Hokkaido Mathematical Journal

NAKAMURA Yoshihisa, SHIMOMURA Akihiro,

Local well-posedness and smoothing effects of strongsolutions for nonlinear Schr\"odinger equations with potentials and magnetic fields.

Hokkaido Mathematical Journal, 34 (2005) pp.37-63

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Abstract

In this paper, we study the existence and the regularity of local strong solutions for the Cauchy problem of nonlinear Schr\"odinger 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 $n$.

MSC(Primary)35B65
MSC(Secondary)35A07, 35Q55, 81Q05
Uncontrolled Keywordsnonlinear Schr\"odinger equations with time-dependent potentials and magnetic fields, well-posedeness of the Cauchy problem with the subcritical/critical power, local smoothing effects