For some classes of differential equations with multiple characteristics, we study the solvability of various boundary value problems in anisotropic Sobolev spaces. The aim of the paper is to prove theorems of existence and uniqueness of regular solutions, i.e. solutions that have all generalized derivatives entering the equation.

Two nonlinear mathematical models on equilibrium of plates in contact with obstacles of two types are studied. It is assumed that the plate contains a bulk rigid inclusion that touches the obstacle in the initial state. The first type of obstacle limits displacements of the plates to a square-shaped section lying on the front surface. The second type of obstacle also restricts displacements on the front surface, but has a pointwise character, i.e. Signorini-type conditions are specified at one given point. The convergence of solutions of a family of variational problems is proved as the parameter that determines the area of the contact zone tends to zero. It is shown that a limit function is the solution to the problem describing the pointwise contact of the plate.

We present a new method for computing the f-vector of a marked order polytope. Namely, given an arbitrary (polytopal) subdivision of an arbitrary convex polytope, we construct a cochain complex (over the two-element field Z2) such that the dimensions of its cohomology groups equal the components of the f-vector of the original polytope. In the case of a marked order polytope and its well-known cubosimplicial subdivision, this cochain complex can be described purely combinatorially, which yields the said computation of the f-vector. Of independent interest may be our combinatorial description of the said cubosimplicial subdivision (which was originally constructed geometrically).

The Cauchy problem for one system unsolvable with respect to the highest time derivative is considered. This system is pseudohyperbolic. The unique solvability of the Cauchy problem in Sobolev spaces is proved and estimates for the solution are obtained.

Using compactness methods for functions from the scale of Banach spaces, we prove the solvability of the problem with nonlinear latent heat of matter fusion in Stefan’s condition. An initial boundary value problem in a non-cylindrical domain with a given curved boundary of class W¹/₂ is preliminarily investigated, for which we obtain uniform estimates necessary for the main problem. Then we consider a problem in which the coefficient of the latent specific heat of fusion in the condition on the unknown boundary is a function of the size of the thawing zone s(t). This technique can also be applied to more general equations. The studied problem describes the processes of transition of matter from one state to another. As a result, the regular global in time solvability of the one-phase Stefan problem for the nonlinear parabolic equation is established. The initial data belong only to the class W¹/₂, while the phase transition boundary defined together with the solution belongs to W¹/₄.

In this article we examine in Sobolev spaces well-posedness questions of inverse problems of recovering the heat transfer coefficient from a collection of integrals of a flux with weight over the boundary. Under some conditions we demonstrate that a unique solution to the problem exists locally in time and depends continuously on the data. The method is constructive and the proposed approach allows us to construct new numerical methods for determining a solution. The proof employs a priori bounds and the contraction mapping principle.

The unique solvability of a Cauchy-type problem and linear inverse coefficient problems for an evolution equation in a Banach space with a first-order Riemann– Liouville integro-differential operator with a regular kernel is investigated. The operator at the unknown function in the equation is assumed to be closed. The conditions for the existence and uniqueness of a solution of the Cauchy type problem for a linear inhomogeneous equation are obtained. A criterion of correct solvability is found for the inverse problem with a stationary unknown coefficient and with an integral overdetermination condition in the Riemann–Stieltjes sense, which includes the condition of final overdetermination as a special case. The conditions for the solvability and stability of a solution of the inverse problem with a nonstationary unknown coefficient and an abstract overdetermination condition on the interval are found. The abstract results obtained are used in the study of linear inverse initial boundary value problems for equations with a firstorder Riemann–Liouville type regular integro-differential operator in a time variable, with polynomials with respect to a self-adjoint elliptic differential operator in spatial variables and with an unknown coefficient.

We consider an adaptation to the piecewise smooth Chua system of the previously developed high-precision numerical method for constructing approximations to unstable solutions of dynamic systems with quadratic nonlinearities on their attractors. Also, a modification of the Benettin–Wolf algorithm for calculating the characteristic Lyapunov exponents of the considered piecewise smooth system is obtained for the mode under consideration. A method based on the least squares method is developed, which makes possible to calculate the averaged estimate of the highest Lyapunov exponent based on the data on the behavior of the linearized dynamic system using a high-precision method over large time intervals. The following results are obtained for hidden attractors in the Chua system: 1) the fractal dimension of the hidden chaotic attractor based on the Poincar´e return statistics, 2) the values of the characteristic Lyapunov exponents for a stable cycle and a chaotic attractor with the use of the developed modification of the Benettin–Wolf algorithm; its efficiency is increased due to parallel computing.