


Vol 234, No 4 (2018)
- Year: 2018
- Articles: 12
- URL: https://journal-vniispk.ru/1072-3374/issue/view/14961
Article



Model Elliptic Boundary-Value Problems for Pseudodifferential Operators in Canonical Nonsmooth Domains
Abstract
We consider a simplest elliptic pseudodifferential equation in a multi-dimensional cone (multi-dimensional angle) and describe all possible structures of its solutions related to the wave factorization of the elliptic symbol. Depending on the index of wave factorization, we consider various statements of well-posed boundary-value problems. The existence of solutions is studied in Sobolev–Slobodetskii spaces.



Vibrations of a Fluid Containing a Wide Spaced Net with Floats Under Its Free Surface
Abstract
We consider the problem of low-frequency vibrations of a heavy viscous incompressible fluid occupying a vessel. Under the free surface of the fluid, there is a wide spaced net with floats forming a nonperiodic structure. On the walls of the vessel and the surface of the floats the adhesion condition (zero Dirichlet condition) is imposed. For this problem, which is formulated in terms of a quadratic operator pencil, we construct a limit (homogenized) pencil and establish a homogenization theorem in the case of a “fairly small” number of floats. It is shown that asymptotically, this structure does not affect free vibrations of the fluid.



Maximum Principle for Nonlinear Parabolic Equations
Abstract
A maximum principle is obtained for solutions of parabolic equations of the form
where



Mixed Dirichlet–Steklov Problem for the Biharmonic Equation in Weighted Spaces
Abstract
We address the uniqueness of weak solutions of the mixed Dirichlet–Steklov problem for the biharmonic equation in the exterior of a compact set in the class of functions having a finite Dirichlet integral with the weight |x|a. Depending on the parameter a, we establish some uniqueness theorems and obtain precise formulas for the dimension of the space of solutions.



Behavior of Stabilizing Solutions of the Riccati Equation
Abstract
Sufficient conditions are found for the existence of stabilizing solutions of the Riccati differential equation y′ = (y − y1(x)) (y − y2(x)) with given y1(x) and y2(x). For various types of stabilizing solutions, the number of points of extremum is examined.



Some Problems of Distributed and Boundary Control for Systems with Integral Aftereffect
Abstract
We consider the problem of exact control for a system described by an equation with integral “memory.” It is shown that, under certain conditions, this system can be brought to rest in finite time by distributed control bounded in absolute value, and, in a special one-dimensional case, by control applied to an end-point of the interval. We consider different types of kernels in the integral term of the equation and describe some relationships between problems of controllability of some hyperbolic and parabolic systems.






Lyapunov Characteristics of Oscillation, Rotation, and Wandering of Solutions of Differential Systems
Abstract
A number of Lyapunov exponents are defined for solutions of linear systems on the half-line. These exponents are responsible for such properties of the solutions as oscillation, rotation, and wandering and are defined in terms of certain functionals applied to the solutions on finite intervals as a result of two operations: upper or lower averaging in time and minimization over all bases in the phase space. We consider important special cases of systems: those of a low order, autonomous systems, those associated with equations of an arbitrary order. We obtain a set of relations (equalities and inequalities) between the said exponents, together with their refined values in special cases. It is shown that this set is complete in the sense that it cannot be extended or strengthened by any other meaningful relation.



Two-Sided Semi-Local Smoothing Splines
Abstract
A semi-local smoothing spline of degree n and class Cp is a function defined on an interval, having p continuous derivatives on that interval, and coinciding with a polynomial of degree n on the subintervals forming its partition. The domain of each polynomial is a subinterval on which m + 1 values of the approximated function are given, but in order to construct the polynomial, it is necessary to know M ≥ m + 1 values (m and M are determined by the class and the degree of the spline). The additional values can be borrowed from the adjacent subintervals. When constructing an S-spline in the periodic case, the problem of additional values is solved on the basis of periodicity, but in the nonperiodic case, one is expected to define the lacking values of a function beyond the domain. The present paper is aimed at nonperiodic two-sided S-splines whose construction does not require additional data.



Spectrum and Stabilization in Hyperbolic Problems
Abstract
We study the connection between the stabilization of solutions of a mixed hyperbolic problem and spectral properties of the corresponding elliptic boundary value problem. We consider the first mixed problem for the wave equation in bounded and unbounded domains in ℝn, determine the class of its energy solutions, and represent the solutions in terms of the Bochner–Stieltjes integral. We study how the spectrum of the elliptic operator affects the behavior of local energy of a solution and describe a method which allows us to study the stabilization of solutions with the help of estimates in the spectral parameter for solutions of the stationary problem on the upper half-plane.



Integrable Systems on the Tangent Bundle of a Multi-Dimensional Sphere
Abstract
This paper contains a systematic exposition of some results on the equations of motion of a dynamically symmetric n-dimensional rigid body in a nonconservative field of forces. Similar bodies are considered in the dynamics of actual rigid bodies interacting with a resisting medium under the conditions of jet flow past the body with a nonconservative following force acting on the body in such a way that its characteristic point has a constant velocity, which means that the system has a nonintegrable servo-constraint.


