Internal modes and radiation damping for quadratic Klein-Gordon in 3D

TL;DR

This paper studies the long-term behavior of solutions to 3D quadratic Klein-Gordon equations with an external potential, demonstrating decay of internal modes and radiation damping under certain conditions.

Contribution

It extends previous results to quadratic nonlinearities, showing decay rates and asymptotic behavior of solutions near zero, including radiation damping phenomena.

Findings

Decay rate of internal mode amplitude: approximately t^{-1/2}

Dispersive decay of the continuous component: approximately t^{-1}

Extension of radiation damping results to quadratic Klein-Gordon equations

Abstract

We consider Klein-Gordon equations with an external potential $V$ and a quadratic nonlinearity in $3+1$ space dimensions. We assume that $V$ is regular and decaying and that the (massive) Schr\"odinger operator $H=-\Delta+V+m^2$ has a positive eigenvalue $\lambda^2<m^2$ with associated eigenfunction $\phi.$ This is a so-called internal mode and gives rise to time-periodic and spatially localized solutions of the linear flow. We address the classical question of whether such solutions persist under the full nonlinear flow, and describe the behavior of all solutions in a suitable neighborhood of zero. Provided a natural Fermi-Golden rule holds, our main result shows that a solution to the nonlinear Klein-Gordon equation can be decomposed into a discrete component $a(t)\phi$ where $a(t)$ decays over time, and a continuous component $v$ which has some weak dispersive properties. We obtain precise asymptotic information on these components such as the sharp rates of decay $\vert a(t) \vert \approx t^{-1/2}$ and ${\| v(t) \|}_{L^\infty_x} \approx t^{-1}$, (where the implicit constants are independent of the small size of the data) as well as the growth of a natural weighted norm of the profile of $v.$ In particular, our result extends the seminal work of Soffer-Weinstein for the cubic Klein-Gordon, and shows that radiation damping also occurs in the quadratic case.