Journal of the Mathematical Society of Japan

Critical nonlinear Schrödinger equations in higher space dimensions

Nakao HAYASHI, Chunhua LI, and Pavel I. NAUMKIN

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Abstract

We study the critical nonlinear Schrödinger equations \[ i\partial _{t}u+\frac{1}{2}\Delta u = \lambda \vert u\vert^{{2}/{n}}u, \quad (t,x) \in \mathbb{R}^{+}\times \mathbb{R}^{n}, \] in space dimensions $n\geq 4$, where $\lambda \in \mathbb{R}$. We prove the global in time existence of solutions to the Cauchy problem under the assumption that the absolute value of Fourier transform of the initial data is bounded below by a positive constant. Also we prove the two side sharp time decay estimates of solutions in the uniform norm.

Note

The firtst author was partially supported by JSPS KAKENHI Grant Numbers JP25220702, JP15H03630. The second author was partially supported by NNSFC Grant No.11461074. The third author was partially supported by CONACYT and PAPIIT project IN100616.

Article information

Source
J. Math. Soc. Japan, Volume 70, Number 4 (2018), 1475-1492.

Dates
Received: 16 January 2017
Revised: 13 April 2017
First available in Project Euclid: 27 July 2018

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1532678876

Digital Object Identifier
doi:10.2969/jmsj/77127712

Mathematical Reviews number (MathSciNet)
MR3868214

Zentralblatt MATH identifier
07009709

Subjects
Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10] 35B40: Asymptotic behavior of solutions

Keywords
critical NLS equations higher space dimensions large time asymptotics

Citation

HAYASHI, Nakao; LI, Chunhua; NAUMKIN, Pavel I. Critical nonlinear Schrödinger equations in higher space dimensions. J. Math. Soc. Japan 70 (2018), no. 4, 1475--1492. doi:10.2969/jmsj/77127712. https://projecteuclid.org/euclid.jmsj/1532678876


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