# Volume 58, № 2, 2006

### On the behavior of orbits of uniformly stable semigroups at infinity

Gorbachuk M. L., Gorbachuk V. I.

Ukr. Mat. Zh. - 2006. - 58, № 2. - pp. 148–159

For uniformly stable bounded analytic $C_0$-semigroups $\{T(t)\} t ≥ 0$ of linear operators in a Banach space $B$, we study the behavior of their orbits $T (t)x, x ∈ B$, at infinity. We also analyze the relationship between the order of approaching the orbit $T (t)x$ to zero as $t → ∞$ and the degree of smoothness of the vector $x$ with respect to the operator $A^{-1}$ inverse to the generator A of the semigroup $\{T(t)\}_{t \geq 0}$. In particular, it is shown that, for this semigroup, there exist orbits approaching zero at infinity not slower than $e^{-at^{\alpha}}$, where $a > 0,\; 0 < \alpha < \pi/(2 (\pi - 0 )),\; \theta$ is the angle of analyticity of $\{T(t)\}_{t \geq 0}$, and the collection of these orbits is dense in the set of all orbits.

### On the analyticity of solutions of $\overrightarrow{2b}$-parabolic systems

Ukr. Mat. Zh. - 2006. - 58, № 2. - pp. 160-167

It is proved that if the coefficients of a $\overrightarrow{2b}$ -parabolic system admit analytic extension to a complex region in the space variables, then the fundamental matrix of solutions of the Cauchy problem and regular solutions of the system also possess the same property.

### On the convergence of functions from a Sobolev space satisfying special integral estimates

Ukr. Mat. Zh. - 2006. - 58, № 2. - pp. 168–183

For sequences of functions from a Sobolev space satisfying special integral estimates, we, in one case, establish a lemma on the choice of pointwise convergent subsequences and, in a different case, prove a theorem on convergence of the corresponding sequences of generalized derivatives in measure. These results are applied to the problem of existence of the entropy solutions of nonlinear equations with degenerate coercivity and *L* ^{1}-data.

### Topological methods in the theory of operator inclusions in Banach spaces. I

Ukr. Mat. Zh. - 2006. - 58, № 2. - pp. 184–194

We develop topological methods for the investigation of operator inclusions in Banach spaces, prove the generalized Ky Fan inequality, and study the critical points of many-valued mappings in topological spaces.

### Asymptotic behavior of eigenvalues and eigenfunctions of the Fourier problem in a thick multilevel junction

Ukr. Mat. Zh. - 2006. - 58, № 2. - pp. 195–216

A spectral boundary-value problem is considered in a plane thick two-level junction $\Omega_{\varepsilon}$, which is the union of a domain $\Omega_{0}$ and a large number $2N$ of thin rods with thickness of order $\varepsilon = \mathcal{O} (N^{-1})$. The thin rods are divided into two levels depending on their length. In addition, the thin rods from each level are $\varepsilon$-periodically alternated. The Fourier conditions are given on the lateral boundaries of the thin rods. The asymptotic behavior of the eigenvalues and eigenfunctions is investigated as $\varepsilon \rightarrow 0$, i.e., when the number of the thin rods infinitely increases and their thickness tends to zero. The Hausdorff convergence of the spectrum is proved as $\varepsilon \rightarrow 0$, the leading terms of asymptotics are constructed and the corresponding asymptotic estimates are justified for the eigenvalues and eigenfunctions.

### Improved scales of spaces and elliptic boundary-value problems. I

Mikhailets V. A., Murach A. A.

Ukr. Mat. Zh. - 2006. - 58, № 2. - pp. 217–235

We study improved scales of functional Hilbert spaces over *R ^{n}* and smooth manifolds with boundary. The isotropic Hörmander-Volevich-Paneyakh spaces are elements of these scales. The theory of elliptic boundary-value problems in these spaces is developed.

### Averaging of the Dirichlet problem for a special hyperbolic Kirchhoff equation

Ukr. Mat. Zh. - 2006. - 58, № 2. - pp. 236–249

We prove a statement on the averaging of a hyperbolic initial-boundary-value problem in which the coefficient of the Laplace operator depends on the space $L^2$-norm of the gradient of the solution. The existence of the solution of this problem was studied by Pokhozhaev. In a space domain in $ℝ^n,\; n ≥ 3$, we consider an arbitrary perforation whose asymptotic behavior in a sense of capacities is described by the Cioranesku-Murat hypothesis. The possibility of averaging is proved under the assumption of certain additional smoothness of the solutions of the limiting hyperbolic problem with a certain stationary capacitory potential.

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

Shishkov A. E., Taranets R. M.

Ukr. Mat. Zh. - 2006. - 58, № 2. - pp. 250–271

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.

### Initial-boundary-value problems for quasilinear degenerate hyperbolic equations with damping. Neumann problem

Ukr. Mat. Zh. - 2006. - 58, № 2. - pp. 272–282

We study the behavior of the total mass of the solution of Neumann problem for a broad class of degenerate parabolic equations with damping in spaces with noncompact boundary. New critical indices for the investigated problem are determined.

### Sign changes in rational *L*_{w}^{1}-approximation

_{w}

Blatt H. P., Grothmann R., Kovacheva R. K.

Ukr. Mat. Zh. - 2006. - 58, № 2. - pp. 283–287

Let $f \in L_{1}^{w}[-1, 1]$, let $r_{n, m}(f)$ be a best rational $L_{1}^{w}$-approximation for $f$ with respect to real rational functions of degree at most n in the numerator and of degree at most m in the denominator, let $m = m(n)$, and let $\lim_{n\rightarrow \infty}(n - m(n)) = \infty$. Then we show that the counting measures of certain subsets of sign changes of $f - r_{n,m}(f)$ converge weakly to the equilibrium measure on $[-1, 1]$ as $n\rightarrow \infty$. Moreover, we prove discrepancy estimates between these counting measures and the equilibrium measure.

### Еhird summer school algebra, analysis and topology

Ukr. Mat. Zh. - 2006. - 58, № 2. - pp. 288-289