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.

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).

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.

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).

Автономное комплексное уравнение Гинзбурга–Ландау (Г.–Л.) и его глобальный аттрактор были изучены в ряде работ и книг (см., например, [1] – [5] и цитированную там литературу). В настоящей заметке исследуется неавтономное уравнение Г.–Л. и его траекторный аттрактор. При этом основное внимание уделяется тем случаям, когда единственность решения задачи Коши для этого уравнения к настоящему времени не установлена. 1. Рассматривается комплексное уравнение Гинзбурга–Ландау с коэффициентами и возбуждающей силой, зависящими от времени:

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.

The paper gives a comprehensive study of Inertial Manifolds for hyperbolic relaxations of an abstract semilinear parabolic equation in a Hilbert space. A new scheme of constructing Inertial Manifolds for such type of problems is suggested and optimal spectral gap conditions which guarantee their existence are established. Moreover, the dependence of the constructed manifolds on the relaxation parameter in the case of the parabolic singular limit is also studied.

We consider reaction-diffusion systems with random rapidly oscillating coefficient. We do not assume any Lipschitz condition for the nonlinear function in the system, so, the uniqueness theorem for the corresponding initial-value problem may not hold for the considered reaction-diffusion system. Under the assumption that the random function is ergodic and statistically homogeneous in space variables we prove that the trajectory attractors of these systems tend in a weak sense to the trajectory attractors of the homogenized reaction-diffusion systems whose coefficient is the average of the corresponding term of the original systems.

We consider complex Ginzburg-Landau (GL) type equations of the form: \begin{document}${\partial _t}u = (1 + \alpha i)\Delta u + R{\mkern 1mu} u + (1 + \beta i)|u{|^2}u + g,$ \end{document} where \begin{document}$R$\end{document} , \begin{document}$β$\end{document} , and \begin{document}$g$\end{document} are random rapidly oscillating real functions. Assuming that the random functions are ergodic and statistically homogeneous in space variables, we prove that the trajectory attractors of these systems tend to the trajectory attractors of the homogenized equations whose terms are the average of the corresponding terms of the initial systems. Bibliography: 52 titles.

We consider the damped and driven Navier-Stokes system with stress free boundary conditions and the damped Euler system in a bounded domain Omega subset of R-2. We show that the damped Euler system has a (strong) global attractor in H-1 (Omega). We also show that in the vanishing viscosity limit the global attractors of the Navier-Stokes system converge in the non-symmetric Hausdorff distance in H-1 (Omega) to the strong global attractor of the limiting damped Euler system (whose solutions are not necessarily unique). (C) 2017 Elsevier B.V. All rights reserved.

Given $\rho\in[0,1]$, we consider for $\varepsilon\in(0,1]$ the nonautonomous viscoelastic equation with a singularly oscillating external force $$ \partial_{tt} u-\kappa(0)\Delta u - \int_0^\infty \kappa'(s)\Delta u(t-s) d s +f(u)=g_{0}(t)+\varepsilon ^{-\rho }g_{1}(t/\varepsilon ) $$ together with the {\it averaged} equation $$ \partial_{tt} u-\kappa(0)\Delta u - \int_0^\infty \kappa'(s)\Delta u(t-s) d s +f(u)=g_{0}(t). $$ Under suitable assumptions on the nonlinearity and on the external force, the related solution processes $S_\varepsilon(t,\tau)$ acting on the natural weak energy space ${\mathcal H}$ are shown to possess uniform attractors ${\mathcal A}^\varepsilon$. Within the further assumption $\rho<1$, the family ${\mathcal A}^\varepsilon$ turns out to be bounded in ${\mathcal H}$, uniformly with respect to $\varepsilon\in[0,1]$. The convergence of the attractors ${\mathcal A}^\varepsilon$ to the attractor ${\mathcal A}^0$ of the averaged equation as $\varepsilon\to 0$ is also established.

with weak solutions having finite energy and enstrophy. We show that these (possibly non-unique) solutions satisfy the energy and enstrophy equality. It is shown that this system has a strong global and a strong trajectory attractor in the Sobolev space H-1. A similar result on the strong attraction holds in the spaces H-1 boolean AND {u : broken vertical bar broken vertical bar curl u broken vertical bar broken vertical bar L-p < infinity} for p >= 2.