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February 2005 Local well-posedness and smoothing effects of strongsolutions for nonlinear Schrödinger equations with potentials and magnetic fields
Yoshihisa NAKAMURA, Akihiro SHIMOMURA
Hokkaido Math. J. 34(1): 37-63 (February 2005). DOI: 10.14492/hokmj/1285766208

Abstract

In this paper, we study the existence and the regularity of local strong solutions for the Cauchy problem of nonlinear Schröodinger equations with time-dependent potentials and magnetic fields. We consider these equations when the nonlinear term is the critical and/or power type which is, for example, equal to $\lambda |u|^{p-1} u$ with some $1 \le p < \infty$, $\lambda \in {\bf C}$. We prove local well-posedness of strong solutions under the additional assumption $1 \le p < \infty$ for space dimension $n = 4$, $1 \le p \le 1+4/(n-4)$ for $n \ge 5$, and local smoothing effects of it under the additional assumption $1 \le p \le 1+2/(n-4)$ when $n \ge 5$ without any restrictions on $n$.

Citation

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Yoshihisa NAKAMURA. Akihiro SHIMOMURA. "Local well-posedness and smoothing effects of strongsolutions for nonlinear Schrödinger equations with potentials and magnetic fields." Hokkaido Math. J. 34 (1) 37 - 63, February 2005. https://doi.org/10.14492/hokmj/1285766208

Information

Published: February 2005
First available in Project Euclid: 29 September 2010

zbMATH: 1067.35111
MathSciNet: MR2130771
Digital Object Identifier: 10.14492/hokmj/1285766208

Subjects:
Primary: 35B65
Secondary: 35A07, 35Q55, 81Q05

Rights: Copyright © 2005 Hokkaido University, Department of Mathematics

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Vol.34 • No. 1 • February 2005
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