六维空间中径向二次型波动方程的解孤子分解
偏微分方程分析
2022-12-05 v2
摘要
我们考虑六维二次型半线性波动方程。这一能量临界问题存在一个基态解，它是唯一（模尺度变换）的正驻波解。我们证明任何保持能量范数有界的球对称解，渐近演化为若干个解耦的经调制的基态之和，加上一项辐射项。作为该方法的副产品，我们证明了不辐射任何辐射的多孤子解的不存在性。证明遵循由最后三位作者针对大奇数维所开创的方法，将问题化简为排除此类非辐射多孤子的存在，通过推导控制其调制参数的有限维常微分方程组导出矛盾。相比之下，六维中的困难在于某些能量通道估计的失效以及相关的线性共振的存在。主要新颖之处在于使用了新的能量通道估计，以及对小能量非辐射解的分类。
引用
@article{arxiv.2201.01848,
title = {Soliton resolution for the radial quadratic wave equation in six space dimensions},
author = {Charles Collot and Thomas Duyckaerts and Carlos Kenig and Frank Merle},
journal= {arXiv preprint arXiv:2201.01848},
year = {2022}
}
备注
In this version 2, the article has been shortened in comparison with version 1. Version 2 only keeps the main theorems and their proofs. The results of Section 3, as well as the results of Section 4 and 5, of version 1, have been extended and are now contained in two other articles, see arXiv preprint arXiv:2211.16075 and arXiv:2211.16085 respectively