Attractors of parabolic and hyperbolic equations, the nature of their compactness and attraction (Q1117392)

Consider the quasilinear parabolic system \[ (1)\quad \partial_ tu=a\Delta u-f(x,u)+\lambda u+\Sigma b_ i(x)\partial_ iu-g(x);\quad (\partial u/\partial \nu)|_{\partial \Omega}=0,\quad \Omega \subseteq R^ n \] and the quasilinear hyperbolic equation with dissipation \[ (2)\quad \partial_ t^ 2u+\gamma \partial_ tu=\Delta u-f(x,u)-g(x),\quad x\in {\bar \Omega}\subseteq R^ n,\quad u|_{\partial \Omega}=0. \] Attractors of both (1) and (2) are studied. The character of attraction to attractors and the character of the attractor's compactness are described exactly.