Asymptotic behaivours of solutions for nonlinear wave equations

非线性波动方程解的渐近行为

• 批准号: 17340040
• 负责人: NAKAO Mitsuhiro
• 金额: $ 4.76万
• 依托单位: Kyushu University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2005
• 资助国家: 日本
• 起止时间: 2005 至 2007
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-17340040/
• 关键词: Nonlinear wave equations; Global attractors; Energy decay; Nonlinear degenerate parabolic equations; Smoothing effect; 非線形熱方程式; 非線形方物形方程式; エネルギー評価; 外部問題; グローバル・アトラクラー

The main object of this project is to study the asymptotic behaviours of solutions of nonlinear wave equations through the investigation of global attractors. As related problems we also intended to investigate the energy decay problem for the wave equations and global attractors for nonlinear parabolic equations.First we considered the problem for the equations in bounded domains and established new results concerning the existence, sire and some absorbing properties of global attractors.Secondly, we considered the exterior problem fix Klein-Gordon type nonlinear wave equations and established a parallel results to the problem in bounded domains. In exterior domains the Sobolev spaces are not embedded I compactly into $1,^p$ spaces. This difficulty was overcome by the discover y that the local energy of solutions are controlled as small as we can near infinity when time also goes to infinity.In a joint work with Professor Y. Zhijiag from China we proved the existence and some exponential type absorbing of global attractors for some quasi-linear wave equations. This result generalize a known one for one space dimension to general dimensions.As related problems we give several results on global attractors for degenerate type quasi-linear parabolic equations which include estimates on smoothing effects. These are joint works with Prof C. Chen from China and Dr NT, Aris from Indonesia.

该项目的主要目标是通过研究全局吸引子来研究非线性波动方程解的渐近行为。作为相关问题，我们还打算研究波动方程的能量衰减问题和非线性抛物线方程的全局吸引子。首先，我们考虑了有界域中的方程问题，并建立了关于全局吸引子的存在性、源性和一些吸收性质的新结果。其次，我们考虑了修正Klein-Gordon型非线性波动方程的外部问题，并建立了有界域问题的并行结果。在外部域中，Sobolev 空间没有紧凑地嵌入到 $1,^p$ 空间中。这个困难是通过发现当时间也趋于无穷大时解的局部能量被控制在接近无穷大的小而克服的。在与来自中国的Y.Zhijiag教授的合作中，我们证明了存在性和一些指数型吸收一些拟线性波动方程的全局吸引子。这一结果将一个空间维度的已知结果推广到一般维度。作为相关问题，我们给出了简并型拟线性抛物线方程的全局吸引子的几个结果，其中包括对平滑效应的估计。这些是与来自中国的 C. Chen 教授和来自印度尼西亚的 NT, Aris 博士的合作成果。

项目成果

Local attractors for nonlinear wave equations with some dissipative terms

具有一些耗散项的非线性波动方程的局部吸引子

• 发表时间: 2007
• 期刊: Far EastJ.Math.Sci. 27
• 作者: Nakao M.;Axis N.;Nakao M.
• 通讯作者: Nakao M.

Energy decay for the wave equation in exterior domains with a localized and a nonlinear boundary dissipations

具有局部和非线性边界耗散的外部域波动方程的能量衰减

• 发表时间: 2007
• 期刊: Nonlinear Analysis, TMA 66
• 作者: Nakao M.;Bae J.
• 通讯作者: Bae J.

Total energy decay for the wave equation in exterior domains with a localized dissipation near infinity

局域耗散接近无穷大的外部域波动方程的总能量衰减

• 发表时间: 2007
• 期刊: Journal of Mathematical Analysis and Applications 326
• 作者: Nakao M.;Axis N.;Nakao M.;K.Mochizuki
• 通讯作者: K.Mochizuki

Energy decay and periodic solutions for the wave equation in exterior domains with a localized and a nonlinear boundary dissipations

具有局部和非线性边界耗散的外域波动方程的能量衰减和周期解

• 发表时间: 2007
• 期刊: Nonlinear Analysis, TMA 66
• 作者: Nakao M.;Bae J
• 通讯作者: Bae J

Global attractors for nonlinear wave equations with some nonlinear dissipations

具有一些非线性耗散的非线性波动方程的全局吸引子

• 发表时间: 2006
• 期刊: Journal of Differential Equations 227
• 作者: Nakao M.;Axis N.;Nakao M.;K.Mochizuki;M.Nakao
• 通讯作者: M.Nakao