## Analysis & PDE

• Anal. PDE
• Volume 4, Number 3 (2011), 405-460.

### Scattering threshold for the focusing nonlinear Klein–Gordon equation

#### Abstract

We show scattering versus blow-up dichotomy below the ground state energy for the focusing nonlinear Klein–Gordon equation, in the spirit of Kenig and Merle for the $H1$ critical wave and Schrödinger equations. Our result includes the $H1$ critical case, where the threshold is given by the ground state for the massless equation, and the 2D square-exponential case, where the mass for the ground state may be modified, depending on the constant in the sharp Trudinger–Moser inequality. The main difficulty is the lack of scaling invariance in both the linear and the nonlinear terms.

#### Article information

Source
Anal. PDE, Volume 4, Number 3 (2011), 405-460.

Dates
Revised: 11 May 2010
Accepted: 8 June 2010
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.apde/1513731147

Digital Object Identifier
doi:10.2140/apde.2011.4.405

Mathematical Reviews number (MathSciNet)
MR2872122

Zentralblatt MATH identifier
1270.35132

#### Citation

Ibrahim, Slim; Masmoudi, Nader; Nakanishi, Kenji. Scattering threshold for the focusing nonlinear Klein–Gordon equation. Anal. PDE 4 (2011), no. 3, 405--460. doi:10.2140/apde.2011.4.405. https://projecteuclid.org/euclid.apde/1513731147

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