Volume 219, Nº 2 (2016)

We study the periodic boundary value problem for a modified Kuramoto–Sivashinsky equation which can serve as a mathematical model describing formation of nanorelief on the surface of a planar target under the action of ion flux. We show that, as in the case of the traditional Kuramoto–Sivashinsky equation, it is possible to obtain spatially inhomogeneous solutions under the condition that the homogeneous equilibrium states can change the stability. We consider local bifurcations. We find sufficient conditions for the existence of shortwave solutions. Bibliography: 11 titles.

For multidimensional discrete mappings of the form s_k+1 = g(s_k, p) we develop the method of symbolic CTQ-analysis and study the main properties of the T-alphabet. We formulate basic principles for the analytic study of discrete mappings in terms of the proposed formalism. Bibliography: 15 titles. Illustrations: 5 figures.

We develop the method of singular space-time transformations for systems with impact and friction. We consider a mathematical model of a Painlevé problem concerning impacts of an absolutely rigid bar with a rough surface. Using the method of singular space-time change of variables, we obtain the model of an absolutely rigid body by passing to the limit as the generalized rigidity parameter tends to infinity.

We study the homogenization of the Riemann–Hilbert boundary value problem for the Beltrami equation with an oscillating ε-periodic coefficient, where ε > 0 is a small parameter. The homogenized problem has a similar form, but with a constant coefficient. We prove error estimates of homogenization in the L₂- and W₂¹ -norms of order \( O\left(\sqrt{\varepsilon}\right) \). Bibliography: 9 titles.

We prove the existence and uniqueness of a weak solution to the initial-boundary value problem for a fourth order pseudohyperbolic equation in a cylinder with nonlocal boundary conditions containing the first and second order time-derivatives amd the integral of the sought solution. Bibliography: 3 titles.

We consider the system of differential equations describing the initial step of capture of nonlinear oscillations in autoresonance under weak dissipation. We study the stability in probability of resonance solutions with unboundedly growing amplitude under persistent random perturbations. Bibliography: 11 titles. Illustrations: 1 Figure

We study the regularity of solutions to problems of minimization of integral functionals, the Dirichlet and Neumann problems for elliptic operators of order 2m in domains with Hölder boundary on a compact Riemannian manifold. We establish interactions between the smoothness of the right-hand side, the regularity of the boundary, and the smoothness of solutions to the problems under consideration. Bibliography: 10 titles.