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

TL;DR

This paper establishes the first $H^1$ scattering results for nonlinear Klein-Gordon equations with metastable states, providing new methods to analyze decay rates and solution behavior.

Contribution

It introduces a novel approach to study $L^2_x$ norms and derives the sharp decay rate for solutions, advancing understanding of metastability in Klein-Gordon equations.

Findings

Proved the first $H^1$ scattering result for these equations.

Developed a new method for analyzing $L^2_x$ norms.

Derived the sharp decay rate $1/(1+t)^{1/4}$ using oscillation analysis.

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 $L^2_x$ 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 $1/(1+t)^{\frac{1}{4}}$.