Том 234, № 4 (2018)

We propose an asymptotic classification for solutions of the equation y^IV(x) − po|y|^k sgn y = 0, 0 < k < 1, po > 0, and establish the existence of its periodic solutions.

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.

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.

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.

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.

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.