# Pukalskyi I. D.

### Boundary-value problem with impulsive action for a parabolic equation with degeneration

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 5. - pp. 645-655

UDC 517.984.54

For a second-order parabolic equation, we consider a problem with oblique derivative and impulsive action. The coefficients of the equation and the boundary condition have power singularities of any order in the time and space variables on some set of points.

We establish conditions for the existence and uniqueness of the solution of the problem in Hölder spaces with power weight.

### Boundary-Value Problem with Impulsive Conditions and Degeneration for Parabolic Equations

Isaryuk I. M., Pukalskyi I. D.

Ukr. Mat. Zh. - 2015. - 67, № 10. - pp. 1348-1357

We consider the second boundary-value problem for a parabolic equation with power singularities in the coefficients of space variables and impulsive conditions in the time variable. By using the maximum principle and *a priori* estimates, we establish the existence and uniqueness of the solution of posed problem in Hölder spaces with power weights.

### Nonlocal Parabolic Problem with Degeneration

Isaryuk I. M., Pukalskyi I. D.

Ukr. Mat. Zh. - 2014. - 66, № 2. - pp. 208–215

We study the problem for a second-order linear parabolic equation with nonlocal integral condition in the time variable and power singularities in the coefficients of any order with respect to the time and space variables. By using the maximum principle and *a priori* estimates, we establish the existence and uniqueness of the solution of this problem in Hölder spaces with power weights.

### Nonlocal Dirichlet problem for linear parabolic equations with degeneration

Ukr. Mat. Zh. - 2007. - 59, № 1. - pp. 109–121

In the spaces of classical functions with power weight, we prove the correct solvability of the Dirichlet problem for parabolic equations with nonlocal integral condition with respect to the time variable and an arbitrary power order of degeneration of coefficients with respect to the time and space variables.

### Boundary-Value Problem for Linear Parabolic Equations with Degeneracies

Ukr. Mat. Zh. - 2005. - 57, № 3. - pp. 377–387

In spaces of classical functions with power weight, we prove the correct solvability of a boundary-value problem for parabolic equations with an arbitrary power order of degeneracy of coefficients with respect to both time and space variables.

### Cauchy Problem for Nonuniformly Parabolic Equations with Degeneracy

Ukr. Mat. Zh. - 2003. - 55, № 11. - pp. 1520-1529

In spaces of classical functions with power weight, we prove the existence and uniqueness of a solution of the Cauchy problem for nonuniformly parabolic equations without restrictions on the power order of degeneracy of the coefficients. We obtain an estimate for the solution of the problem in the corresponding spaces.

### One-Sided Nonlocal Boundary-Value Problem for Singular Parabolic Equations

Ukr. Mat. Zh. - 2001. - 53, № 11. - pp. 1521-1532

In spaces of classical functions with power weight, we prove the existence and uniqueness of a solution of a one-sided nonlocal boundary-value problem for parabolic equations with an arbitrary power order of degeneracy of coefficients. We obtain an estimate for the solution of this problem in the corresponding spaces.

### Green function of a parabolic boundary-value problem and the optimization problem

Ukr. Mat. Zh. - 2000. - 52, № 4. - pp. 567-571

We establish necessary and sufficient conditions for the choice of optimal control over systems de-scribed by a general parabolic problem with restricted internal control.

### Nonlocal Neumann problem for a degenerate parabolic equation

Ukr. Mat. Zh. - 1999. - 51, № 9. - pp. 1232–1243

In the spaces of classical functions with power weight, we prove the existence and uniqueness of a solution of the nonlocal Neumann problem for nonuniformly parabolic equations without restrictions on the power order of coefficient degeneration. We find an estimate of the solution of this problem in the spaces considered.

### Applications of the green function of parabolic boundary-value problems to optimal control problems

Matychuk M. I., Pukalskyi I. D.

Ukr. Mat. Zh. - 1985. - 37, № 6. - pp. 738–744

### Solution of a parabolic boundary-value problem with singularities in the coefficients of the boundary operator

Matychuk M. I., Pukalskyi I. D.

Ukr. Mat. Zh. - 1982. - 34, № 5. - pp. 637—640

### Directional derivative problem for a degenerating parabolic equation

Ukr. Mat. Zh. - 1981. - 33, № 4. - pp. 459–466