## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Asymptotic Analysis for Nonlinear Dispersive and Wave Equations, K. Kato, T. Ogawa and T. Ozawa, eds. (Tokyo: Mathematical Society of Japan, 2019), 259 - 309

### On the scattering problem of mass-subcritical Hartree equation

#### Abstract

In this article, we consider mass-subcritical Hartree equation. Scattering problem is treated in the framework of weighted spaces. We first establish basic properties such as local-wellposedness and criteria for finite-time blowup and scattering. Then, the first result is that uniform in time bound in critical weighted norm implies scattering. The proof is based on the concentration compactness/rigidity argument initiated by Kenig and Merle. By using the argument, existence of a threshold solution between small scattering solutions and other solutions is also deduced for the focusing model, which is the second result. The threshold is neither ground state nor any other standing wave solutions, as is known for the power type NLS equation.

#### Article information

**Dates**

Received: 1 September 2016

Revised: 25 October 2016

First available in Project Euclid:
31 October 2019

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1572545248

**Digital Object Identifier**

doi:10.2969/aspm/08110259

**Zentralblatt MATH identifier**

07176824

**Subjects**

Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]

**Keywords**

Hartree equation nonlinear Schrödinger equation Schrödinger-Poisson system Schrödinger-Newton system Choquard equation scattering threshold solution concentration compactness rigidity argument

#### Citation

Masaki, Satoshi. On the scattering problem of mass-subcritical Hartree equation. Asymptotic Analysis for Nonlinear Dispersive and Wave Equations, 259--309, Mathematical Society of Japan, Tokyo, Japan, 2019. doi:10.2969/aspm/08110259. https://projecteuclid.org/euclid.aspm/1572545248