Fermi's Golden Rule and $H^1$ Scattering for Nonlinear Klein-Gordon Equations with Metastable States

Abstract

In this paper, we explore the metastable states of nonlinear Klein-Gordon equations with potentials. These states come from the instability of a bound state under a nonlinear Fermi's golden rule. In [16], Soffer and Weinstein studied the instability mechanism and obtained an anomalously slow-decaying rate $1/(1+t{)}^{\frac{1}{4}}$. Here we develop a new method to study the $L_x^2$ norm of solutions to Klein-Gordon equations. With this method, we prove the first $H^1$ scattering result for Klein-Gordon equations with metastable states. By exploring the oscillations, we also give another more robust and more intuitive approach to derive the sharp decay rate $1/(1+t{)}^{\frac{1}{4}}$.

47 pages. To appear in Discrete & Continuous Dynamical Systems - A

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• Attractors of nonlinear Hamiltonian PDEs