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Asymptotic behaivours of solutions for nonlinear wave equations

非线性波动方程解的渐近行为

基本信息

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

项目摘要

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.

本项目的主要目的是通过研究整体吸引子来研究非线性波动方程解的渐近性态。作为相关问题，我们还研究了波动方程的能量衰减问题和非线性抛物型方程的整体吸引子.首先，我们考虑了有界区域上的波动方程的能量衰减问题，建立了关于整体吸引子的存在性、吸引子的存在性和一些吸收性质的新结果.其次，我们还研究了非线性抛物型方程的能量衰减问题.考虑了Klein-Gordon型非线性波动方程的外问题，在有界区域上建立了该问题的一个平行结果.在外部区域中，Sobolev空间不是I紧嵌入到$1，^p $空间中。这一困难被克服了发现y的地方能源的解决方案控制尽可能小，我们可以接近无穷大时，时间也走向无穷大。在一个联合工作与教授Y。Zhijiag等证明了一类拟线性波动方程的整体吸引子的存在性和指数型吸收。作为相关问题，我们给出了退化型拟线性抛物方程整体吸引子的几个结果，其中包括光滑效应的估计。这些都是与C教授的合作作品。来自中国的Chen和来自印度尼西亚的NT博士，阿里斯。