Abstract
We consider the Cauchy problem for the cubic nonlinear Schrödinger equation \begin{equation} \begin{cases} iu_{t}+\frac{1}{2}u_{xx}=u^{3},\text{ }x \in \mathbf{R},\text{ }t>0, \\ u(0,x)=u_{0}(x),\text{ }x\in \mathbf{R.} \end{cases} \label{A} \end{equation} The aim of the present paper is to consider problem (0.1) in low-order Sobolev spaces, when the initial data $u_{0}\in \mathbf{H}^{\alpha }\cap \mathbf{H}^{0,\alpha }$ with $\alpha >\frac{1}{2}.$ In our previous paper [7] we proved the global existence of solutions to (0.1) if the initial data $u_{0}\in \mathbf{H}^{2}\cap \mathbf{H}^{0,2}$. Also we find the large-time asymptotics of solutions.
Citation
Nakao Hayashi. Pavel I. Naumkin. "Global existence for the cubic nonlinear Schrödinger equation in lower order Sobolev spaces." Differential Integral Equations 24 (9/10) 801 - 828, September/October 2011. https://doi.org/10.57262/die/1356012886
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