# Sidenko N. R.

### 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.

### $G$-convergence of periodic parabolic operators with a small parameter by the time derivative

Ukr. Mat. Zh. - 1993. - 45, № 4. - pp. 525–538

In this paper, we consider a sequence $\mathcal{P}^k$ of divergent parabolic operators of the second order, which are periodic in time with period $T = \text{const}$, and a sequence $\mathcal{P}^k_{\psi}$ of shifts of these operators by an arbitrary periodic vector function $ \psi \in X = \{L^2((0, T) \times \Omega)\}^n$ where $\Omega$ is a bounded Lipschitz domain in the space $\mathbb{R}^n$. The compactness of the family $\{P_{Ψ^k} ¦ Ψ \in X, k \in ℕ\}$ in $k$ with respect to strong $G$-convergence, the convergence of arbitrary solutions of the equations with the operator $\mathcal{P}^k_{\psi}$, and the local character of the strong $G$-convergence in $Ω$ are proved under the assumptions that the matrix of coefficients of $L^2$ is uniformly elliptic and bounded and that their time derivatives are uniformly bounded in the space $L^2(Ω; L^2(0,T))$.

### Asymptotics of solutions of a time-periodic boundary-value problem for a singularly perturbed nonlinear parabolic equation with rapidly oscillating coefficients

Ukr. Mat. Zh. - 1984. - 36, № 2. - pp. 165 - 171

### Averaging of a boundary-value problem, periodic with respect to time, for a singularly perturbed weakly nonlinear parabolic equation

Ukr. Mat. Zh. - 1983. - 35, № 4. - pp. 441—447

### Nonstationary problem of thermal conductivity in a system of diathermally separated bodies

Ukr. Mat. Zh. - 1973. - 25, № 4. - pp. 492—501