On existence of global solutions of Schrödinger equations with subcritical nonlinearity for $\widehat {L}^p$-initial data

Abstract:

We construct a local theory of the Cauchy problem for the nonlinear Schrödinger equations \begin{eqnarray*} && iu_t + u_{xx} \pm |u|^{{\alpha }-1}u =0, \qquad x \in \mathbb {R}, \quad t \in \mathbb {R},\\ && u(0,x)=u_0 (x) \end{eqnarray*} with $\alpha \in (1,5)$ and $u_0 \in \widehat {L}^p (\mathbb {R})$ when $p$ lies in an open neighborhood of $2$. Moreover we prove the global existence for the initial value problem when $p$ is sufficiently close to $2$.