We consider the motion of the mechanical system consisting of the case (a solid body) and the inner mass (a material point). The inner mass circulates inside the case on a circle centered at the center of mass of the case. We suppose that absolute value of the velocity of circular motion of the inner mass is constant. The case moves translationally and rectilinearly on a flat horizontal surface with forces of viscous friction and dry Coulomb friction on it. The inner mass moves in vertical plane. We perform the full qualitative investigation of the dynamics of this system. We prove that there always exist a unique motion of the case with periodic velocity. We study all possible types of such a periodic motion. We establish that for any initial velocity, the case either reaches the periodic mode of motion in a finite time or asymptotically approaches to it depending on the parameters of the problem.

By means of the modified method of test functions, we obtain sufficient conditions of absence of nontrivial solutions for some classes of semilinear elliptic inequalities of higher order and quasilinear elliptic inequalities containing nonhomogeneous terms (independent of the unknown function).

In this paper, we use the contemporary proof (by Abrosimov and Mednykh) of the Sforza formula for volume of an arbitrary non-Euclidean tetrahedron to derive new formulas that express volumes of hyperbolic tetrahedra of special kind (orthoschemes and tetrahedra with the symmetry group S 4) via dihedral angles.

In this paper, we investigate qualitative properties of solutions of boundary-value problems for strongly elliptic differential-difference equations. Earlier results establish the existence of generalized solutions of these problems. It was proved that smoothness of such solutions is preserved in some subdomains but can be violated on their boundaries even for infinitely smooth function on the right-hand side. For differential-difference equations on a segment with continuous right-hand sides and boundary conditions of the first, second, or the third kind, earlier we had obtained conditions on the coefficients of difference operators under which there is a classical solution of the problem that coincides with its generalized solution. Also, for the Dirichlet problem for strongly elliptic differential-difference equations, the necessary and sufficient conditions for smoothness of the generalized solution in Holder spaces on the boundaries between subdomains were obtained. The smoothness of solutions inside some subdomains except for ε-neighborhoods of angular points was established earlier as well. However, the problem of smoothness of generalized solutions of the second and the third boundary-value problems for strongly elliptic differential-difference equations remained uninvestigated. In this paper, we use approximation of the differential operator by finite-difference operators in order to increase the smoothness of generalized solutions of the second and the third boundary-value problems for strongly elliptic differential-difference equations in the scale of Sobolev spaces inside subdomains. We prove the corresponding theorem.

In this paper, we consider initial-boundary value problem on semiaxis for generalized Kawahara equation with higher-order nonlinearity. We obtain the result on existence and uniqueness of the global solution. Also, if the equation contains the absorbing term vanishing at infinity, we prove that the solution decays at large time values.