Asymptotic behaivours of solutions for nonlinear wave equations

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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.