2019
Том 71
№ 11

# Volume 71, № 11, 2019 (Current Issue)

Article (Ukrainian)

### A method for the construction of exact solutions to the nonlinear heat equation $u_t = \left(F(u)u_x \right)_x +G(u)u_x +H(u)$

Ukr. Mat. Zh. - 2019. - 71, № 11. - pp. 1443 -1454

UDC 517.9
We propose a method for the construction of exact solutions to the nonlinear heat equation based on the classical method of separation of variables and its generalization. We consider substitutions used to reduce the nonlinear heat equation to a system of two ordinary differential equations and construct the classes of exact solutions by the method of generalized separation of variables.

Article (Russian)

### On the crossing of maximal subgroups of finite groups

Ukr. Mat. Zh. - 2019. - 71, № 11. - pp. 1455-1465

UDC 517.542
We establish the structure of normghal subgroups in $\theta$-Frattini extensions, where $\theta$ is a subgroup functor. For a local Fitting structure $\frak F$ containing all nilpotent groups, it is shown that, in a solvable group, the crossing of $\frak F$-abnormal maximal $\theta$-subgroups not containing $\frak F$-radicals and not belonging to $\frak F$ coincides with the crossing of $\frak F$-abnormal maximal $\theta$-subgroups and belongs to the structure of $\frak F.$

Article (Russian)

### Asymptotic behavior of the solutions of boundary-value problems for singularly perturbed integro-differential equations

Ukr. Mat. Zh. - 2019. - 71, № 11. - pp. 1466-1479

UDC 517.928
We study the asymptotic behavior of the solutions of a boundary-value problem with boundary jumps for linear integrodifferential equations of the third order with small parameters at the two highest derivatives. The asymptotic convergence of the solution of a singularly perturbed integrodifferential boundary-value problem to the solution of the corresponding modified degenerate boundary-value problem is proved.

Article (English)

### Systems of variational inequalities and multiple-set split equality fixed-point problems for countable families of multivalued type-one mappings of the demicontractive type

Ukr. Mat. Zh. - 2019. - 71, № 11. - pp. 1480-1501

UDC 517.9
Our main aim is to introduce an iterative algorithm for the approximation of a common solution to a split-equality problem for finite families of variational inequalities and the split equality fixed-point problem. By using our iterative algorithm, we state and prove a strong convergence theorem for the approximation of an element in the intersection of the set of solutions of the split-equality problem for finite families of variational inequalities and the set of solutions of the split equality fixed-point problem for countable families of multivalued type-one mappings of the demicontractive type. Finally, we apply our result to study related problems. Our result supplements and extends some recent results in the literature.

Article (English)

### A parabolic equation for the fractional Laplacian in the whole space: blow-up of nonnegative solutions

Ukr. Mat. Zh. - 2019. - 71, № 11. - pp. 1502-1518

UDC 517.9
The main aim of the present paper is to investigate under what conditions the nonnegative solutions blow-up for the parabolic problem $\dfrac{\partial u}{\partial t} = - (-\triangle)^{\frac{\alpha}{2}}u + \dfrac{c}{|x|^{\alpha}}u$ in $\mathbb{R}^{d}\times (0 , T),$ where $0<\alpha<\min(2,d),$ $(-\triangle)^{\frac{\alpha}{2}}$ is the fractional Laplacian on $\mathbb{R}^{d}$ and the initial condition $u_{0}$ is in $L^{2}(\mathbb{R}^{d}).$

Article (Ukrainian)

### Non-periodic locally soluble groups with non-Dedekind locally nilpotent norm of decomposable subgroups

Ukr. Mat. Zh. - 2019. - 71, № 11. - pp. 1519-1528

UDC 512.544
We study the relations between the properties of nonperiodic groups and the norms of their decomposable subgroups. The influence of restrictions imposed on the norm of decomposable subgroups and on the properties of the group is analyzed under the condition that this norm is non-Dedekind and locally nilpotent. We also describe the structure of nonperiodic locally soluble groups for which the norm of decomposable subgroups possesses the indicated properties.

Article (Ukrainian)

### Generalization of resonance equations for the Laguerre- and Legendre-type polynomials to the fourth-order equations

Ukr. Mat. Zh. - 2019. - 71, № 11. - pp. 1529-1538

UDC 517.587
A recurrent algorithm for finding particular solutions of а fourth-order resonance equation connected with the generalization of Laguerre and Legendre polynomials is constructed and substantiated. For this purpose, we use the general theorem on the representation of partial solutions of resonance equations in Banach spaces, which was proved by V. L. Makarov in 1976. An example of general solution to the resonant equations with a differential operator for the Laguerre-type polynomials is presented.

Article (English)

### Cohomology of $q$-deformed Witt – Virasoro superalgebras of the Hom type

Ukr. Mat. Zh. - 2019. - 71, № 11. - pp. 1539-1552

UDC 512.5
We study Virasoro-type extensions of the $q$-deformed Witt Hom – Lie superalgebras. Moreover, we provide the cohomology of the $q$-deformed Witt – Virasoro superalgebras of the Hom type.

Article (Ukrainian)

### Deterministic diffusion

Ukr. Mat. Zh. - 2019. - 71, № 11. - pp. 1553-1569

UDC 517.938
In this paper, we present a series of definitions and properties of lifting dynamical systems (LDS) corresponding to the notion of deterministic diffusion. We give heuristic explanations of the mechanism of formation of deterministic diffusion in LDS and the anomalous deterministic diffusion in the case of transportation in long billiard channels with spatially periodic structures and nonideal reflection law. The expressions for the coefficient of deterministic diffusion are obtained.

Article (Ukrainian)

### Linear differential-functional equations with absolutely unstable solutions

Ukr. Mat. Zh. - 2019. - 71, № 11. - pp. 1570-1578

UDC 517.929
For linear differential-functional equations of retarded and neutral types with infinitely many deviations and self-adjoint operator coefficients, we present necessary and sufficient conditions for the absolute instability of the zero solutions.

Brief Communications (Russian)

### On the spectral properties of the one-dimensional Stark operator on the half-line

Ukr. Mat. Zh. - 2019. - 71, № 11. - pp. 1579-1584

UDC 517.9
We consider a one-dimensional Stark operator on a half-line with the Dirichlet boundary condition at zero. The asymptotic behavior of the eigenvalues at infinity is found.