On qualitative properties of solutions of nonlinear equations of Sobolev type

The work deals with the study of qualitative properties of solutions of some equations of Sobolev type. To Sobolev equations, unlike the classical case, the Cauchy–Kovalevskaya theorem does not apply. However, one can apply many methods of functional analysis to study these equations. These equations can be used to model many physical processes, for instance, in the theory of semiconductors and in hydrodynamics. The main results of the work are as follows. First, we consider Cauchy problem. Under some conditions we obtain asymptotic behavior of solution for large times. Second, we consider initial boundary problems. We find sufficient conditions for solvability in time unique solvability, and also local solvability (but not global). In the case of local solvability we find two-sided estimates of the time of existence of solution in terms of explicit, implicit and quadrature formulas.