1123
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发表时间：
2008
影响因子：
1
通讯作者：
Nobu Kishimoto
中科院分区：
文献类型：
--
作者：
Nobu Kishimoto
证明了一维二次非线性薛定谔方程的局部适定性
$ iu_t+u_{x x}=\bar u^2$
在$H^s(\mathbb R)$中表示$s\ge -1$，在$H^{-1}$下面表示不适。I. Bejenaru和T. Tao给出了另一个二次非线性$u^2$的相同结果，二次非线性薛定谔方程的Sharp适定性和病态性结果，J. Funct。但解的函数空间很大程度上依赖于非线性的特殊性质$u^2$。我们构造了适合于非线性的解空间$\bar u^2$。
We prove the local well-posedness of a 1-D quadratic nonlinear Schrodinger equation
$ iu_t+u_{x x}=\bar u^2$
in $H^s(\mathbb R)$ for $s\ge -1$ and ill-posedness below $H^{-1}$. The same result for another quadratic nonlinearity $u^2$ was given by I. Bejenaru and T. Tao, Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrodinger equation, J. Funct. Anal. 233 (2006), but the function space of solutions depended heavily on the special property of the nonlinearity $u^2$. We construct the solution space suitable for the nonlinearity $\bar u^2$.