# Volume 47, № 2, 1995

### Method for symmetrization and estimation of solutions of the Neumann problem for the equation of a porous medium in domains with noncompact boundary for infinitely increasing time

Ukr. Mat. Zh. - 1995. - 47, № 2. - pp. 147–157

We consider the initial boundary-value Neumann problem for the equation of a porous medium in a domain with noncompact boundary. By using a symmetrization method, we obtain exact*L* _{p}-estimates, 1≤*p*≤∞, for solutions as t→∞.

### Two-phase contact Stefan problem

Ukr. Mat. Zh. - 1995. - 47, № 2. - pp. 158–167

The existence of the classical solution of the many-dimensional two-phase Stefan problem is proved for any finite time interval in the case of contact of an unknown (free) boundary with the known one.

### On the triviality condition for the kernel of the linearized first boundary-value problem for a system of equations of magnetic hydrodynamics

Ukr. Mat. Zh. - 1995. - 47, № 2. - pp. 168–173

The first boundary-value problem for a system of equations of magnetic hydrodynamics in a cylinder of infinite length with ideally conducting surface is considered under conditions in the case where it has a two-dimensional solution. Sufficient conditions are obtained for the problem linearized in the neighborhood of this solution to have the trivial solution.

### Stochastic differential equations on imbedded manifolds

Gikhman I. I., Klychkova I. E.

Ukr. Mat. Zh. - 1995. - 47, № 2. - pp. 174–179

We construct a solution of a stochastic differential equation on an imbedded manifold in the case where the ambient manifold is a Euclidean space.

### Equiasymptotic stability of integral sets

Ukr. Mat. Zh. - 1995. - 47, № 2. - pp. 180–185

We give the definition of the equiasymptotic stability of the integral set of a system of ordinary differential equations and prove several theorems.

### Partial stability and stabilization of dynamical systems

Ukr. Mat. Zh. - 1995. - 47, № 2. - pp. 186–193

We prove the theorem on necessary and sufficient conditions of partial instability and the theorem on partial stabilization of nonlinear dynamical systems. We obtain sufficient conditions of controllability for systems linear with respect to control. We also study the problem of control and stabilization of an angular motion of a solid body by rotors.

### Averaging of the Neumann problems for nonlinear elliptic equations in domains with accumulators

Ukr. Mat. Zh. - 1995. - 47, № 2. - pp. 194–212

We study the asymptotic behavior of solutions of the Neumann problems for nonlinear elliptic equations in domains with accumulators, which simulate porous media. An effective description is given for an averaged problem, which, in the case of simple accumulators, is a problem for the system of a functional equation and a differential equation; in the case of double accumulators, it is a problem for the system of two functional equations and a differential equation.

### On weak convergence of solutions of random perturbed evolution equations

Ukr. Mat. Zh. - 1995. - 47, № 2. - pp. 213–219

We consider the weak convergence of measures generated by solutions of linear evolution equations depending on diffusion processes to the Gaussian measure as a small parameter tends to zero.

### Asymptotic behavior of Lebesgue functions of two variables

Ukr. Mat. Zh. - 1995. - 47, № 2. - pp. 220–226

We describe the asymptotic behavior of Lebesgue functions generated by the rhombic partial Fourier sums.

### Large deviation theorems in the problem of testing two simple hypotheses

Lin'kov Yu. N., Medvedeva M. I.

Ukr. Mat. Zh. - 1995. - 47, № 2. - pp. 227–235

Limit theorems on large deviations of the logarithm of the likelihood ratio are proved for the problem of distinguishing two simple hypotheses in the general scheme of statistical experiments under the null hypothesis and under an alternative hypothesis. The theorems obtained are applied to the investigation of a decrease in the probability of errors of the first and second kind for the Neumann-Pearson criterion.

### Limit theorems for solutions of stochastic equations with periodic coefficients

Ukr. Mat. Zh. - 1995. - 47, № 2. - pp. 236–241

We prove theorems on the behavior of solutions of stochastic equations with periodic coefficients and integral functionals of these solutions as time infinitely increases.

### Estimation of solutions of nonautonomous systems

Ukr. Mat. Zh. - 1995. - 47, № 2. - pp. 242–248

We study the problem of using the direct Lyapunov method to get estimates for solutions of a system of ordinary differential system in general form. Theorems on asymptotic stability and behavior of solutions are proved.

### On convergence theorems for homeomorphisms of Sobolev class

Ukr. Mat. Zh. - 1995. - 47, № 2. - pp. 249–259

We prove a new convergence theorem for homeomorphisms of Sobolev class with a locally summable upper bound of deformations. This theorem allows us to generalize the known Strebel and Bers-Boyarskii convergence theorems.

### Identification of boolean functions by methods of linear algebra

Skobelev V. G., Speranskii D. V.

Ukr. Mat. Zh. - 1995. - 47, № 2. - pp. 260–268

We prove that the problem of identification of a Boolean function by using methods of the theory of linear spaces over finite fields is solvable.

### On averaging of nonlinear elliptic problems with inhomogeneous boundary conditions

Ukr. Mat. Zh. - 1995. - 47, № 2. - pp. 269–276

A sequence of solutions of nonlinear elliptic problems is considered in the case where the Dirichlet conditions are given on the one part of the boundary and the Neumann conditions are given on the other part. The boundary-value problem is constructed.

### Solvability of boundary-value problems for quasilinear elliptic and parabolic equations in unbounded domains in classes of functions growing at infinity

Ukr. Mat. Zh. - 1995. - 47, № 2. - pp. 277–289

For divergent elliptic equations with the natural energetic space*W* _{p} ^{m} (Ω),*m*≥1,*p*>2, we prove that the Dirichlet problem is solvable in a broad class of domains with noncompact boundaries if the growth of the right-hand side of the equation is determined by the corresponding theorem of Phragmén-Lindelöf type. For the corresponding parabolic equation, we prove that the Cauchy problem is solvable for the limiting growth of the initial function % MathType!MTEF!2!1!+- $$u_0 (x) \in L_{2.loc} (R^n ): \int\limits_{|x|< \tau } {u_0^2 dx \leqslant c\tau ^{n + 2mp/(p - 2)} \forall \tau< \infty } $$

### Estimates of the extinction probability of the Galton-Watson process

Ukr. Mat. Zh. - 1995. - 47, № 2. - pp. 290–291

We generalize Quine's result on the convergence of the extinction probability of the supercritical Galton-Watson process to one if the average number of offsprings tends to one without any restrictions on factorial moments.

### Exactness of the cordes condition of the Hölder property for the gradient of nondivergent elliptic systems

Ukr. Mat. Zh. - 1995. - 47, № 2. - pp. 292–294

We give an example of a nondivergent elliptic system showing that the classical Cordes condition of the Hölder property for the gradient of solution of a nondivergent elliptic equation is exact if properly extended onto systems.

### Conditions of existence of univalently stressed contours in the first main problem of the theory of elasticity for a plane with cut

Ukr. Mat. Zh. - 1995. - 47, № 2. - pp. 295–296

A definition of a univalently stressed contour in a compressible isotropic plane with a curvilinear cut is given, which extends the notion of an equiresistant contour. Conditions for elliptic and square contours to be univalently stressed are formulated and proved.