Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ

We consider an even-order ordinary differential operator with a spectral parameter and integral conditions. An a priori estimate of the solutions to the problem is obtained for sufficiently large values of the spectral parameter. The discreteness and the sectoral structure of the spectrum of the corresponding operators are proved.

The paper proves the existence and uniqueness theorem of a strong solution for a inhomogeneous incompressible Kelvin-Voigt fluid motion model. It is not assumed that the initial condition for the fluid density is separated from zero. To prove the existence of a solution, an approximation problem is considered, its solvability and strong a priori estimates for its solutions, independent of the approximation parameter, are established. After that, a passage to the limit is carried out as the approximation parameter tends to zero and it is shown that solutions to the approximation problem converge to a strong solution of the original problem as the approximation parameter tends to zero. The uniqueness of the solution is established using the Gronwall–Bellman inequality.

In this paper, we study the topological structure of a solution set to the Cauchy problem for semilinear differential inclusions of fractional order α ∈ (1, 2) in Banach spaces. It is assumed that the linear part of the inclusions is a linear closed operator generating a strongly continuous and uniformly bounded family of cosine operator functions. The nonlinear part is represented by a upper semicontinuous multivalued operator of Caratheodory type. It is established that the set of solutions to the problem is an Rδ-set.

The nonlinear Schro¨dinger equation of a general form is investigated, in which the chromatic dispersion and the potential are given by two arbitrary functions. The equation under consideration is a natural generalization of a wide class of related nonlinear equations that are often encountered in various sections of theoretical physics, including nonlinear optics, superconductivity, and plasma physics. Exact solutions of the nonlinear Schro¨dinger equation of general form are found, which are expressed in quadratures. One-dimensional reductions are described, which reduce the studied partial differential equation to simpler ordinary differential equations or systems of such equations. The exact solutions obtained in this work can be used as test problems intended to assess the accuracy of numerical methods for integrating nonlinear equations of mathematical physics.

The main methods and approaches of the theory of discontinuous systems are applied to the construction of the theory of functional-differential equations with a discontinuous right-hand side. In particular, methods for describing sets of discontinuity points and sliding modes of discontinuous systems with delay by using a special class of invariantly differentiable functionals are considered.