Kmit I. Nonlocal boundary value problems for hyperbolic systems with singularities

The thesis is devoted to hyperbolic systems of first-order equations with nonlocal reflection boundary conditions or nonlocal integral conditions, admitting singularities of various type in the initial data. The author obtains results on qualitative properties of one-dimensional first-order hyperbolic operators. Specifically, she shows an effect of smoothing solutions for a large class of mixed problems, constructs parametrices for one-dimensional first-order hyperbolic operators, develops a general approach to proving the Fredholm property for such operators, and for the first time establishes the Fredholm alternative for periodic dissipative problems for general linear hyperbolic systems of first order. Furthermore, the thesis contains general results about the correct solvability and regularity of solutions for one-dimensional first-order hyperbolic equations and systems in algebras of generalized functions, in the space of distributions, and in terms of weak solutions to the regularized problems.