We consider the Cauchy problem for one system unsolvable with respect to the highest time derivative. The system under study belongs to the class of pseudohyperbolic systems and describes transverse flexural-torsional vibrations of an elastic rod. We prove the unique solvability of the Cauchy problem in Sobolev spaces and obtain estimates for the solution.

 We study the solvability in anisotropic Sobolev spaces of nonlocal boundary problems for the third order quasi-parabolic equations with an integrally-disturbed Samarskii condition. A uniqueness and existence theorem is proved for regular solutions (i. e. the solutions that have all generalized derivatives that were used in equation).

Mikhailichenko constructed a complete classification of two-dimensional geometries of maximum mobility, which contains, in addition to well-known geometries, three geometries of the Helmholtz type (actually Helmholtz, pseudo-Helmholtz, and dual Helmholtz). Each of these geometries is specified by a function of a pair of points (an analogue of the Euclidean distance) and is a geometry of local maximum mobility, that is, it allows a three-parameter group of movements. The groups of motions of these geometries are uniquely associated with non-unimodular matrix three-dimensional Lie groups, the study of which is the subject of this article.

In this work, left-invariant metrics of the studied matrix Lie groups are constructed, and Levi-Civita connections are found, as well as curvature on these Lie groups. Geodesics on such Lie groups are studied.

We consider a class of systems of difference equations with time-varying delay and periodic coefficients in linear terms. Conditions for the asymptotic stability of the zero solution are established and estimates characterizing stabilization rates of solutions at infinity are obtained.

We consider a model of reptile population dynamics in which the gender of the future individuals depends on the environment temperature. The model is described by a system of delay differential equations in which the delay parameter is responsible for the time spent by individuals in immature age. We study the case of complete extinction of the entire population and the case of stabilization of the population size at a constant value. In each case Lyapunov–Krasovskii functionals are constructed, with the help of which we establish estimates characterizing the rate of extinction of the population in the first case and the rate of stabilization of the population size in the second. Using the obtained estimates, it is possible to evaluate the time for which the population size will reach the equilibrium state.

In the half-plane R²₊ we consider a stationary system of two-velocity hydrodynamics with one pressure and homogeneous divergent and inhomogeneous boundary conditions for two velocities. Such system is overridden. The solution to this system is reduced to the sequential solution of two boundary value problems: the Stokes problem for one velocity and pressure and an overdetermined boundary value problem for the vector Poisson equation for the other speed. With an appropriate choice of function spaces, the existence and uniqueness are proven for generalized solution with the corresponding stability estimate.

The problem of propagation of a Rayleigh surface wave in an infinite half space is studied within the framework of the micropolar theory of elasticity. It is assumed that the deformed state of the medium is described by independent vectors of displacement and rotation (a Cosserat medium). A general solution describing the propagation of a surface Rayleigh wave is obtained. Using the method of constructing majorants, it is shown that there are no surface Rayleigh waves when boundary conditions are specified on the surface corresponding to the main problems of the classical theory of elasticity: “rigid embedding”, “sliding embedding”, and “rigid mesh”. For the cases of boundary conditions “free surface” and “elastic constraint,” corresponding to the problems of the classical theory of elasticity, it is shown by the method of constructing majorants that there is a surface Rayleigh wave when moment stresses are zero on the surface, while the phase velocity of the wave tends to a finite limit at high wave frequencies; when the rotation vector is equal to zero on the surface, sufficient conditions are found for the parameters of the Cosserat medium for the existence of surface Rayleigh waves, while the phase velocity of the wave tends to a finite limit at high wave frequencies. A qualitative analysis of the obtained dispersion relations showed that the Rayleigh surface wave has dispersion; the elastic constraint leads to the absence of a surface wave at low frequencies. In the case of a micropolar medium made of polyurethane foam, numerical values of the parameters of the wave and deformation of the medium are constructed. The attenuation of the displacement vector with depth in the micropolar theory of elasticity is slower than the attenuation in the classical theory of elasticity. A significant difference in the values of the displacement vector in the classical and micropolar environments is observed in the direction of elastic constraint.