A dynamic boundary-value problem without initial conditions for almost linear parabolic equations

We study a dynamic boundary-value problem without initial conditions for linear and almost linear parabolic equations. First, we establish conditions for the existence of a unique solution of a problem without initial conditions for a certain abstract implicit evolution equation in the class of functions with exponential behavior as t → −∞. Then, using these results, we prove the existence of a unique solution of the original problem in the class of functions with exponential behavior at infinity.     

Inverse problem for the heat equation with degeneration

We consider the inverse problem of determining the time-dependent thermal diffusivity that is equal to zero at the initial moment of time. We establish conditions for the existence and uniqueness of a classical solution of the problem under consideration. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 11, pp. 1563–1570, November, 2005.     

Inverse problem for a parabolic equation with strong power degeneration

We consider the inverse problem of determining the time-dependent coefficient of the leading derivative in a full parabolic equation under the assumption that this coefficient is equal to zero at the initial moment of time. We establish conditions for the existence and uniqueness of a classical solution of the problem under consideration. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 11, pp. 1487–1500, November, 2006.     

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

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 the space variables. The case of arbitrarily many space variables is considered. We establish conditions for the existence and uniqueness of a solution of this problem independent of the behavior of the solution as |x| → + ∞. The indicated classes of existence and uniqueness are the spaces of locally integrable functions, and, furthermore, the dimension of the domain does not limit the order of nonlinearity. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 11, pp. 1523–1531, November, 2007.     

Third-order linear differential equation with three additional conditions and formula for Green’s function

In this paper, we investigate a third-order linear differential equation with three additional conditions. We find a solution to this problem and give a formula and an existence condition for Green’s function. We compare two Green’s functions for two such problems with different additional conditions: nonlocal and classical boundary conditions. Formula applications are shown by examples.     

On a Time-Dependent Fluid-Solid Coupling in 3D with Nonstandard Boundary Conditions

We study the motion of a Stokes fluid through an elastic half cylinder with thickness. The fluid is driven by a small time-dependent pressure drop between the outflow and the inflow sections. Nonstandard boundary conditions are imposed at the tube’s ends and we consider small displacements of the elastic structure, such that the domains involved are not moving in time. We prove existence and uniqueness of a weak solution for this three dimensional fluid-elastic structure interaction problem. Under appropriate regularity assumptions on the data we obtain strong energy estimates and show the existence of a pressure with improved properties.     

Strong Solutions to Non-stationary Channel Flows of Heat-Conducting Viscous Incompressible Fluids with Dissipative Heating

We study an initial-boundary-value problem for time-dependent flows of heat-conducting viscous incompressible fluids in channel-like domains on a time interval (0,T). For the parabolic system with strong nonlinearities and including the artificial (the so called “do nothing”) boundary conditions, we prove the local in time existence, global uniqueness and smoothness of the solution on a time interval (0,T ^∗), where 0<T ^∗≤T.     

Properties of a Solution of an Inhomogeneous Hyperbolic Equation with Random Right-Hand Side

We consider an inhomogeneous hyperbolic equation with zero initial and boundary conditions and a random centered sample-continuous Gaussian right-hand side. We establish conditions for the existence of a solution of the first boundary-value problem of mathematical physics in the form of a series uniformly convergent in probability in terms of a covariance function. An estimate for the distribution of the supremum of a solution of this problem is obtained. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 4, pp. 474–482, April, 2005.     

Solutions of the Navier–Stokes equations with various types of boundary conditions

In this paper we consider the initial boundary value problem of the Navier–Stokes system with various types of boundary conditions. We study the global-in-time existence and uniqueness of a solution of this system. In particular, suppose that the problem is solvable with some given data (the initial velocity and the external body force). We prove that there exists a unique solution for data which are small perturbations of the previous ones.     

Determination of the coefficient of thermal conductivity and heat capacity per unit volume in a multilayer medium

We consider the inverse problem of simultaneously determining two time-dependent thermophysical characteristics—the coefficient of thermal conductivity and the heat capacity per unit volume—for a body having the shape of a layer situated between two other layers with known thermophysical characteristics. The necessary measurements are carried out on their outside boundaries. The problem is reduced to a system of nonlinear equations for which the existence of a solution is established by using Schauder's fixed-point theorem. We find conditions that guarantee that the solution of the inverse problem is unique. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 2, 1997, pp. 153–159.     

A boundary-value problem for parabolic equations with general nonlocal conditions

We study a boundary-value problem with general two-point conditions with respect to the time coordinate, and periodic conditions on the spatial coordinates for Shilov-parabolic equations with constant coefficients. We construct the solution in the form of a Fourier series. We establish conditions for existence and uniqueness of a classical solution of the problem. We prove quantitative theorems on a lower bound for the small denominators that arise in solving the problem. Translated fromMatematichni Methody i Fiziko-mekhanichni Polya, Vol. 38, 1995.     

On the existence and uniqueness of a generalized solution of the mixed problem for the wave equation with the second and third boundary conditions

We prove the uniqueness of a generalized solution of an initial-boundary value problem for the wave equation with boundary conditions of the third and second kind. In addition, we find a closed-form expression for the analytic solution of that problem with zero initial data. The result plays an important role in the investigation of the boundary control problem. We show how to use the obtained solution for the investigation of the boundary control problem in the case of subcritical time intervals for which the solution of the boundary control problem, if it exists at all, is unique. We obtain necessary and sufficient conditions for the existence of a unique solution in a class admitting the existence of finite energy.     

On time decay estimates for solutions of the second mixed problem for a quasilinear parabolic equation

We obtain conditions for the existence of a global solution of the second mixed problem for a quasilinear parabolic equation in an unbounded domain. The estimates for the decay of a solution that depend on the geometry of the domain are established. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 51, No. 1, pp. 25–32, January–March, 2008.     

On the existence of vanishing at infinity symmetric solutions to the plane stationary exterior Navier–Stokes problem

We study a nonhomogeneous boundary-value problem for the steady-state Navier–Stokes equations in a two-dimensional exterior domain with two orthogonal symmetry axes. The existence of a solution which tends to zero uniformly at infinity is proved under suitable parity conditions on the data of the problem. The result is obtained for arbitrary values of the flux of the boundary datum.     

Singular Cauchy problem for the equation of flow of thin viscous films with nonlinear convection

For multidimensional equations of flow of thin capillary films with nonlinear diffusion and convection, we prove the existence of a strong nonnegative generalized solution of the Cauchy problem with initial function in the form of a nonnegative Radon measure with compact support. We determine the exact upper estimate (global in time) for the rate of propagation of the support of this solution. The cases where the degeneracy of the equation corresponds to the conditions of “strong” and “weak” slip are analyzed separately. In particular, in the case of “ weak” slip, we establish the exact estimate of decrease in the L ²-norm of the gradient of solution. It is well known that this estimate is not true for the initial functions with noncompact supports. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 2, pp. 250–271, February, 2006.     

On the Well-Posedness of a Two-Point Boundary-Value Problem for a System with Pseudodifferential Operators

We investigate the problem of the well-posedness of a boundary-value problem for a system of pseudodifferential equations of arbitrary order with nonlocal conditions. The equation and boundary conditions contain pseudodifferential operators whose symbols are defined and continuous in a certain domain H ⊂ ℝ _σ ^m . A criterion for the existence and uniqueness of solutions and for the continuous dependence of the solution on the boundary function is established. __________ Published in Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 8, pp. 1131 – 1136, August, 2005.     

On the structure of the general solution of a degenerate linear system of second-order differential equations

We establish sufficient conditions for the existence of a general Cauchy-type solution and conditions for the solvability of the Cauchy problem for a system of second-order differential equations. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 2, pp. 292–298, February, 1998.     

Mixed problem for a semilinear ultraparabolic equation in an unbounded domain

We establish conditions for the existence and uniqueness of a solution of the mixed problem for the ultraparabolic equation {fx1870-01} in an unbounded domain with respect to the variables x. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 12, pp. 1661–1673, December, 2007.     

On the existence of a solution of a quasilinear mixed problem for a quasiwave equation

We establish conditions for the existence and uniqueness of a solution of a quasilinear mixed problem for a quasiwave equation. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 9, pp. 1182–1185, September, 1997.     

Problem with nonlocal conditions for weakly nonlinear hyperbolic equations

For weakly nonlinear hyperbolic equations of order n, n≥3, with constant coefficients in the linear part of the operator, we study a problem with nonlocal two-point conditions in time and periodic conditions in the space variable. Generally speaking, the solvability of this problem is connected with the problem of small denominators whose estimation from below is based on the application of the metric approach. For almost all (with respect to the Lebesgue measure) coefficients of the equation and almost all parameters of the domain, we establish conditions for the existence of a unique classical solution of the problem. Institute of Applied Problems in Mechanics and Mathematics, Ukrainian Academy of Sciences, L’viv. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 2, pp. 186–195, February, 1997.