2019
Том 71
№ 9

# Aldashev S. A.

Articles: 15
Brief Communications (English)

### Well-posedness of the Dirichlet problem in a cylindrical domain for three-dimensional elliptic equations with degeneration of type and order

Ukr. Mat. Zh. - 2017. - 69, № 9. - pp. 1270-1274

The paper shows the unique solvability of the classical Dirichlet problem in cylindrical domain for three-dimensional elliptic equations with degeneration type and order.

Brief Communications (Russian)

### Well-posedness of mixed problems for multidimensional hyperbolic equations with wave operator

Ukr. Mat. Zh. - 2017. - 69, № 7. - pp. 992-999

We establish the unique solvability and obtain the explicit expression for the classical solution of the mixed problem for multidimensional hyperbolic equations with wave operator.

Brief Communications (Russian)

### Well-Posedness of the Dirichlet and Poincaré Problems for the Wave Equation in a Many-Dimensional Domain

Ukr. Mat. Zh. - 2014. - 66, № 10. - pp. 1414–1419

We determine a many-dimensional domain in which the Dirichlet and Poincaré problems for the wave equation are uniquely solvable.

Brief Communications (Russian)

### Well-posedness of the Dirichlet and Poincare problems for a multidimensional Gellerstedt equation in a cylindric domain

Ukr. Mat. Zh. - 2012. - 64, № 3. - pp. 426-432

We prove the unique solvability of the Dirichlet and Poincare problems for a multidimensional Gellerstedt equation in a ´cylindric domain. We also obtain a criterion for the unique solvability of these problems.

Brief Communications (Russian)

### Eigenvalues and eigenfunctions of the Gellerstedt problem for the multidimensional Lavrent?ev?Bitsadze equation

Ukr. Mat. Zh. - 2011. - 63, № 6. - pp. 827-832

Eigenvalues and eigenfunctions of the Hellerstedt problems for the Lavrentiev - Bitsadze multidimensional equation are found.

Article (Russian)

### Existence of eigenfunctions of the Tricomi spectral problem for some classes of multidimensional mixed hyperbolic–parabolic equations

Ukr. Mat. Zh. - 2010. - 62, № 6. - pp. 723 – 732

We show that there exists a countable set of eigenfunctions of the Tricomi spectral problem for multidimensional mixed hyperbolic–parabolic equations.

Brief Communications (Russian)

### Nonuniqueness of the solution of the gellerstedt space problem for one class of many-dimensional hyperbolic-elliptic equations

Ukr. Mat. Zh. - 2010. - 62, № 2. - pp. 265–269

It is shown that the solution of the Gellerstedt space problem is not unique for one class of multidimensional hyperbolic-elliptic equations.

Brief Communications (Russian)

### Criterion for the uniqueness of a solution of the Darboux-Protter problem for multidimensional Hyperbolic equations with Chaplygin operator

Ukr. Mat. Zh. - 2004. - 56, № 8. - pp. 1119–1127

We obtain a criterion for the uniqueness of a regular solution of the Darboux-Protter problem for multidimensional hyperbolic equations with Chaplygin operator. We also prove a theorem on the uniqueness of solutions of the dual problem.

Article (Russian)

### Criterion for the Uniqueness of a Solution of the Darboux–Protter Problem for Degenerate Multidimensional Hyperbolic Equations

Ukr. Mat. Zh. - 2003. - 55, № 11. - pp. 1569-1575

We obtain a criterion for the uniqueness of a regular solution of the Darboux–Protter problem for degenerate multidimensional hyperbolic equations.

Brief Communications (Russian)

### Darboux–Protter Spectral Problems for One Class of Multidimensional Hyperbolic Equations

Ukr. Mat. Zh. - 2003. - 55, № 1. - pp. 100-107

For a multidimensional hyperbolic equation with a wave operator in the principal part, we show that the Darboux–Protter spectral problem has the countable set of eigenfunctions, and its dual problem is the Volterra problem.

Article (Russian)

### Some problems for multidimensional integro-differential equations of hyperbolic type

Ukr. Mat. Zh. - 2000. - 52, № 5. - pp. 590-595

We prove the well-posedness of the Cauchy, Goursat, and Darboux problems for multidimensional in-tegro-differential equations of the hyperbolic type encountered in biology.

Article (Russian)

### Many-dimensional Dirichlet and Tricomi problems for one class of hyperbolic-elliptic equations

Ukr. Mat. Zh. - 1997. - 49, № 12. - pp. 1587–1593

For the generalized many-dimensional Lavrent’ev-Bitsadze equation, we prove the unique solvability of the Dirichlet and Tricomi problems. We also establish the existence and uniqueness of a solution of the Dirichlet problem in the hyperbolic part of a mixed domain.

Brief Communications (Russian)

### On the well-posedness of derichlet problems for the many-dimensional wave equation and lavrent’ev-bitsadze equation

Ukr. Mat. Zh. - 1996. - 48, № 5. - pp. 702-706

We prove the unique solvability of the Dirichlet problems for the many-dimensional wave equation and Lavrent’ev-Bitsadze equation.

Article (Russian)

### Well-posedness of many-dimensional Darboux problems for degenerating hyperbolic equations

Ukr. Mat. Zh. - 1994. - 46, № 10. - pp. 1304–1311

Forthe equation $$\sum^{m}_{i=t}t^{k_i}U_{x_i,x_i} - U_n + \sum^{m}_{i=t} a_i(x,t)U_{x_i} + b(x, t)u_t + c(x,t)u = 0,$$ $$k_i = \text{const} ≥ 0,\; i = l ..... m, x = (x_1,..., x_m),\; m_>2,$$ we find a many-dimensional analog of the well-known "Gellerstedt condition" $$a_i(x,t) = O(1)t^{\alpha},\; i = 1,..., m,\, \alpha >\frac{k_1}{2} - 2.$$ We prove that if this condition is satisfied, then the Darboux problems are uniquely solvable.

Article (Ukrainian)

### Some boundary-value problems for linear multidimensional second-order hyperbolic equations

Ukr. Mat. Zh. - 1991. - 43, № 4. - pp. 415-420