论文标题
内部模式引起的增长$ 3 $ d非线性klein-gordon方程
Internal mode-induced growth in $3$d nonlinear Klein-Gordon equations
论文作者
论文摘要
本说明通过证明$ 3 $ d的内部模式的非线性klein-gordon方程的解决方案的散射语句来补充论文\ cite {lp}。我们表明，小解决方案在频率空间中的一维集合周围显示出增长，并在短暂的短暂时间后成为$ l^{\ infty} $的顺序。动力学是由内部模式反馈到场（连续光谱）组件的方程式中驱动的。 证明的主要部分包括显示出适合辐射场“良好”组件的合适小度。这是分为两个步骤完成的：首先，使用\ cite {lp}开发的机械，我们将问题减少到界定某个二次正常形式校正。然后，我们通过建立一些具有奇异核的双线性操作员的精制估计来控制后者。
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.
