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

IF 0.6 4区 数学 Q3 MATHEMATICS
Tristan L'eger, F. Pusateri
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引用次数: 3

Abstract

This note complements the paper [19] by proving a scattering statement for solutions of nonlinear Klein-Gordon equations with an internal mode in 3d. We show that small solutions exhibit growth around a one-dimensional set in frequency space and become of order one in L∞ 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 [19], 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.
$3$d非线性Klein-Gordon方程的内模诱导增长
本文通过证明三维内模非线性Klein-Gordon方程解的散射命题,对论文[19]进行了补充。我们证明了小解在频率空间中围绕一维集合生长,并在短的瞬态时间后在L∞上变为1阶。动力学是由内部模态反馈到场(连续谱)分量方程中驱动的。证明的主要部分包括为辐射场的“好”组成部分显示适当的小。这分两步完成:首先,使用[19]中开发的机器,我们将问题简化为确定二次范式修正的边界。然后,我们通过建立一些奇异核双线性算子的改进估计来控制后者。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Rendiconti Lincei-Matematica e Applicazioni
Rendiconti Lincei-Matematica e Applicazioni MATHEMATICS, APPLIED-MATHEMATICS
CiteScore
1.30
自引率
0.00%
发文量
27
审稿时长
>12 weeks
期刊介绍: The journal is dedicated to the publication of high-quality peer-reviewed surveys, research papers and preliminary announcements of important results from all fields of mathematics and its applications.
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