Scattering threshold for the focusing nonlinear Klein-Gordon equation

TL;DR

This paper establishes a dichotomy between scattering and blow-up for the focusing nonlinear Klein-Gordon equation below the ground state energy, extending known results to critical and 2D exponential cases.

Contribution

It generalizes the scattering versus blow-up dichotomy to the Klein-Gordon equation, including the critical and 2D exponential regimes, despite the lack of scaling invariance.

Findings

Dichotomy between scattering and blow-up established

Results include $H^1$ critical and 2D exponential cases

Thresholds depend on ground state solutions and Trudinger-Moser constant

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-Merle for the $H^1$ critical wave and Schr\"odinger equations. Our result includes the $H^1$ 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 lack of scaling invariance both in the linear and nonlinear terms.