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arXiv 1703.04888math.AP
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三维临界非线性Klein-Gordon方程的修正散射

Modified scattering for the Klein-Gordon equation with the critical nonlinearity in three dimensions

Satoshi Masaki, Jun-ichi Segata

摘要:

本文考虑三维空间中具有临界非线性项的非线性Klein-Gordon方程的终态问题。我们证明了对于给定的渐近轮廓,存在一个解决方案(NLKG)收敛到给定的渐近轮廓t到无穷大。这里的渐近轮廓是由线性Klein-Gordon方程的解的首项与对数相位校正。一个合适的近似解的构建是基于傅里叶级数展开的非线性在我们以前的文件和平滑修改的相位校正由Ginibre-Ozawa的组合。

英文摘要:

In this paper, we consider the final state problem for the nonlinear Klein-Gordon equation (NLKG) with a critical nonlinearity in three space dimensions. We prove that for a given asymptotic profile, there exists a solution to (NLKG) which converges to given asymptotic profile as t to infinity. Here the asymptotic profile is given by the leading term of the solution to the linear Klein-Gordon equation with a logarithmic phase correction. Construction of a suitable approximate solution is based on the combination of Fourier series expansion for the nonlinearity used in our previous paper and smooth modification of phase correction by Ginibre-Ozawa.

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