# Lavrenyuk S. P.

### Mixed problem for a semilinear ultraparabolic equation in an unbounded domain

Lavrenyuk S. P., Oliskevych M. O.

Ukr. Mat. Zh. - 2007. - 59, № 12. - pp. 1661–1673

We establish conditions for the existence and uniqueness of a solution of the mixed problem for the ultraparabolic equation $$u_t + \sum^m_{i=1}a_i(x, y, t) u_{y_i} - \sum^n_{i,j=1} \left(a_{ij}(x, y, t) u_{x_i}\right)_{x_j} + \sum^n_{i,j=1} b_{i}(x, y, t) u_{x_i} + b_0(x, y, t, u) =$$ $$= f_0(x, y, t, ) - \sum^n_{i=1}f_{i, x_i} (x, y, t) $$
in an unbounded domain with respect to the variables *x*.

### Mixed problem for a nonlinear hyperbolic equation in a domain unbounded with respect to space variables

Lavrenyuk S. P., Pukach P. Ya.

Ukr. Mat. Zh. - 2007. - 59, № 11. - pp. 1523–1531

We investigate the first mixed problem for a quasilinear hyperbolic equation of the second order with power nonlinearity in a domain unbounded with respect to space variables. We consider the case of an arbitrary number of space variables. We obtain conditions for the existence and uniqueness of the solution of this problem independent of the behavior of solution as $|x| \rightarrow +\infty$. The indicated classes of the existence and uniqueness are defined as spaces of local integrable functions. The dimension of the domain in no way limits the order of nonlinearity.

### Mixed problem for a nonlinear ultraparabolic equation that generalizes the diffusion equation with inertia

Lavrenyuk S. P., Protsakh N. P.

Ukr. Mat. Zh. - 2006. - 58, № 9. - pp. 1192–1210

We consider a mixed problem for a nonlinear ultraparabolic equation that is a nonlinear generalization of the diffusion equation with inertia and the special cases of which are the Fokker-Planck equation and the Kolmogorov equation. Conditions for the existence and uniqueness of a solution of this problem are established.

### Variational Ultraparabolic Inequalities

Lavrenyuk S. P., Protsakh N. P.

Ukr. Mat. Zh. - 2004. - 56, № 12. - pp. 1616-1628

In a bounded domain of the space ℝ^{ n +2}, we consider variational ultraparabolic inequalities with initial condition. We establish conditions for the existence and uniqueness of a solution of this problem. As a special case, we establish the solvability of mixed problems for some classes of nonlinear ultraparabolic equations with nonclassical and classical boundary conditions.

### Galerkin Method for First-Order Hyperbolic Systems with Two Independent Variables

Lavrenyuk S. P., Oliskevych M. O.

Ukr. Mat. Zh. - 2002. - 54, № 10. - pp. 1356-1371

We investigate a mixed problem for a weakly nonlinear first-order hyperbolic system with two independent variables in bounded and unbounded domains. Assuming that the nonlinearities are monotonic, we obtain conditions for the existence and uniqueness of a generalized solution; these conditions do not depend on the behavior of a solution as *x* → +∞.

### Mixed Problem for an Ultraparabolic Equation in Unbounded Domain

Lavrenyuk S. P., Protsakh N. P.

Ukr. Mat. Zh. - 2002. - 54, № 8. - pp. 1053-1066

We investigate a mixed problem for a nonlinear ultraparabolic equation in a certain domain *Q* unbounded in the space variables. This equation degenerates on a part of the lateral surface on which boundary conditions are given. We establish conditions for the existence and uniqueness of a solution of the mixed problem for the ultraparabolic equation; these conditions do not depend on the behavior of the solution at infinity. The problem is investigated in generalized Lebesgue spaces.

### On a Parabolic Variational Inequality That Generalizes the Equation of Polytropic Filtration

Ukr. Mat. Zh. - 2001. - 53, № 7. - pp. 867-878

We obtain conditions for the existence and uniqueness of a solution of a parabolic variational inequality that is a generalization of the equation of polytropic elastic filtration without initial conditions. The class of uniqueness of a solution of this problem consists of functions that increase not faster than *e* ^{−μt }, μ > 0, as *t* → −∞.

### A Mixed Problem for One Pseudoparabolic System in an Unbounded Domain

Domans'ka G. P., Lavrenyuk S. P.

Ukr. Mat. Zh. - 2001. - 53, № 1. - pp. 123-129

We prove the existence and uniqueness of a solution of a mixed problem for a system of pseudoparabolic equations in an unbounded (with respect to space variables) domain.

### On certain nonlinear pseudoparabolic variational inequalities without initial conditions

Lavrenyuk S. P., Ptashnik B. I.

Ukr. Mat. Zh. - 1999. - 51, № 3. - pp. 328–337

We consider a nonlinear pseudoparabolic variational inequality in a tube domain semibounded in variable*t*. Under certain conditions imposed on coefficients of the inequality, we prove the theorems of existence and uniqueness of a solution without any restriction on its behavior as*t*→−∞.

### Pseudoparabolic variational inequalities without initial conditions

Lavrenyuk S. P., Ptashnik M. B.

Ukr. Mat. Zh. - 1998. - 50, № 7. - pp. 919–929

We consider a pseudoparabolic variational inequality in a cylindrical domain semibounded in a variable *t*. Under certain conditions imposed on the coefficients of the inequality, we prove theorems on the unique existence of a solution for a class of functions with exponential growth as *t* → ∞.

### Systems of parabolic variational inequalities without initial conditions

Ukr. Mat. Zh. - 1997. - 49, № 4. - pp. 540–547

We consider a parabolic variational inequality without initial conditions. We construct a class of existence and uniqueness for a solution of this inequality. This class is defined by the exponential decrease or increase of solutions as *t*→−∞, depending on the coefficients of the inequality.

### On the reduction method for non-linear systems

Lavrenyuk S. P., Parasyuk E. N.

Ukr. Mat. Zh. - 1973. - 25, № 4. - pp. 550—554