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

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.