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Fermi’s golden rule and \(H^1\) scattering for nonlinear Klein-Gordon equations with metastable states. (English) Zbl 1431.35145

Summary: 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 [Invent. Math. 136, No. 1, 9–74 (1999; Zbl 0910.35107)], A. Soffer and M. I. 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 evolution of \(L^2_x\) norm of solutions to Klein-Gordon equations. With this method, we prove a \(H^1\) 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 \(1/(1+t)^{\frac{1}{4}}\).

MSC:

35Q40 PDEs in connection with quantum mechanics
35B34 Resonance in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35L70 Second-order nonlinear hyperbolic equations
35P25 Scattering theory for PDEs
35C08 Soliton solutions

Citations:

Zbl 0910.35107
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References:

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