**
**### Mathematics

1. Verzhbitskii M. A., Pyatkov S. G.

On some inverse problems of determining boundary regimes
**Abstract.** We consider the inverse problem of determining unknown functions occurring in boundary conditions together with the solution to the initial-boundary value problem for a second-order parabolic equation. The overdetermination conditions are integrals of the solution with weight. The existence and uniqueness theorem for solutions to this inverse problem is established.

**Keywords:** inverse problem, parabolic equation, boundary and initial conditions, Sobolev space, existence and uniqueness theorem, solvability.

2. Zikirov O. S., Kholikov D. K.

On some problem for a loaded pseudoparabolic equation of the third order
**Abstract.** We study solvability of a non-local problem with integral condition for the loaded pseudoparabolic equation of the third order. The existence and uniqueness of the classical solution of the considered problem is proved by Riemann's method.

**Keywords:** loaded equation, Riemann function, non-local condition, pseudoparabolic equation.

3. Iskhokov S. A., Gadoev M. G., Petrova M. N.

On some spectral properties of a class of degenerate elliptic differential operators
**Abstract.** Some spectral properties are investigated for a class of degenerate-elliptic operators A with singular matrix coeﬃcients generated by noncoercive sesquilinear forms. Operator A is considered in the Hilbert space L

_{2}(Ω)

^{l}, where

Ω ⊂ R

^{n} is a limit-tube domain and

l > 0 is an integer.

**Keywords:** spectral properties, degenerate-elliptic operator, noncoercitive sesquilinear form, limit-cylindrical (x) domain, resolvent of generalized Dirichlet problem.

4. Lazarev N. P.

Optimal size control of a rigid inclusion in equilibrium problems for inhomogeneous three-dimensional bodies with a crack
**Abstract.** We consider equilibrium problems for an inhomogeneous three-dimensional body with a crack at the inclusion-matrix interface. The matrix of the plate is assumed to be elastic. The boundary condition on the crack curve is given in the form of inequality and describes mutual nonpenetration of the crack faces. We analyze the dependence of solutions on the size of the rigid inclusion. It is shown that as the size of the rigid inclusion's volume tends to zero the solutions of the corresponding equilibrium problems converge to the solution of the equilibrium problem for a body containing a thin rigid delaminated inclusion. The existence of the solution to the optimal control problem is proved. For that problem, the size parameter of the rigid inclusion is chosen as the control function, while the cost functional is an arbitrary continuous functional.

**Keywords:** crack, rigid inclusion, variational inequality, energy functional, nonlinear boundary conditions.

5. Lyubanova A. Sh.

Inverse problems for nonlinear stationary equations
**Abstract.** Identification of the unknown constant coeﬃcient in the main term of the partial differential equation -kMψ

_{1}(u) + g(x)ψ

_{2}(u) = f(x) with the Dirichlet boundary condition is investigated. Here

_{i}(u), i = 1, 2 is a nonlinear increasing function of

u and

M is a second-order linear elliptic operator. The coeﬃcient

k is recovered on the base of additional integral boundary data. The existence and uniqueness of the solution to the inverse problem with a function u and a positive real number k is proved.

**Keywords:** inverse problem, boundary value problem, second-order elliptic equation, existence and uniqueness theorem, filtration.

6. Mitrokhin S. I.

On a study of the spectrum of a boundary value problem for the fifth-order differential operator with integrable potential
**Abstract.** A boundary value problem for the fifth-order differential operator with separated boundary conditions is considered. The potential of the operator is a summable function on the segment. For large values of the spectral parameter we obtain the asymptotic behavior of the corresponding differential equation. The equation on eigenvalues of the considered operator and the indicator diagram of this equation are studied. A new method for finding an asymptotics of eigenvalues of the studied operator is offered.

**Keywords:** boundary value problem, differential operator, separated boundary conditions, summable potential, asymptotics of the eigenvalues, eigenfunction

7. Popov S. V.

On behavior of the Cauchy-type integral at the endpoints of the integration contour and its application to boundary value problems for parabolic equations with changing direction of time
**Abstract.** We consider N. I. Muskhelishvili's theorem about the behavior of Cauchy-type integrals at the endpoints of the integration contour and the discontinuity points of the density and its application to boundary value problems for 2n-parabolic equations with changing direction of time. For parabolic equations with changing direction of time, the smoothness of initial and boundary data does not imply in general that the solution belongs to the Holder spaces. Application of the theory of singular equations makes it possible to specify necessary and suﬃcient conditions for the solution to belong to the Holder spaces. Moreover, under general gluing conditions, using unified approach we can show that for such equations the nonintegral exponent of the space may essentially affect both the number of solvability conditions and the smoothness of the solutions. To prove the solvability of boundary value problems for such equations, we consider continuous bonding gluing conditions with the 2n-1

-th derivative. Note that in the case of n=3

the smoothness of the initial data and solvability conditions determine that the solution belongs to smoother Holder spaces near the endpoints of the integration contour.

**Keywords:** Cauchy-type integral, Muskhelishvili’s theorem, parabolic equation with changing direction of time, bonding gluing condition, Holder space, singular integral equation.

8. Skvortsova M. A.

Stability of solutions in the predator-prey model with delay
**Abstract.** We consider a system of delay differential equations describing the interaction between two populations the predators and prey. We study the asymptotic stability of stationary solutions to this system. Using the modified Lyapunov-Krasovskii functional we establish estimates for solutions characterizing the rate of convergence to the stationary solutions.

**Keywords:** predator-prey model, delay differential equations, asymptotic stability, characteristic quasipolynomial, estimates for solutions, modified Lyapunov–Krasovskii functional.