This paper is devoted to studying the reaction - diffusion systems with rapidly oscillating coefficients in the equations and in boundary conditions in domains with locally periodic oscillating boundary; on this boundary a Robin boundary condition is imposed. We consider the supercritical case, when the homogenization changes the Robin boundary condition on the oscillating boundary is to the homogeneous Dirichlet boundary condition in the limit as the small parameter, which characterizes oscillations of the boundary, tends to zero. In this case, we prove that the trajectory attractors of these systems converge in a weak sense to the trajectory attractors of the limit (homogenized) reaction - diffusion systems in the domain independent of the small parameter. For this aim we use the homogenization theory, asymptotic analysis and the approach of V.V. Chepyzhov and M.I. Vishik concerning trajectory attractors of dissipative evolution equations. The homogenization method and asymptotic analysis are used to derive the homogenized reaction - diffusion system and to prove the convergence of solutions. First we define the appropriate auxiliary functional spaces with weak topology, then, we prove the existence of trajectory attractors for these systems and formulate the main Theorem. Finally, we prove the main convergence result with the help of auxiliary lemmas.

We study homogenization of random reaction-diffusion systems with rapidly oscillating terms in domains with rapidly oscillating boundary and containing a small parameter epsilon that characterizes the oscillation rate of coefficients in equations and in the boundary conditions. We consider reactiondiffusion systems that obey some general dissipative conditions. We study the asymptotic behavior of trajectory attractors of the considered systems as epsilon -> 0+. We combine homogenization methods and the theory of trajectory attractors. We construct trajectory attractors 2t epsilon for the considered random reactiondiffusion systems and the trajectory attractor 2t for the corresponding limit (homogenized) deterministic reaction-diffusion system including the limit boundary conditions. After that, we prove that, with probability one, the random trajectory attractor 2t epsilon converges to the (non-random) trajectory attractor 2t as epsilon -> 0 in the weak topology of some natural functional space that contains the trajectory spaces of the reaction-diffusion systems.

In the present paper, reaction–diffusion systems (RD-systems) with rapidly oscillating coefficients and righthand sides in equations and in boundary conditions were considered in domains with locally periodic oscillating (wavering) boundary. We proved a weak convergence of the trajectory attractors of the given systems to the trajectory attractors of the limit (homogenized) RD-systems in domain independent of the small parameter, characterizing the oscillation rate. We consider the critical case in which the type of boundary condition was preserved. For this aim, we used the approach of Chepyzhov and Vishik concerning trajectory attractors of evolutionary equations. Also, we applied the homogenization (averaging) method and asymptotic analysis to derive the limit (averaged) system and to prove the convergence. Defining the appropriate axillary functional spaces with weak topology, we proved the existence of trajectory attractors for these systems. Then, we formulated the main theorem and proved it with the help of auxiliary lemmata.

A reaction -diffusion system in a domain with randomly located obstacles was considered. When studying the problem, we sat the homogeneous Dirichlet condition on the outer boundary of the domain and the Neumann condition on the boundary of the cavities. Under such assumptions, it was proven that random trajectory attractors of this system with random coefficients converge in some weak topology to the deterministic trajectory attractor of a homogenized reaction -diffusion system with deterministic coefficients in a homogeneous domain without obstacles. In the case of uniqueness, we obtained weak convergence of random global attractors to a deterministic global attractor.

We consider the Ginzburg-Landau equation in the perforated domain, with rapidly oscillating coefficients. We derive the homogenized Ginzburg-Landau equation with a 'strange term' (potential) and prove that the trajectory attractors of the given equation tend in a weak sense to the trajectory attractors of the homogenized one. Assuming additional conditions to be satisfied for the coefficients, we provide also a convergence of the global attractor.

In this paper the Ginzburg-Landau equation is considered in locally periodic porous medium, with rapidly oscillating terms in the equation and boundary conditions. It is proved that the trajectory attractors of this equation converge in a weak sense to the trajectory attractors of the limit Ginzburg-Landau equation with an additional potential term. For this aim we use an approach from the papers and monographs of V.V. Chepyzhov and M.I. Vishik concerning trajectory attractors of evolution equations. Also we apply homogenization methods appeared at the end of the XX-th century. First, we apply the asymptotic methods for formal construction of asymptotics, then, we verify the leading terms of asymptotic series by means of the methods of functional analysis and integral estimates. Defining the appropriate axillary functional spaces with weak topology, we derive the limit (homogenized) equation and prove the existence of trajectory attractors for this equation. Then we formulate the main theorem and prove it with the help of axillary lemmas.

In the paper we consider a problem for complex Ginzburg–Landau equations in a medium with locally periodic small obstacles. It is assumed that the obstacle surface can have different conductivity coefficients. We prove that the trajectory attractors of this system converge in a certain weak topology to the trajectory attractors of the homogenized Ginzburg–Landau equations with an additional potential (in the critical case), without an additional potential (in the subcritical case) in the medium without obstacles, or disappear (in the supercritical case).

The two-dimensional system of Navier–Stokes equations in a medium with anisotropic variable viscosity and periodic small obstacles is considered. It is proved that the trajectory attractors of the system tend in a certain weak topology to the trajectory attractors of the homogenized system of Navier–Stokes equations with an additional potential in a medium without obstacles.

We study reaction–diffusion systems with rapidly oscillating terms in the coefficients of equations and in the boundary conditions, in media with periodic obstacles. The non-linear terms of the equations only satisfy general dissipation conditions. We construct trajectory attractors for such systems in the strong topology of the corresponding trajectory dynamical systems. By means of generalized Fatou’s lemma we prove the strong convergence of the trajectory attractors of considered systems to the trajectory attractors of the corresponding homogenized reaction–diffusion systems which contain an additional potential.

The theory of embedding of spaces of differentiable functions studies important relations of differential (smoothness) properties of functions in various metrics and has wide application in the theory of boundary value problems of mathematical physics, approximation theory and other fields of mathematics. In this article, we prove the theorems about traces and extensions for functions from Nikolsky-Besov spaces with generalized mixed smoothness and mixed metrics. The proofs of the obtained results is based on the inequality of different dimensions for trigonometric polynomials in Lebesgue spaces with mixed metrics and the embedding theorem of classical Nikolsky-Besov spaces in the space of continuous functions.

In a perforated domain, we consider the two-dimensional system of Navier–Stokes equations with rapidly oscillating terms in the equations and boundary conditions. We prove that the trajectory attractors of this system converge in some weak topology to trajectory attractors of the homogenized Navier–Stokes equations with an additional potential.

We obtain a sharp estimate for the order of the orthoprojection width of the Nikol’skii–Besov class with mixed smoothness and mixed metric in the metric of anisotropic Lorentz spaces.

This article deals with two-dimensional Navier-Stokes system of equations with rapidly oscillating terms in the equations and boundary conditions. Studying the problem in a perforated domain, the authors set homogeneous Dirichlet condition on the outer boundary and the Fourier (Robin) condition on the boundary of the cavities. Under such assumptions it is proved that the trajectory attractors of this system converge in some weak topology to trajectory attractors of the homogenized Navier-Stokes system of equations with an additional potential and nontrivial right hand side in the domain without pores. For this aim, the approaches from the works of A.V. Babin, V.V. Chepyzhov, J.-L. Lions, R. Temam, M.I. Vishik concerning trajectory attractors of evolution equations and homogenization methods appeared at the end of the XX-th century are used. First, we apply the asymptotic methods for formal construction of asymptotics, then, we verify the leading terms of asymptotic series by means of the methods of functional analysis and integral estimates. Defining the appropriate axillary functional spaces with weak topology, we derive the limit (homogenized) system of equations and prove the existence of trajectory attractors for this system. Lastly, we formulate the main theorem and prove it through axillary lemmas.

A system of reaction-diffusion equations in a perforated domain with rapidly oscillating terms in the equations and in the boundary conditions is considered. It is not assumed that the uniqueness theorem conditions are satisfied for the corresponding initial-boundary value problem. We have proved the strong convergence of the trajectory attractors of this system to the trajectory attractors of the homogenized reaction-diffusion system with a `strange term' (potential).

The theory of embedding of spaces of differentiable functions studies the important relations of differential (smoothness) properties of functions in various metrics and has a wide application in the theory of boundary value problems of mathematical physics, approximation theory, and other fields of mathematics. In this article, we prove the embedding theorems for anisotropic spaces Nikol’skii-Besov with a generalized mixed smoothness and mixed metric, and anisotropic Lorentz spaces. The proofs of the obtained results are based on the inequality of different metrics for trigonometric polynomials in Lebesgue spaces with mixed metrics and interpolation properties of the corresponding spaces.

In the paper we study a system of reaction–diffusion equations in a perforated domain with rapidly oscillating terms in the equation and in the boundary conditions. A nonlinear function in the equations may not satisfy the Lipschitz condition and hence, the uniqueness theorem for the corresponding initial–boundary value problem for the considered system of reaction-diffusion equations may not be satisfied. It was proved that the trajectory attractors of this system weakly converge in the corresponding topology to the trajectory attractors of the homogenized reaction-diffusion system with a “strange term” (potential).

We consider reaction-diffusion equation in perforated domain, with rapidly oscillating coefficient in boundary conditions. We do not assume any Lipschitz condition for the nonlinear function in the equation, so, the uniqueness theorem for the corresponding initial boundary value problem may not hold for the considered reaction-diffusion equation. We prove that the trajectory attractors of this equation tend in a weak sense to the trajectory attractors of the homogenized reaction-diffusion equation with a "strange term" (potential). Bibliography: 48 titles. (C) 2020 Elsevier Ltd. All rights reserved.

We study the behavior of attractors of the reaction–diffusion equation in a perforated domain as the small parameter characterizing the perforation tends to zero.

In this paper we study the interpolation properties of Nikol’skii-Besov spaces with a dominant mixed derivative and mixed metric with respect to anisotropic and complex interpolation methods. An interpolation theorem is proved for a weighted discrete space of vector-valued sequences l^α_q(A). It is shown that the Nikol’skii-Besov space under study is a retract of the space l^α_q(Lp). Based on the above results, interpolation theorems were obtained for Nikol’skii-Besov spaces with the dominant mixed derivative and mixed metric.

In this paper we estimate the order of the triginometric width of the Nikol'skii Besov classes Bp"(7,r) with mixed metric in the anisotropic Lorentz space Lge (77`) when 1<p= (p1,...,p,,) < 2 < q= (qi,...,q,,). The concept of a trigonometric width in the one-dimensional case was first introduce by R.S. Ismagilov and he established his estimates for certain classes in the space of continuous functions. For a function of several variables exact orders of trigonometric width of Sobolev class WT,, Nikol'skii class HT in the space Lq are established by E.S. Belinsky, V.E. Majorov, Yu. Makovoz, G.G. Magaril-Ilyaev, V.N. Temlyakov. This problem for the Besov class Bpr, was investigated by A.S. Romanyuk, D.B. Bazarkhanov. The trigonometric width for the anisotropic Nikol'skii-Besov classes Bp';7.- (r) in the metric of the anisotropic Lorentz spaces Lq9(77) was found by K.A. Bekmaganbetov and Ye. Toleugazy.