Paliichuk L. Multivalued analysis for evolution systems of wave type with irregular conditions

The dissertation work is devoted to the study of asymptotic behavior of solutions for classes of dissipative dynamical systems of wave type with irregular restrictions. In a bounded domain the qualitative behavior of weak solutions for autonomous evolution system of wave type with discontinuous nonlinearity in the scalar case is studied. The properties and estimates for weak solutions are established. The existence of Lyapunov type function is obtained. The existence of a compact invariant global attractor for all weak solutions is proved, and its structural properties are obtained. The asymptotic behavior of solutions of a stochastically perturbed dissipative dynamical system is studied. The existence of a random attractor for an abstract non-compact multivalued dynamical system is proved. This allows obtaining the existence of a random attractor for a semi-linear wave equation with a non-smooth nonlinear term disturbed by additive white noise. In a bounded domain the qualitative behavior of weak solutions for autonomous evolution inclusion of wave type with interaction function of sub-gradient type is investigated. The properties and estimates for weak solutions are established. The existence of the Lyapunov type function is obtained. The existence of a compact invariant global attractor for all weak solutions is proved, and its structural properties are obtained. Moreover, the existence of trajectory attractor is proved. The relationship between global, trajectory and the space of complete trajectories is established. The finite-dimensionality within a small parameter of the dynamics of solutions for the investigated object is established. In addition, the sufficient conditions for the existence of a uniform compact globular attractor in a non-autonomous case are determined. The algorithm for solving problems of global behavior investigation of state functions for problems with irregular restrictions is built. The obtained theoretical results are applied to the study of piezoelectric system. The work is theoretical and practical. Its results are substantially complemente and generalize a mathematical apparatus for studying the qualitative behavior of solutions for classes of autonomous evolution problems of wave type with irregular conditions in bounded domains in cases where the conditions on the parameters of the problem do not guarantee the uniqueness of the solution of the corresponding Cauchy problems. These results can be used in mathematical modeling of complex evolution processes with nonsmooth or discontinuous interaction functions. The obtained theoretical results can be used in the control processes to reduce or compensate the undesirable effects, to substantiate the numerical algorithms for finding weak solutions, to derivate the studied systems at given stationary levels.