Scattering threshold for the focusing nonlinear Klein-Gordon equation

2010/01/10 by Slim Ibrahim, Nader Masmoudi, Ibrahim, Slim +3 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Quantum Chromodynamics and Particle Interactions #Cold Atom Physics and Bose-Einstein Condensates

paper · pdf · doi:10.48550/arxiv.1001.1474

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 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 lack of scaling invariance both in the linear and nonlinear terms.

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• Blow-up, decay, and convergence to equilibrium for focusing damped cubic Klein-Gordon and Duffing equations

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