On existence of global solutions of Schrödinger equations with subcritical nonlinearity for $\widehat {L}^p$-initial data
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- by Ryosuke Hyakuna and Masayoshi Tsutsumi
- Proc. Amer. Math. Soc. 140 (2012), 3905-3920
- DOI: https://doi.org/10.1090/S0002-9939-2012-11314-0
- Published electronically: March 22, 2012
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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 $2$.References
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Bibliographic Information
- Ryosuke Hyakuna
- Affiliation: Department of Applied Mathematics, School of Science and Engineering, Waseda University, Tokyo, Japan
- Masayoshi Tsutsumi
- Affiliation: Department of Applied Mathematics, School of Science and Engineering, Waseda University, Tokyo, Japan
- Received by editor(s): December 14, 2010
- Received by editor(s) in revised form: May 10, 2011
- Published electronically: March 22, 2012
- Communicated by: Matthew J. Gursky
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 140 (2012), 3905-3920
- MSC (2010): Primary 35Q55, 35Q41
- DOI: https://doi.org/10.1090/S0002-9939-2012-11314-0
- MathSciNet review: 2944731