Internal mode-induced growth in $3$d nonlinear Klein-Gordon equations

This note complements the paper \cite{LP} by proving a scattering statement for solutions of nonlinear Klein-Gordon equations with an internal mode in $3$d. We show that small solutions exhibit growth around a one-dimensional set in frequency space and become of order one in $L^{\infty}$ after a short transient time. The dynamics are driven by the feedback of the internal mode into the equation for the field (continuous spectral) component. The main part of the proof consists of showing suitable smallness for a "good" component of the radiation field. This is done in two steps: first, using the machinery developed in \cite{LP}, we reduce the problem to bounding a certain quadratic normal form correction. Then we control this latter by establishing some refined estimates for certain bilinear operators with singular kernels.