Asymptotic behaivours of solutions for nonlinear wave equations

这个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行为。作为相关问题，我们还研究了波动方程和非线性抛物型方程整体吸引子的能量衰减问题。首先，我们考虑了有界域上方程的能量衰减问题，建立了关于整体吸引子的存在性、存在性和某些吸引子性质的新结果。其次，我们考虑了Klein-Gordon型非线性波动方程的外部问题，建立了与有界域上的问题类似的结果。在外域中，Sobolev空间不是紧嵌入到$1，^p$空间中的。当时间也是无穷大时，解的局部能量被控制得尽可能小，克服了这一困难。在与中国的Y.Zhijiag.教授的合作中，我们证明了一类拟线性波动方程整体吸引子的存在性和某些指数型吸引子。作为相关问题，我们给出了几个关于退化型拟线性抛物型方程整体吸引子的结果，其中包括光滑化效应的估计。这些是与来自中国的陈振明教授和来自印度尼西亚的NT，ARIS博士合作的作品。

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.