Vol 55, No 10 (2019)

For parametric families of n-dimensional linear differential systems on the time semiaxis with parameter varying in a metric space, we consider two functions of the parameter defined as the dimension of the subspace of solutions that have the characteristic exponent that, respectively, is less than or does not exceed a given real number. A complete description is derived both for the functions themselves and for the vector function composed of them, for the families of systems continuous in one of the two topologies: uniform or compact-open. In addition, the Lebesgue sets and the sets of points of upper and lower semicontinuity are described for the indicated functions.

We study the question about representing a functional on the space of linear differential systems in the form of k successive limits (k ∈ ℕ) of a sequence of functionals each of which defined by the restriction of the system to a finite interval (depending on the functional) of the time semiaxis. The case in which the functional is a Lyapunov invariant is considered separately.

We study the application of a singular Schrödinger operator to studying a linear second-order differential equation in a Banach space.

We study a nonlinear partial differential equation that models processes occurring in a semiconductor medium. Six classes of exact solutions of this equation expressed via elementary and special functions are obtained. The qualitative behavior of these solutions is analyzed. We show that some of them are bounded and some of them “blow up” (become infinite in finite time).

In the multidimensional case, we study the problem with initial and boundary conditions for the equation of vibrations of a beam with one end clamped and the other hinged. An existence and uniqueness theorem is proved for the posed problem in Sobolev classes. A solution of the problem under consideration is constructed as the sum of a series in the system of eigenfunctions of a multidimensional spectral problem for which the eigenvalues are determined as the roots of a transcendental equation and the system of eigenfunctions is constructed. It is shown that this system of eigenfunctions is complete and forms a Riesz basis in Sobolev spaces. Based on the completeness of the system of eigenfunctions, a theorem about the uniqueness of a solution to the posed initial-boundary value problem is stated.

For a fourth-order generalized Mangeron equation with nonsmooth coefficients defined on a rectangular domain, we consider the Neumann problem with nonclassical conditions that do not require matching conditions. We justify the equivalence of these conditions to classical boundary conditions for the case in which the solution to the problem is sought in an isotropic Sobolev space. The problem is solved by reduction to a system of integral equations whose well-posed solvability is established based on the method of integral representations. The well-posed solvability of the Neumann problem for the generalized Mangeron equation is proved by the method of operator equations.

We study a boundary value problem with periodicity conditions and with a nonlocal Dezin condition for a mixed elliptic-hyperbolic equation in a rectangular domain with power-law degeneracy on the transition line. Necessary and sufficient conditions for the uniqueness of the solution are established, the uniqueness of the solution being proved based on the completeness of the system of eigenfunctions of a one-dimensional eigenvalue problem. The solution is constructed in the form of a series. The problem of small denominators occurs when justifying the convergence of the series. Under some conditions imposed on the given parameters and functions, the convergence of the series is proved in the class of regular solutions.

We study the problem of calculating the preferred differential realization system in the space of similar plant-controller-observer models induced by transformation groups over an identified second-order dynamical system. We prove theorems on the existence of a transforming matrix (an a posteriori basis of the configuration space) in the transformation groups GL_n (ℝ) and SO_n minimizing the mismatch between the positional force matrix and its reference rated parameters. Based on Morse theory, we construct a nonlinear matrix characteristic equation of the optimal SO_n-adjustment process. The results have applications in the differential precise modeling of forced oscillations and generate statements of the problem in the infinite-dimensional case.