Vol 55, No 5 (2019)

We study the behavior of singular solutions of the Emden-Fowler type equation y(ⁿ) = p(x, y, y’,..., y(^n-1)|y|^k sgn y, n > 2, with a regular (k > 1) or singular (0 < k < 1) nonlinearity. A singular solution is a solution that has a vertical (possibly, resonance) asymptote (for k > 1) or a solution that vanishes together with derivatives of order < n at some point or has a point of accumulation of zeros (for 0 < k < 1).

We study the problem of estimating the expression Υ(λ) = sup{|∫₀^xf(t)e^iλω(t)dt|: x ∈ [0, 1]}, where the derivative of the function ω(t) is positive almost everywhere on [0, 1]. In particular, for f ∈ L_p[0, 1], p ∈ (1, 2], we prove the estimate ∥Υ(λ)∥ L_q(ℝ) ≤ C∥f∥L_p, where 1/p + 1/q = 1. The same estimate is obtained in the space Lq(dμ), where dμ is an arbitrary Carleson measure in the open upper half-plane ℂ+. In addition, we estimate more complicated expressions like Υ(λ) that arise when studying the asymptotics of fundamental solution systems for systems of the form y′ = λρ(x)By +A(x)y +C(x, λ)y as |λ| →∞in appropriate sectors of the complex plane.

Let B = B ⊕ C^N be a finite-dimensional extension of a Banach space B, and let B be equipped with the norm ||u|| = (||u||² + ||a||²)^1/2, where u = {u, a}, u ∈ B, a ∈ C^N. The element u is called the projection of u onto B. We find a criterion for the simultaneous completeness and minimality (respectively, for the basis property) of the system {u_k}_k=N+1⁸ of projections under the condition that the system {u_k}_k=1⁸ is complete and minimal (respectively, is a basis) in the space B. This criterion is used to study the basis property of root functions of second- and fourth-order ordinary differential operators in the space L₂.

We propose a method for constructing asymptotic eigenfunctions of the operator ̂L = ∇D(x₁,x₂)∇ defined in a domain Ω ? R ² with coefficient D(x) degenerating on the boundary ∂Ω. Such operators arise, for example, in problems about long water waves trapped by coasts and islands. These eigenfunctions are associated with analogs of Liouville tori of integrable geodesic flows with the metric defined by the Hamiltonian system with Hamiltonian D(x)p² and degenerating on ∂ Ω. The situation is unusual compared, say, with the case of integrable two-dimensional billiards, because the momentum components of trajectories on such “tori” are infinite over the boundary, where D(x) = 0, although their projections onto the plane R² are compact sets, as a rule, diffeomorphic to annuli in R². We refer to such systems as billiards with semi-rigid walls.

We derive statements of linearized boundary value problems in small perturbations arising in continuum mechanics for incompressible viscous media and inviscid media. The known main three-dimensional flow is assumed to be steady-state; along with this flow, a perturbed flow of the same medium induced by the same bulk and surface forces is considered in a domain with unknown moving boundary. The arising linearized statements are reduced to a system of four equations for the perturbations of pressure and velocity components in the unperturbed domain and to a system of homogeneous boundary conditions carried over to the unperturbed boundaries. It turns out that in such statements, the spectral parameter α—the complex vibration frequency—occurs linearly in three equations of motion and one boundary condition. In special cases of the perturbation pattern, reduction is possible to one equation for the stream function amplitude linearly containing the parameter α and four boundary conditions, two of which contain the parameter α (occurring linearly in one condition and quadratically in the other). Examples include a layer of a heavy Newtonian fluid flowing down a sloping plane and vibrations in a two-layer system of heavy perfect fluids.

We study the nonlocal stabilization problem for a hydrodynamic type equation (more precisely, the differentiated Burgers equation) with periodic boundary conditions. The method is based on results in earlier papers, where nonlocal stabilization theory was constructed for a normal semilinear equation obtained from the original differentiated Burgers equation by removing part of the nonlinear terms. In the present paper, nonlocal stabilization of the full original equation by an impulse feedback control is considered. In the future, we intend to extend the results to the 3D Helmholtz equation.

We show that if X is a reflexive Banach space, then a nonautonomous operator Riccati integral equation has a unique strongly continuous self-adjoint nonnegative solution P(t) ∈ L(X,X*).

In neoclassical economic theory, the dynamics of development is mainly described by the Solow production function (Solow model), which determines the growth of the aggregate income of a country or an individual company depending on the main economic factors, such as physical capital, labor, and technological progress, driven by technological revolutions. In the upcoming digital economy, the key role will be played by technological information, which includes the knowledge and know-how needed to create high-quality goods and services that meet individual preferences, rather than mass demand, as before. It follows that the most appropriate mathematical model for describing the economic dynamics in the digital age is the half-forgotten Ramsey–Cass–Koopmans model with optimization of the consumer behavior of a representative household. This paper shows that this model implies the need to ensure the exponential growth of technological information in order for the economy to move along an optimum development trajectory. The main advantage of the Ramsey–Cass–Koopmans model is the transformation of practically significant economic development tasks into the problem of optimizing the corresponding Lagrangian or Hamiltonian, with each Lagrangian (Hamiltonian) generating a certain mode of production of technological information. Thus, it is through the Lagrangian (Hamiltonian) that the functional connection between economic growth and the dynamics of the production of technological information is established. For the first time, a formula has been obtained that links the pace of technological progress with the pace of production of technological information and can be the basis for developing an information model of economic dynamics. Examples of real modes of production of technological information used to calculate the trajectories of technological progress in the information–digital era are given.