## Vol 64, No 1 (2018): Diﬀerential and Functional Diﬀerential Equations

**Year:**2018**Articles:**12**URL:**http://journals.rudn.ru/CMFD/issue/view/1252**DOI:**https://doi.org/10.22363/2413-3639-2018-64-1

Articles

A Stable Diﬀerence Scheme for a Third-Order Partial Diﬀerential Equation

###### Abstract

The nonlocal boundary-value problem for a third order partial diﬀerential equation in a Hilbert space H with a self-adjoint positive deﬁnite operator A is considered. A stable three-step diﬀerence scheme for the approximate solution of the problem is presented. The main theorem on stability of this diﬀerence scheme is established. In applications, the stability estimates for the solution of diﬀerence schemes of the approximate solution of three nonlocal boundary value problems for third order partial diﬀerential equations are obtained. Numerical results for oneand two-dimensional third order partial diﬀerential equations are provided.

**Contemporary Mathematics. Fundamental Directions**. 2018;64(1):1-19

Mixed Problem for a Parabolic System on a Plane and Boundary Integral Equations

###### Abstract

We consider the mixed problem for a one-dimensional (with respect to the spatial variable) second-order parabolic system with Dini-continuous coeﬃcients in a domain with nonsmooth lateral boundaries. Using the method of boundary integral equations, we ﬁnd a classical solution of this problem. We investigate the smoothness of solution as well.

**Contemporary Mathematics. Fundamental Directions**. 2018;64(1):20-36

Entropy in the Sense of Boltzmann and Poincare, Boltzmann Extremals, and the Hamilton-Jacobi Method in Non-Hamiltonian Context

###### Abstract

In this paper, we prove the H-theorem for generalized chemical kinetics equations. We consider important physical examples of such a generalization: discrete models of quantum kinetic equations (Uehling-Uhlenbeck equations) and a quantum Markov process (quantum random walk). We prove that time averages coincide with Boltzmann extremals for all such equations and for the Liouville equation as well. This gives us an approach for choosing the action-angle variables in the Hamilton-Jacobi method in a non-Hamiltonian context. We propose a simple derivation of the Hamilton-Jacobi equation from the Liouville equations in the ﬁnite-dimensional case.

**Contemporary Mathematics. Fundamental Directions**. 2018;64(1):37-59

Investigation of Operator Models Arising in Viscoelasticity Theory

###### Abstract

We study the correct solvability of initial problems for abstract integrodiﬀerential equations with unbounded operator coeﬃcients in a Hilbert space. We do spectral analysis of operator-functions that are symbols of such equations. The equations under consideration are an abstract form of linear integrodiﬀerential equations with partial derivatives arising in viscoelasticity theory and having a number of other important applications. We describe localization and structure of the spectrum of operatorfunctions that are symbols of such equations.

**Contemporary Mathematics. Fundamental Directions**. 2018;64(1):60-73

Generalized Keller-Osserman Conditions for Fully Nonlinear Degenerate Elliptic Equations

###### Abstract

We discuss the existence of entire (i.e. deﬁned on the whole space) subsolutions of fully nonlinear degenerate elliptic equations, giving necessary and suﬃcient conditions on the coeﬃcients of the lower order terms which extend the classical Keller-Osserman conditions for semilinear elliptic equations. Our analysis shows that, when the conditions of existence of entire subsolutions fail, a priori upper bounds for local subsolutions can be obtained.

**Contemporary Mathematics. Fundamental Directions**. 2018;64(1):74-85

Schlesinger’s Equations for Upper Triangular Matrices and Their Solutions

###### Abstract

We consider explicit integral expressions of hypergeometric and hyperelliptic types for solutions of Schlesinger’s equations in classes of upper triangular matrices with eigenvalues that produce arithmetic progressions with the same diﬀerence. These integral representations extend and generalize earlier known results.

**Contemporary Mathematics. Fundamental Directions**. 2018;64(1):86-97

Some Free Boundary Problems Arising in Rock Mechanics

###### Abstract

В этой статье мы рассматриваем несколько физических процессов в механике горных пород, которые описываются задачами со свободной границей. Некоторые из них известны (задачи Муската), другие совершенно новые (подземное выщелачивание и динамика трещин в подземных горных породах).

**Contemporary Mathematics. Fundamental Directions**. 2018;64(1):98-130

Estimates of Solutions of Elliptic Diﬀerential-Diﬀerence Equations with Degeneration

###### Abstract

We consider a second-order diﬀerential-diﬀerence equation in a bounded domain Q ⊂ Rn. We assume that the diﬀerential-diﬀerence operator contains some diﬀerence operators with degeneration corresponding to diﬀerentiation operators. Moreover, the diﬀerential-diﬀerence operator under consideration cannot be expressed as a composition of a diﬀerence operator and a strongly elliptic diﬀerential operator. Degenerated diﬀerence operators do not allow us to obtain the G˚arding inequality. We prove a priori estimates from which it follows that the diﬀerential-diﬀerence operator under consideration is sectorial and its Friedrichs extension exists. These estimates can be applied to study the spectrum of the Friedrichs extension as well. It is well known that elliptic diﬀerential-diﬀerence equations may have solutions that do not belong even to the Sobolev space W 1(Q). However, using the obtained estimates, we can prove some smoothness of solutions, though not in the whole domain Q, but inside some subdomains Qr generated by the shifts of the boundary, where U Qr = Q.

**Contemporary Mathematics. Fundamental Directions**. 2018;64(1):131-147

Boundedness and Finite-Time Stability for Multivalued Doubly-Nonlinear Evolution Systems Generated by a Microwave Heating Problem

###### Abstract

Doubly-nonlinear evolutionary systems are considered. Suﬃcient conditions of the boundedness of solutions of such systems are derived. Analogical results for a one-dimensional microwave heating problem are proved. The notions of global process and of a local multivalued process are introduced. Suﬃcient conditions for the ﬁnite-time stability of a global process and of a local multivalued process are shown. For local multivalued processes suﬃcient conditions for the ﬁnite-time instability are derived. For the one-dimensional microwave heating problem conditions of the ﬁnite-time stability are shown.

**Contemporary Mathematics. Fundamental Directions**. 2018;64(1):148-163

On Homotopic Classiﬁcation of Elliptic Problems with Contractions and K-Groups of Corresponding C∗-Algebras

###### Abstract

We consider calculation of a group of stable homotopic classes for pseudodiﬀerential elliptic boundary problems. We study this problem in terms of topological K-groups of some spaces in the following cases: for boundary-value problems on manifolds with boundaries, for conjugation problems with conditions on a closed submanifold of codimension one, and for nonlocal problems with contractions.

**Contemporary Mathematics. Fundamental Directions**. 2018;64(1):164-179

Uniform Basis Property of the System of Root Vectors of the Dirac Operator

###### Abstract

We study one-dimensional Dirac operator L on the segment [0,π] with regular in the sense of Birkhoff boundary conditions U and complex-valued summable potential P=(pij(x)), i,j=1,2. We prove uniform estimates for the Riesz constants of systems of root functions of a strongly regular operator L assuming that boundary-value conditions U and the number ∫(p1(x)-p4(x))dx are fixed and the potential P takes values from the ball B(0,R) of radius R in the space Lϰ for ϰ>1. Moreover, we can choose the system of root functions so that it consists of eigenfunctions of the operator L except for a finite number of root vectors that can be uniformly estimated over the ball ∥P∥ϰ≤R.

**Contemporary Mathematics. Fundamental Directions**. 2018;64(1):180-193

Identiﬁcations for General Degenerate Problems of Hyperbolic Type in Hilbert Spaces

###### Abstract

In a Hilbert space X, we consider the abstract problem M∗ddt(My(t))=Ly(t)+f(t)z,0≤t≤τ,My(0)=My0, where L is a closed linear operator in X and M∈L(X) is not necessarily invertible, z∈X. Given the additional information Φ[My(t)]=g(t) wuth Φ∈X∗, g∈C1([0,τ];C). We are concerned with the determination of the conditions under which we can identify f∈C([0,τ];C) such that y be a strict solution to the abstract problem, i.e., My∈C1([0,τ];X), Ly∈C([0,τ];X). A similar problem is considered for general second order equations in time. Various examples of these general problems are given.

**Contemporary Mathematics. Fundamental Directions**. 2018;64(1):194-210