Том 220, № 1 (2017)

Let S₊ and S₊^′ be the Schwartz spaces of rapidly decreasing functions and tempered distributions on ℝ₊ , respectively. Let P(S₊^′) be the space of continuous polynomials over S₊^′ and let P ′ (S₊^′) be its strong dual. These spaces have representations in the form of Fock type spaces

\( \varGamma \left({S}_{+}\right):=\underset{n\in {\mathbb{Z}}_{+}}{\oplus}\left({\oplus}_{s,\mathrm{p}}^n{S}_{+}\right)\kern2em \mathrm{and}\kern2em \varGamma \left({S}_{+}^{\prime}\right):=\underset{n\in {\mathbb{Z}}_{+}}{\times}\left({\oplus}_{s,\mathrm{p}}^n{S}_{+}^{\prime}\right), \)

respectively. In the present paper, the Gâteaux differentiability of elements of the spaces P(S₊^′), P ′ (S₊^′), Γ(S₊), and Γ(S₊^′) is investigated. The relationship between the Gâteaux derivative, the operators of creation and annihilation in the Fock type spaces, and the differentiations on Γ(S₊) and Γ(S₊^′) is established.

We study the behavior as t → ∞ of strong solutions of the initial-boundary-value problem for the system of semilinear equations of magnetoelasticity in exterior domains. For the vectors of magnetic induction and displacements, the estimates of decay are obtained in L^∞ and in the Sobolev spaces of fractional order, respectively.

We consider five different formulations of the stationary problem of passage of plane acoustic waves through a rectangular plate. The first of these formulations corresponds to the application of the inertial mass model based on the hypothesis of the nondeformability of a nonfixed rigid plate in the course of its interaction with incident and plane acoustic waves formed in the surrounding half spaces. The other four statements correspond to taking into account (according to the model of the Winkler base) or neglecting the compliance of the support contour of a hingedly supported rectangular plate deformed according to the Kirchhoff model and to the application one- or three-dimensional wave equations for the description of motions of the acoustic media and the construction of the equation of motion of the plate with regard for its certain external damping. The use of these last four statements enables us to obtain smoothened graphic frequency dependences whose shapes agree with the experimental dependences obtained by testing specimens in the acoustic laboratory aimed at finding the soundproofing index of the plate.

We propose an approach to the description of thermoelastic processes in deformable solids with regard for the structural heterogeneity of the material and the geometric heterogeneity of the surface of the body. In formulating the source relations, we use the methods of thermodynamics of nonequilibrium processes and nonlinear continuum mechanics. We take into account the structure of the material by introducing an irreversible component of the vector of mass flow. The geometric subsurface heterogeneity of the body is taken into account by introducing mass sources modeling its properties and by the dependence of the characteristics of materials, including the moduli of elasticity, on density. We study the equilibrium state of the half space. It is shown that two characteristic sizes correspond to the distributions of stresses and density. One of these sizes is connected with the structural heterogeneity of the material and the other is connected with the geometric heterogeneity of the actual surface of the body. We discuss the limits of applicability of the local gradient approach in the case of linearized approximation.