Fermi’s golden rule and H1 scattering for nonlinear Klein-Gordon equations with metastable states

Xinliang An, Avy Soffer

Research output: Contribution to journalArticle


In this paper, we explore the metastable states of nonlinear Klein-Gordon equations with potentials. These states come from 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) 41 . Here we develop a new method to study the evolution of L2 x norm of solutions to Klein-Gordon equations. With this method, we prove a H1 scattering result for Klein-Gordon equations with metastable states. By exploring the oscillations, with a dynamical system approach we also find a more robust and more intuitive way to derive the sharp decay rate 1/(1 + t) 14 .

Original languageEnglish (US)
Pages (from-to)331-373
Number of pages43
JournalDiscrete and Continuous Dynamical Systems- Series A
Issue number1
StatePublished - 2020

All Science Journal Classification (ASJC) codes

  • Analysis
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics


  • And phrases. Nonlinear Klein-Gordon equation
  • Fermi’s golden rule
  • Metastable states
  • scattering scattering.

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