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October, 2018 Critical nonlinear Schrödinger equations in higher space dimensions
Nakao HAYASHI, Chunhua LI, Pavel I. NAUMKIN
J. Math. Soc. Japan 70(4): 1475-1492 (October, 2018). DOI: 10.2969/jmsj/77127712


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.

Funding Statement

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.


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Nakao HAYASHI. Chunhua LI. Pavel I. NAUMKIN. "Critical nonlinear Schrödinger equations in higher space dimensions." J. Math. Soc. Japan 70 (4) 1475 - 1492, October, 2018.


Received: 16 January 2017; Revised: 13 April 2017; Published: October, 2018
First available in Project Euclid: 27 July 2018

zbMATH: 07009709
MathSciNet: MR3868214
Digital Object Identifier: 10.2969/jmsj/77127712

Primary: 35B40 , 35Q55

Keywords: critical NLS equations , higher space dimensions , large time asymptotics

Rights: Copyright © 2018 Mathematical Society of Japan


Vol.70 • No. 4 • October, 2018
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