## Journal of the Mathematical Society of Japan

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

### Critical nonlinear Schrödinger equations in higher space dimensions

Nakao HAYASHI, Chunhua LI, and Pavel I. NAUMKIN

#### 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