Advanced Studies in Pure Mathematics

On the scattering problem of mass-subcritical Hartree equation

Satoshi Masaki

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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

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

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


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