In this survey, we have made an attempt to present the contemporaryideas and methods of investigation of qualitative dynamics of infinite dimensionaldissipative systems. Essential concepts such as dissipativity andasymptotic smoothness of dynamical systems, global and fractal attractors,determining functionals, regularity of asymptotic dynamics are presented.We place the emphasis on the quasi-stability method developed byI. Chueshov and I. Lasiecka. The method is based on an appropriate decompositionof the difference of the trajectories into a stable and a compactparts. The existence of this decomposition has a lot of important consequences:asymptotic smoothness, existence and finite dimensionality of attractors,existence of a finite set of determining functionals, and (under someadditional conditions) existence of a fractal exponential attractor. The restof the paper shows the application of the abstract theory to specific problems.The main attention is paid to the demonstration of the scope of thequasi-stability method.
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