## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Asymptotic Analysis and Singularities — Hyperbolic and dispersive PDEs and fluid mechanics, H. Kozono, T. Ogawa, K. Tanaka, Y. Tsutsumi and E. Yanagida, eds. (Tokyo: Mathematical Society of Japan, 2007), 321 - 328

### Small data scattering for the Klein-Gordon equation with a cubic convolution

#### Abstract

We consider the scattering problem for the Klein-Gordon equation with cubic convolution nonlinearity. We present the method to prove the existence of the scattering operator on a neighborhood of 0 in the weighted Sobolev space $H^{s,\sigma} = (1- \Delta)^{-s/2}\ \langle x \rangle^{-\sigma}\ L_2 (\mathbb{R}^n)$. The method is based on the complex interpolation method of the weighted Sobolev spaces and the Strichartz estimates for the inhomogeneous Klein-Gordon equation.

#### Article information

**Dates**

Received: 31 October 2005

Revised: 10 January 2006

First available in Project Euclid:
16 December 2018

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1545000600

**Digital Object Identifier**

doi:10.2969/aspm/04710321

**Mathematical Reviews number (MathSciNet)**

MR2387241

#### Citation

Sasaki, Hironobu. Small data scattering for the Klein-Gordon equation with a cubic convolution. Asymptotic Analysis and Singularities — Hyperbolic and dispersive PDEs and fluid mechanics, 321--328, Mathematical Society of Japan, Tokyo, Japan, 2007. doi:10.2969/aspm/04710321. https://projecteuclid.org/euclid.aspm/1545000600