Fermi's golden rule and <inline-formula><tex-math id="M1">\begin{document}$ H^1 $\end{document}</tex-math></inline-formula> 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 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 \begin{document}$ 1/(1+t)^{ \frac14} $\end{document} . Here we develop a new method to study the evolution of \begin{document}$ L^2_x $\end{document} norm of solutions to Klein-Gordon equations. With this method, we prove a \begin{document}$ H^1 $\end{document} scattering result for Klein-Gordon equations with metastable states. By exploring the oscillations, with a dynamical system approach we also find a more robust and more intuitive way to derive the sharp decay rate \begin{document}$ 1/(1+t)^{ \frac14} $\end{document} .