Strichartz estimates and smooth attractors of dissipative hyperbolic equations.

In this thesis we prove attractor existence and its smoothness for several classes of damped wave equations with critical nonlinearity. The term "critical" refers to the fact that behaviour of the solutions is determined not only by the energy but also by some more subtle space-time norms which are known as Strichartz norms. One of the main achievements of the work is the construction of the global attractor to the so called weakly damped wave equation with nonlinearity that admits fifth order polynomial growth. This problem was open from the first part of the 90's and its solution required combination of tools from various branches of mathematics. The ideas that we have developed we apply to several classes of wave equation with non-local damping. In this case the amount of energy dissipation that occurs in a fixed bunch of space depends on the solution in the whole region where the problem is considered. Though this model may seem to be more complicated at first sight, in fact, in this case solutions of the corresponding problem possess better regularizing properties. Finally we would like to remark that the developed ideas have general nature and thus open new opportunities for further investigations. In particular, the newly discovered techniques and ideas have already been successfully implemented for the construction of the global attractor in problems of phase separation