On existence of global solutions of Schrödinger equations with subcritical nonlinearity for \(\widehat L^p\)-initial data. (English) Zbl 1283.35126
Summary: We construct a local theory of the Cauchy problem for the nonlinear Schrödinger equations
\[
\begin{aligned} & iu_t + u_{xx}\pm| u|^{{\alpha }-1}u =0,\quad x\in\mathbb R,\quad t\in\mathbb R,\\ & u(0,x)=u_0 (x)\end{aligned}
\]
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.
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35Q41 | Time-dependent Schrödinger equations and Dirac equations |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
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