# Volume 58, № 8, 2006

### Regularity of nonlinear flows on noncompact Riemannian manifolds: Differential geometry versus stochastic geometry or what kind of variations is natural?

Antoniouk A. Val., Antoniouk A. Vict.

Ukr. Mat. Zh. - 2006. - 58, № 8. - pp. 1011–1034

It is shown that the geometrically correct investigation of regularity of nonlinear differential flows on manifolds and related parabolic equations requires the introduction of a new type of variations with respect to the initial data. These variations are defined via a certain generalization of a covariant Riemannian derivative to the case of diffeomorphisms. The appearance of curvature in the structure of high-order variational equations is discussed and a family of a priori nonlinear estimates of regularity of any order is obtained. By using the relationship between the differential equations on manifolds and semigroups, we study $C^{∞}$ regular properties of solutions of the parabolic Cauchy problems with coefficients increasing at infinity. The obtained conditions of regularity generalize the classical coercivity and dissipation conditions to the case of a manifold and correlate (in a unified way) the behavior of diffusion and drift coefficients with the geometric properties of the manifold without traditional separation of curvature.

### Influence of the inhomogeneity of porous media on the instantaneous compactification of the support of solution of the filtration problem

Ukr. Mat. Zh. - 2006. - 58, № 8. - pp. 1035–1044

We study the phenomenon of instantaneous compactification and the initial behavior of the support of solution of the filtration equation for inhomogeneous porous media.

### On some properties of Buhmann functions

Ukr. Mat. Zh. - 2006. - 58, № 8. - pp. 1045–1067

We study functions introduced by Buhmann. The exact exponent of smoothness of these functions is obtained and the problem of positivity of their Hankel transforms is analyzed.

### Groups with weak maximality condition for nonnilpotent subgroups

Kurdachenko L. A., Semko N. N.

Ukr. Mat. Zh. - 2006. - 58, № 8. - pp. 1068–1083

A group $G$ satisfies the weak maximality condition for nonnilpotent subgroups or, shortly, the condition Wmax-(non-nil), if $G$does not possess the infinite ascending chains $\{H_n | n \in N\}$ of nonnilpotent subgroups such that the indexes $|H_{n+i} :\; H_n |$ are infinite for all $n \in N$. In the present paper, we study the structure of hypercentral groups satisfying the weak maximality condition for nonnilpotent subgroups.

### Boundedness of weak solutions of a nondiagonal singular parabolic system of three equations

Ukr. Mat. Zh. - 2006. - 58, № 8. - pp. 1084–1096

The $L^{∞}$-estimates of weak solutions are established for a quasilinear nondiagonal parabolic system of singular equations whose matrix of coefficients satisfies special structural conditions. A procedure based on the estimation of linear combinations of the unknowns is used.

### Construction of scattering operators by the method of binary Darboux transformations

Pochynaiko M. D., Sidorenko Yu. M.

Ukr. Mat. Zh. - 2006. - 58, № 8. - pp. 1097–1115

By using the binary Darboux transformations, we construct scattering operators for a Dirac system with special potential depending on 2*n* arbitrary functions of a single variable. It is shown that one of the operators coincides with the scattering operator obtained by Nyzhnyk in the case of degenerate scattering data. It is also demonstrated that the scattering operator for the Dirac system is either obtained as a composition of three Darboux self-transformations or factorized by two operators of binary transformations of special form. We also consider several cases of reduction of these operators.

### Multiparameter inverse problem of approximation by functions with given supports

Nesterenko A. N., Radzievskii G. V.

Ukr. Mat. Zh. - 2006. - 58, № 8. - pp. 1116–1127

Let $L_p(S),\;0 < p < +∞$, be a Lebesgue space of measurable functions on $S$ with ordinary quasinorm $∥·∥_p$. For a system of sets $\{B t |t ∈ [0, +∞)^n \}$ and a given function $ψ: [0, +∞) n ↦ [ 0, +∞)$, we establish necessary and sufficient conditions for the existence of a function $f ∈ L_p(S)$ such that $\inf \{∥f − g∥^p_p| g ∈ L_p(S),\;g = 0$ almost everywhere on $S\B t } = ψ (t), t ∈ [0, +∞)^n$. As a consequence, we obtain a generalization and improvement of the Dzhrbashyan theorem on the inverse problem of approximation by functions of the exponential type in $L_2$.

### Classification of linear representations of the Galilei, Poincaré, and conformal algebras in the case of a two-dimensional vector field and their applications

Blazhko L. M., Serov N. I., Zhadan T. O.

Ukr. Mat. Zh. - 2006. - 58, № 8. - pp. 1128–1145

We present the classification of linear representations of the Galilei, Poincaré, and conformal algebras nonequivalent under linear transformations in the case of a two-dimensional vector field. The obtained results are applied to the investigation of the symmetry properties of systems of nonlinear parabolic and hyperbolic equations.

### Asymptotic behavior of the eigenvalues of a boundary-value problem for a second-order elliptic operator-differential equation

Ukr. Mat. Zh. - 2006. - 58, № 8. - pp. 1146–1152

We study the asymptotic behavior of the eigenvalues of a boundary-value problem with spectral parameter in the boundary conditions for a second-order elliptic operator-differential equation. The asymptotic formulas for the eigenvalues are obtained.