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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Authors: Ryosuke Hyakuna and Masayoshi Tsutsumi
Journal: Proc. Amer. Math. Soc. 140 (2012), 3905-3920
MSC (2010): Primary 35Q55, 35Q41
Published electronically: March 22, 2012
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Abstract: We construct a local theory of the Cauchy problem for the nonlinear Schrödinger equations

    $\displaystyle iu_t + u_{xx} \pm \vert u\vert^{{\alpha }-1}u =0, \qquad x \in \mathbb{R}, \quad t \in \mathbb{R},$  
    $\displaystyle u(0,x)=u_0 (x)$  

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$.

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Additional 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

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11314-0
PII: S 0002-9939(2012)11314-0
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
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.