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This article is cited in 6 scientific papers (total in 6 papers) Strong convergence of attractors of reaction-diffusion system with rapidly oscillating terms in an orthotropic porous medium K. A. Bekmaganbetovab, V. V. Chepyzhovc, G. A. Chechkinde a Kazakhstan Branch of Lomonosov Moscow State University, Nur-Sultan b Institute of Mathematics and Mathematical Modeling, Ministry of Education and Science, Republic of Kazakhstan, Almaty c Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), Moscow d Lomonosov Moscow State University e Institute of Mathematics with Computing Centre, Ufa Federal Research Centre, Russian Academy of Sciences, Ufa
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     IntroductionIn this paper, we study the attractors behaviour of the initial-boundary value problem for a system of nonlinear differential equations in perforated domains, depending on small parameter that approaches zero. In contrast to [1], we menage to prove the strong convergence of attractors. The problems in perforated domains have attracted a lot of attention of mathematicians in recent years see, for example, [2]–[7]). Periodic, locally periodic, almost periodic, as well as, random structures have been studied. In [8]–[14] one can find a detailed bibliography concerning the matter. We study the strong convergence and the limit behaviour of attractors as the small parameter, characterizing, in particular, also a perforation, tends to zero. To analyze this phenomenon, we use the averaging methods (see, for example, [2], [9], [10], [12], [14]–[16]), and subtle analysis of trajectory and global attractors. Note several papers on averaging of attractors that have been published recently (see [17]–[19]). In [17] and [19] the homogenization was studied for attractors of scalar evolution equations with dissipation in a periodically perforated domain. Attractors describe the behaviour of solutions to dissipative nonlinear evolution equations as time tends to infinity and, also, characterize the stability and instability for limit structures of the corresponding dynamical systems (see, e.g., the monographs [20]–[22] and the references therein). We are interested in asymptotic behaviour of trajectory and global attractors of a reaction-diffusion system with rapidly oscillating terms in a perforated domain that describe the behaviour of orthotropic media. The Bogolyubov averaging principle [23] was used in [24]–[26], the first papers on homogenization for attractors of evolution equations with rapidly oscillating terms. The homogenization of global attractors for parabolic equations with oscillating parameters has been considered in [21], [27]–[30]. Some questions related to homogenization of uniform global attractors for dissipative wave equations were studied in [31]–[34], in presence of time oscillations, and in [21], [35]–[37], in presence of space oscillations. Similar problems for autonomous and non-autonomous 2D Navier–Stokes equations were studied in [21], [36], [38], and [39]. The papers [39]–[43] deal with partial differential equations containing singular oscillating terms. Dissipative partial differential equations and the theory of their trajectory attractors were developed and described in [21], [44], and [45]. In the case of solution non-uniqueness this approach is especially effective both in the case when the uniqueness theorem for the corresponding initial problem is not proved yet (for example for the inhomogeneous 3D Navier–Stokes system), or in the case when such theorem does not hold (for the reaction-diffusion equations considered in the present paper). We prove in the paper that the trajectory attractor $\mathfrak{A}_{\varepsilon}$ of the reaction-diffusion system in perforated domain converges strongly as $\varepsilon \to 0$ to the trajectory attractor $\overline{\mathfrak{A}}$ of the homogenized system in an appropriate functional space. The parameter $\varepsilon$ here characterizes also the diameter of cavities and the distance between them in a perforated medium. In Section 1, we define the main concepts and formulate the known theorems about trajectory attractors of autonomous evolution equations. In Section 2 we determine the geometric structure of a perforated domain, next we formulate the problem to study, and describe necessary functional spaces. Section 3 is devoted to the construction of trajectory attractors for the reaction-diffusion systems in the strong topology for a fixed parameter $\varepsilon$. In Section 4, we study the homogenization of attractors for autonomous reaction-diffusion system with rapidly oscillating terms in a perforated domain and demonstrate the appearance of a ‘strange term’ (potential) in the homogenized system (concerning the appearance of a ‘strange term’, we refer the reader to the breakthrough papers [2] and [3]). § 1. Trajectory attractors of evolution equationsIn this section, we describe a general scheme for constructing trajectory attractors for autonomous evolution equations. We consider an abstract autonomous evolution equation
$$
\begin{equation}
\frac{\partial u}{\partial t}=A(u),\qquad t\geqslant 0.
\end{equation}
\tag{1}
$$
A nonlinear operator $A(\,{\cdot}\,)\colon E_1\to E_0$ is assumed to be given, where $E_1$ and $E_0$ are Banach spaces and $E_1\subseteq E_0$. For example, $A(u)=\lambda\Delta u-af(u)+g$ (see Section 2). We study global solutions $u(s)$ of the equation (1) as functions of the variable $s\in \mathbb{R}_+$. Here $s\equiv t$ denotes the time variable. A set of solutions of (1) is called a trajectory space $\mathcal{K}^+$ of the equation (1). Let us describe a trajectory space $\mathcal{K}^+$ in more details. First of all, we consider solutions $u(s)$ of (1) defined on a fixed time segment $[t_1,t_2]$ in $\mathbb{R}$. We study solutions of (1) belonging to a certain Banach space $\mathcal{F}_{t_1,t_2}$ that depends on $t_1$ and $t_2$. The space $\mathcal{F}_{t_1,t_2}$ consists of functions $f(s),s\in [t_1,t_2]$ such that $f(s)\in E$ for almost all $s\in [t_1,t_2]$, where $E$ is a Banach space. We assume that $E_1\subseteq E\subseteq E_0$. For example, the space $\mathcal{F}_{t_1,t_2}$ can be the space $C([t_1,t_2];E)$ or the space $L_{p}(t_1,t_2;E)$, $p\in [1,\infty ]$, or the intersection of such spaces (see Section 2). We assume that $\Pi_{t_1,t_2}\mathcal{F}_{\tau_1,\tau_2}\subseteq \mathcal{F}_{t_1,t_2}$ and
$$
\begin{equation}
\| \Pi_{t_1,t_2}f\|_{\mathcal{F}_{t_1,t_2}}\leqslant C(t_1,t_2,\tau_1,\tau_2) \|f\|_{\mathcal{F}_{\tau_1,\tau_2}}\quad \forall\, f\in \mathcal{F}_{\tau_1,\tau_2},
\end{equation}
\tag{2}
$$
where $[t_1,t_2]\subseteq [\tau_1,\tau_2]$ and $\Pi_{t_1,t_2}$ denotes the restriction operator on the segment $[t_1,t_2]$. The constant $C(t_1,t_2,\tau_1,\tau_2)$ is independent on $f$. Usually, the homogeneous case is considered where $C(t_1,t_2,\tau_1,\tau_2)=C(t_2-t_1,\tau_2-\tau_1)$. For $h\in \mathbb{R}$ we denote by $S(h)$ the translation operator It is clear that if the independent variable $s$ of the function $f(\,{\cdot}\,)$ belongs to the segment $[t_1,t_2]$, then the variable $s$ of the function $S(h)f(\,{\cdot}\,)$ belongs to the segment $[t_1-h,t_2-h]$ for $h\in \mathbb{R}$. Assume that the mapping $S(h)$ is an isomorphism from $F_{t_1,t_2}$ into $F_{t_1-h,t_2-h}$ and
$$
\begin{equation}
\| S(h)f\|_{\mathcal{F}_{t_1-h,t_2-h}}= \| f\|_{\mathcal{F}_{t_1,t_2}}\quad \forall\, f\in \mathcal{F}_{t_1,t_2}.
\end{equation}
\tag{3}
$$
This assumptions looks very natural, for example, in the case of time homogeneous spaces. We assume that $f(s)\in \mathcal{F}_{t_1,t_2}$ implies $A(f(s))\in \mathcal{D}_{t_1,t_2}$, where $\mathcal{D}_{t_1,t_2}$ is a wider Banach space than $\mathcal{F}_{t_1,t_2}$ and $\mathcal{F}_{t_1,t_2}\subseteq \mathcal{D}_{t_1,t_2}$. The derivative $\partial f(t)/\partial t$ is considered as a distribution with values in the space $E_0$, that is $\partial f/\partial t\in D'((t_1,t_2);E_0)$. We also assume that $\mathcal{D}_{t_1,t_2}\subseteq D'((t_1,t_2);E_0)$ for all $(t_1,t_2)\subset \mathbb{R}$. A function $u(s)\in \mathcal{F}_{t_1,t_2}$ is called a solution of the equation (1) in the space $\mathcal{F}_{t_1,t_2}$ (on the interval $(t_1,t_2)$), if the equality $\partial u(s)/\partial t=A(u(s))$ holds in the distribution sense of the space $D'((t_1,t_2);E_0)$. We also define the space
$$
\begin{equation}
\mathcal{F}_+^{\mathrm{loc}}=\{f(s),\,s\in \mathbb{R}_+\mid \Pi _{t_1,t_2}f(s)\in \mathcal{F}_{t_1,t_2}\ \forall\, [t_1,t_2]\subset \mathbb{R}_+\}.
\end{equation}
\tag{4}
$$
For example, if $\mathcal{F}_{t_1,t_2}=C([t_1,t_2];E)$, then $\mathcal{F}_+^{\mathrm{loc}}=C(\mathbb{R}_+;E)$, and if $\mathcal{F}_{t_1,t_2}=L_{p}(t_1,t_2;E)$, then $\mathcal{F}_+^{\mathrm{loc}}=L_{p}^{\mathrm{loc}}(\mathbb{R}_+;E)$. A function $u(s)\in \mathcal{F}_+^{\mathrm{loc}}$ is called a solution of the equation (1) in $\mathcal{F}_+^{\mathrm{loc}}$, if $\Pi_{t_1,t_2}u(s)\in \mathcal{F}_{t_1,t_2}$ and $\Pi_{t_1,t_2}u(s)$ is a solution of the equation (1) for every time segment $[t_1,t_2]\subset \mathbb{R}_+$. Let $\mathcal{K}^+$ be a set of solutions of the equation (1) that belongs to $\mathcal{F}_+^{\mathrm{loc}}$, which is not necessary the set of all solutions of this equation from $\mathcal{F}_+^{\mathrm{loc}}$. The elements of $\mathcal{K}^+$ are called trajectories, and the set $\mathcal{K}^+$ itself is called the trajectory space of (1). We assume that the trajectory space $\mathcal{K}^+$ is translation invariant in the following sense: if $u(s)\in \mathcal{K}^+$, then $u(h+s)\in \mathcal{K}^+$ for any $h\geqslant 0$. This property is quite natural for a solution of an autonomous equation in a homogeneous space. Now consider the translation operators $S(h)$ in $\mathcal{F}_+^{\mathrm{loc}}$: It is clear that the mappings $\{S(h),\, h\geqslant 0\}$ form a semigroup of operators in $\mathcal{F}_+^{\mathrm{loc}}$: $S(h_1)S(h_2)=S(h_1+h_2)$ for $h_1,h_2\geqslant 0$ and $S(0)=\mathrm{Id}$ is the identity operator. We replace the variable $h$ with the time variable $t$. The semigroup $\{S(t),\, t\geqslant 0\}$ is called the translation semigroup. Due to the translational invariance assumption, the translation semigroup maps the trajectory space $\mathcal{K}^+$ to itself:
$$
\begin{equation}
S(t)\mathcal{K}^+\subseteq \mathcal{K}^+\quad \forall\, t\geqslant 0.
\end{equation}
\tag{5}
$$
We study the attraction properties of the translation semigroup $\{S(t)\}$ acting on the trajectory space $\mathcal{K}^+\subset \mathcal{F}_+^{\mathrm{loc}}$ as $t\to +\infty$. We define some topology in the space $\mathcal{F}_+^{\mathrm{loc}}$. Let $\rho_{t_1,t_2}(\,{\cdot}\,,{\cdot}\,)$ be a metric defined on $\mathcal{F}_{t_1,t_2}$ for all segments $[t_1,t_2]\subset \mathbb{R}$. Similar to (2) and (3) we assume that
$$
\begin{equation*}
\begin{gathered} \, \rho_{t_1,t_2}(\Pi_{t_1,t_2}f,\Pi_{t_1,t_2}g)\,{\leqslant}\, D(t_1,t_2,\tau_1,\tau_2)\rho_{\tau_1,\tau_2}(f,g)\quad \! \forall\, f,g\,{\in}\, \mathcal{F}_{\tau_1,\tau_2},\ \ [t_1,t_2]\,{\subseteq}\, [\tau_1,\tau_2], \\ \rho_{t_1-h,t_2-h}(S(h)f,S(h)g) =\rho_{t_1,t_2}(f,g)\quad \forall\, f,g\in \mathcal{F}_{t_1,t_2},\quad [t_1,t_2]\subset \mathbb{R},\quad h\in \mathbb{R}. \end{gathered}
\end{equation*}
\notag
$$
For homogeneous spaces we usually have $D(t_1,t_2,\tau_1,\tau_2)=D(t_2-t_1, \tau_2-\tau_1)$. Denote by $\Theta_{t_1,t_2}$ the corresponding metric space on $\mathcal{F}_{t_1,t_2}$. For example, $\rho_{t_1,t_2}$ can be the metric generated by the norm $\|\,{\cdot}\,\|_{\mathcal{F}_{t_1,t_2}}$ of the Banach space $\mathcal{F}_{t_1,t_2}$. In applications, it can happen that the metric $\rho_{t_1,t_2}$ generates a weaker topology $\Theta_{t_1,t_2}$ than the strong convergence topology of the Banach space $\mathcal{F}_{t_1,t_2}$. Denote by $\Theta_+^{\mathrm{loc}}$ the space $\mathcal{F}_+^{\mathrm{loc}}$, equipped with topology of local convergence on $\Theta_{t_1,t_2}$ for every $[t_1,t_2]\subset \mathbb{R}_+$. Precisely speaking, by definition, a sequence of functions $\{f_k(s)\}\subset \mathcal{F}_+^{\mathrm{loc}}$ converges to a function $f(s)\in \mathcal{F}_+^{\mathrm{loc}}$ as $k\to \infty $ in $\Theta_+^{\mathrm{loc}}$, if $\rho_{t_1,t_2}(\Pi_{t_1,t_2}f_k,\Pi_{t_1,t_2}f)\to0$ as $k\to \infty $ for any segment $[t_1,t_2]\subset \mathbb{R}_+$. It is easy to prove that the topology $\Theta_+^{\mathrm{loc}}$ is metrizable, for example, using the Fréchet metric
$$
\begin{equation}
\rho_+(f_1,f_2):=\sum_{m\in \mathbb{N}}2^{-m} \frac{\rho_{0,m}(f_1,f_2)}{1+\rho_{0,m}(f_1,f_2)}.
\end{equation}
\tag{6}
$$
The metric space $\Theta_+^{\mathrm{loc}}$ is clearly complete if all the spaces $\Theta_{t_1,t_2}$ are complete. We note that the translation semigroup $\{S(t)\}$ is continuous in the topology $\Theta_+^{\mathrm{loc}}$. This assertion follows directly from the definition of the topology in $\Theta_+^{\mathrm{loc}}$. We introduce the following Banach space:
$$
\begin{equation}
\mathcal{F}_+^{\mathrm b}:=\{f(s)\in \mathcal{F}_+^{\mathrm{loc}}\mid \| f\|_{\mathcal{F}_+^{\mathrm b}}<+\infty \},
\end{equation}
\tag{7}
$$
with the norm
$$
\begin{equation}
\| f\|_{\mathcal{F}_+^{\mathrm b}}:=\sup_{h\geqslant 0}\| \Pi_{0,1}f(h+s)\|_{\mathcal{F}_{0,1}}.
\end{equation}
\tag{8}
$$
For example, if $\mathcal{F}_+^{\mathrm{loc}}=C(\mathbb{R}_+;E)$, then the space $\mathcal{F}_+^{\mathrm b}=C^{\mathrm b}(\mathbb{R}_+;E)$ with the norm $\| f\|_{\mathcal{F}_+^{\mathrm b}}=\sup_{h\geqslant 0}\| f(h)\|_{E}$, and if $\mathcal{F}_+^{\mathrm{loc}}=L_{p}^{\mathrm{loc}}(\mathbb{R}_+;E)$, then $\mathcal{F}_+^{\mathrm b}=L_{p}^{\mathrm b}(\mathbb{R}_+;E)$ with the norm $\| f\|_{\mathcal{F}_+^{\mathrm b}}=\bigl( \sup_{h\geqslant 0}\int_{h}^{h+1}\| f(s)\|_{E}^{p}\, ds\bigr)^{1/p}$. We note that $\mathcal{F}_+^{\mathrm b}\subseteq \Theta_+^{\mathrm{loc}}$. We need the Banach space $\mathcal{F}_+^{\mathrm b}$ to define bounded sets in the trajectory space $\mathcal{K}^+$. Constructing a trajectory attractor in $\mathcal{K}^+$, we do not use the corresponding uniform metric in the Banach space $\mathcal{F}_+^{\mathrm b}$ to define attraction to the attractor. Instead, we use the metric of local convergence $\Theta_+^{\mathrm{loc}}$, which is essentially weaker. We assume that $\mathcal{K}^+\subseteq \mathcal{F}_+^{\mathrm b}$, that is, any trajectory $u(s)\in \mathcal{K}^+$ of the equation (1) has finite norm (8). We formulate the definitions of an attracting set and a trajectory attractor of the translation semigroup $\{S(t)\}$ acting on $\mathcal{K}^+$. Definition 1.1. A set $\mathcal{P}\subseteq \Theta_+^{\mathrm{loc}}$ is called an attracting set of the translation semigroup $\{S(t)\}$ acting on $\mathcal{K}^+$, in the topology $\Theta_+^{\mathrm{loc}}$, if for any set $\mathcal{B}\subseteq \mathcal{K}^+$, bounded in $\mathcal{F}_+^{\mathrm b}$, the set $\mathcal{P}$ attracts $S(t)\mathcal{B}$ as $t\to +\infty $ in the topology $\Theta_+^{\mathrm{loc}}$, that is, for any $\varepsilon$-neighborhood $O_{\varepsilon}(\mathcal{P})$ in $\Theta_+^{\mathrm{loc}}$ there exists $t_1\geqslant 0$ such that $S(t)\mathcal{B}\subseteq O_{\varepsilon}(\mathcal{P})$ for all $t\geqslant t_1$. The attraction property of $\mathcal{P}$ can be formulated in the following equivalent form: for any set $\mathcal{B}\subseteq \mathcal{K}^+$ bounded in $\mathcal{F}_+^{\mathrm b}$ and for any $M>0$
$$
\begin{equation*}
\operatorname{dist}_{\Theta_{0,M}}\bigl(\Pi_{0,M}S(t)\mathcal{B},\Pi_{0,M}\mathcal{P}\bigr)\to 0,\qquad t\to +\infty,
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\operatorname{dist}_{\mathcal{M}}(X,Y):=\sup_{x\in X} \operatorname{dist}_{\mathcal{M}}(x,Y)=\sup_{x\in X} \inf_{y\in Y}\rho_{\mathcal{M}}(x,y)
\end{equation*}
\notag
$$
denotes the (nonsymmetric) Hausdorff semidistance from a set $X$ to a set $Y$ in the metric space $\mathcal{M}$. Definition 1.2 (see [21]). A set $\mathfrak{A}\subseteq \mathcal{K}^+$ is called a trajectory attractor of the translation semigroup $\{S(t)\}$ on $\mathcal{K}^+$ in the topology $\Theta_+^{\mathrm{loc}}$, if (i) $\mathfrak{A}$ is bounded in $\mathcal{F}_+^{\mathrm b}$ and compact in $\Theta_+^{\mathrm{loc}}$; (ii) $\mathfrak{A}$ is strictly invariant with respect to the translation semigroup: $S(t)\mathfrak{A}=\mathfrak{A}$ for all $t\geqslant 0$; (iii) $\mathfrak{A}$ is an attracting set of the translation semigroup $\{S(t)\}$ on $\mathcal{K}^+$ in the topology $\Theta_+^{\mathrm{loc}}$, that is, for every $M>0$ Remark 1.1. Following the terminology of [20], we can say that a trajectory attractor $\mathfrak{A}$ is a global $(\mathcal{F}_+^{\mathrm b},\Theta_+^{\mathrm{loc}})$-attractor of the translation semigroup $\{S(t)\}$ acting on $\mathcal{K}^+$, that is, $\mathfrak{A}$ attracts $S(t)\mathcal{B}$ as $t\to +\infty $ in the topology $\Theta_+^{\mathrm{loc}}$, where $\mathcal{B}$ is an arbitrary bounded (in $\mathcal{F}_+^{\mathrm b}$) set from $\mathcal{K}^+$: We formulate the main theorems on existence and the structure of a trajectory attractor for the equation (1). Theorem 1.1 (cf. [20], [21], [44]). Let the trajectory space $\mathcal{K}^+$ of the equation (1) satisfies condition (5). Suppose that the translation semigroup $\{S(t)\}$ has an attracting set $\mathcal{P}\subseteq\mathcal{K}^+$ that is bounded in $\mathcal{F}_+^{\mathrm b}$ and compact in $\Theta_+^{\mathrm{loc}}$. Then the translation semigroup $\{S(t),\,t\geqslant 0\}$ acting on $\mathcal{K}^+$ has a trajectory attractor $\mathfrak{A}\subseteq \mathcal{P}$. The set $\mathfrak{A}$ is bounded in $\mathcal{F}_+^{\mathrm b}$ and compact in $\Theta_+^{\mathrm{loc}}$. We describe the structure of the trajectory attractor of the equation (1) using complete trajectories. Consider (1) on the entire time axis
$$
\begin{equation}
\frac{\partial u}{\partial t}=A(u),\qquad t\in \mathbb{R}.
\end{equation}
\tag{9}
$$
We have already defined the trajectory space $\mathcal{K}^+$ of the equation (9) on $\mathbb{R}_+$. We now expand this definition to the entire time axis $\mathbb{R}$. If a function $f(s)$, $s\in\mathbb{R}$, is given on the time axis, then the translations $S(h)f(s)=f(s+h)$ are also defined for negative $h$. A function $u(s)$, $s\in \mathbb{R}$, is called a complete trajectory of the equation (9), if $\Pi_+u(s+h)\in \mathcal{K}^+$ for any $h\in \mathbb{R}$. Here, $\Pi_+=\Pi_{0,\infty}$ denotes the restriction operator on the semi-axis $\mathbb{R}_+$. We have introduced the spaces $\mathcal{F}_+^{\mathrm{loc}},\mathcal{F}_+^{\mathrm b}$, and $\Theta_+^{\mathrm{loc}}$. Similarly, we define the spaces $\mathcal{F}^{\mathrm{loc}}$, $\mathcal{F}^{\mathrm b}$, and $\Theta^{\mathrm{loc}}$:
$$
\begin{equation*}
\begin{aligned} \, \mathcal{F}^{\mathrm{loc}} &:=\{f(s),\, s\in \mathbb{R}\mid \Pi_{t_1,t_2}f(s)\in \mathcal{F}_{t_1,t_2}\ \forall\, [t_1,t_2]\subseteq \mathbb{R}\}; \\ \mathcal{F}^{\mathrm b} &:=\{f(s)\in \mathcal{F}^{\mathrm{loc}}\mid \| f\|_{\mathcal{F}^{\mathrm b}}<+\infty \}, \end{aligned}
\end{equation*}
\notag
$$
where
$$
\begin{equation}
\| f\|_{\mathcal{F}^{\mathrm b}}:=\sup_{h\in \mathbb{R}}\| \Pi_{0,1}f(h+s)\|_{\mathcal{F}_{0,1}}.
\end{equation}
\tag{10}
$$
The topological space $\Theta^{\mathrm{loc}}$ coincides (as a set) with $\mathcal{F}^{\mathrm{loc}}$ and, by definition, $f_k(s)\to f(s)$ $(k\to \infty)$ in $\Theta^{\mathrm{loc}}$, if $\Pi_{t_1,t_2}f_k(s)\to \Pi_{t_1,t_2}f(s)$ $(k\to \infty)$ in $\Theta_{t_1,t_2}$ for every $[t_1,t_2]\subseteq \mathbb{R}$. It is obvious that $\Theta^{\mathrm{loc}}$ is a metric space as well as $\Theta_+^{\mathrm{loc}}$. Definition 1.3. The kernel $\mathcal{K}$ in the space $\mathcal{F}^{\mathrm b}$ of the equation (9) is the union of all complete trajectories $u(s)$, $s\in\mathbb{R}$, of (9) bounded in $\mathcal{F}^{\mathrm b}$ in the norm (10): Theorem 1.2. Let the assumptions of Theorem 1.1 hold. Then The set $\mathcal{K}$ is compact in $\Theta^{\mathrm{loc}}$ and bounded in $\mathcal{F}^{\mathrm b}$. The complete proofs of Theorems 1.1 and 1.2 are presented in [21] and [44]. To prove that a ball from the space $\mathcal{F}_+^{\mathrm b}$ is compact in $\Theta_+^{\mathrm{loc}}$, we will use the following Lemma 1.1. Let $E_0$ and $E_1$ be Banach spaces such that $E_1\subset E_0$. Consider the Banach spaces
$$
\begin{equation*}
\begin{aligned} \, W_{p_1,p_0}(0,M;E_1,E_0)&=\{\psi(s),\, s\in 0,M\mid \psi(\,{\cdot}\,)\in L_{p_1}(0,M;E_1), \\ &\qquad\qquad\psi'(\,{\cdot}\,)\in L_{p_0}(0,M;E_0)\}, \\ W_{\infty,p_0}(0,M;E_1,E_0)&=\{\psi(s),\, s\in 0,M\mid \psi(\,{\cdot}\,)\in L_{\infty}(0,M;E_1), \\ &\qquad\qquad\psi'(\,{\cdot}\,)\in L_{p_0}(0,M;E_0)\} \end{aligned}
\end{equation*}
\notag
$$
(where $p_1\geqslant 1$ and $p_0>1$), with the norms
$$
\begin{equation*}
\begin{aligned} \, \| \psi \|_{W_{p_1,p_0}}& :=\biggl( \int_0^M\| \psi (s)\|_{E_1}^{p_1}\, ds\biggr)^{1/p_1} +\biggl( \int_0^M\| \psi'(s)\|_{E_0}^{p_0}\, ds\biggr)^{1/p_0}, \\ \| \psi \|_{W_{\infty,p_0}}& :=\operatorname{ess\,sup} \{\|\psi(s)\|_{E_1}\mid s\in [0,M]\} +\biggl( \int_0^M\| \psi'(s)\|_{E_0}^{p_0}\, ds\biggr)^{1/p_0}. \end{aligned}
\end{equation*}
\notag
$$
Lemma 1.1 (Oben–Lions–Simon, see [46]). Assume that $E_1\Subset E\subset E_0$. Then the following embeddings are compact: In the next sections, we study reaction-diffusion systems and their trajectory attractors depending on a small parameter $\varepsilon >0$. Definition 1.4. We say that trajectory attractors $\mathfrak{A}_{\varepsilon}$ converge to a trajectory attractor $\overline{\mathfrak{A}}$ as $\varepsilon\to 0$ in the topological space $\Theta_+^{\mathrm{loc}}$, if for any neighborhood $\mathcal{O}(\mathfrak{A})$ in $\Theta_+^{\mathrm{loc}}$ there exists $\varepsilon_1> 0$ such that $\mathfrak{A}_{\varepsilon}\subseteq \mathcal{O}(\overline{\mathfrak{A}})$ for all $\varepsilon<\varepsilon_1$, that is, for any $M>0$ § 2. Notations and problem statementLet $\Omega$ be a bounded domain in $\mathbb{R}^n$, $n\geqslant 3$, containing the origin with the piecewise smooth boundary $\partial\Omega$. Let $G_0$ be a domain belonging to $Y=(-1/2, 1/2)^n$ such that $\overline{G}_0$ is a compact set diffeomorphic to a ball. Assuming $\delta>0$ and $Z$ a set, we introduce the following notation: $\delta Z=\{x\colon \delta^{-1}x\in Z\}$. Let $\varepsilon>0$ be small enough to have For $j\in\mathbb{Z}^n$, we define
$$
\begin{equation*}
P^j_\varepsilon=\varepsilon j, \qquad Y^j_\varepsilon=P^j_\varepsilon+\varepsilon Y, \qquad G^j_\varepsilon=P^j_\varepsilon+\varepsilon^{n/{(n-2)}}G_0.
\end{equation*}
\notag
$$
Denote the domain $\widetilde{\Omega}_\varepsilon=\{x\in\Omega\colon \rho(x, \partial\Omega)>\sqrt{n}\,\varepsilon\}$ and a set of admissible indices
$$
\begin{equation*}
\Upsilon_\varepsilon=\bigl\{j\in\mathbb{Z}^n\colon G^j_\varepsilon \cap \overline{\widetilde{\Omega}}_\varepsilon\neq\varnothing\bigr\}.
\end{equation*}
\notag
$$
We note that $|\Upsilon_\varepsilon|\cong d\varepsilon^{-n}$, where $d>0$ is a constant. Consider the domain
$$
\begin{equation*}
{\Omega}_\varepsilon=\Omega\setminus\overline{G}_\varepsilon,\quad \text{where}\quad G_\varepsilon=\bigcup_{j\in \Upsilon_\varepsilon} G^j_\varepsilon.
\end{equation*}
\notag
$$
We introduce the following notations for cylindrical domains:
$$
\begin{equation*}
Q_\varepsilon=\Omega_\varepsilon\times (0,+\infty), \qquad Q=\Omega\times(0,+\infty).
\end{equation*}
\notag
$$
We study the asymptotic behaviour of trajectory attractors for the following initial-boundary value problem:
$$
\begin{equation}
\begin{alignedat}{2} &\frac{\partial u_\varepsilon}{\partial t} =\lambda\Delta u_\varepsilon- a\biggl(x, \frac x\varepsilon\biggr)f(u_\varepsilon) +g\biggl(x, \frac x\varepsilon\biggr), &\quad x&\in \Omega_\varepsilon, \\ &\frac{\partial u_\varepsilon}{\partial \nu} +\varepsilon^{n/(n-2)} B^j_\varepsilon(x) u_\varepsilon =0, &\quad x&\in \partial G^j_\varepsilon,\ j\in \Upsilon_\varepsilon,\ t\in(0,+\infty), \\ &u_\varepsilon =0, &\quad x&\in \partial\Omega, \\ &u_\varepsilon= U(x), &\quad x&\in\Omega_\varepsilon,\ t=0, \end{alignedat}
\end{equation}
\tag{13}
$$
where $u_\varepsilon(x,t)=(u^1_\varepsilon,\dots, u^N_\varepsilon)^\top$ is the unknown vector function, $f=(f^1,\dots, f^N)^\top$ is the known nonlinear function, $g=(g^1,\dots,g^N)^\top$ is the known right-hand side function, $\lambda$ is the $N\times N$-matrix with constant coefficients having a positive symmetric part: $(\lambda+\lambda^{\top})/2 \geqslant \beta I$ (here $\beta>0$ and $I$ is the identity matrix of order $N$), and $\nu$ is the outward normal vector to the boundaries $G^j_\varepsilon$. The function $a(x,y)\in C(\overline{\Omega} \times \mathbb{R}^n)$ satisfies the inequalities $0<a_0 \leqslant a(x,y) \leqslant A_0$ with some constants $a_0$, $A_0$. We assume that the function $a_\varepsilon(x)=a(x,x/\varepsilon)$ has the mean $\overline{a}(x)$ as $\varepsilon \to 0{+}$ in the space $L_{\infty,*w}(\Omega)$, that is,
$$
\begin{equation}
\int_{\Omega}a\biggl(x, \frac x\varepsilon\biggr)\varphi(x)\, dx \to \int_{\Omega}\overline{a}(x)\varphi(x)\, dx,\qquad \varepsilon \to 0{+},
\end{equation}
\tag{14}
$$
for every $\varphi\in L_1(\Omega)$. For the function $g(x,y)$, we assume that for every $\varepsilon>0$ the functions $g^i_\varepsilon(x)=g^i(x,x/\varepsilon)\in H^{-1}(\Omega)$ and have the means $\overline{g}^{\,i}(x)$ in the space $V'=H^{-1}(\Omega)$ as $\varepsilon \to 0{+}$, that is,
$$
\begin{equation*}
g^i\biggl(x, \frac x\varepsilon\biggr) \rightharpoondown \overline{g}^{\,i}(x) \qquad (\varepsilon \to 0{+}) \quad \text{weakly in }V'
\end{equation*}
\notag
$$
or
$$
\begin{equation}
\int_{\Omega}g^i\biggl(x, \frac x\varepsilon\biggr) \varphi(x)\, dx\to\int_{\Omega}\overline{g}^{\,i}(x)\varphi(x)\, dx,\qquad \varepsilon \to 0{+},
\end{equation}
\tag{15}
$$
for any function $\varphi\in V=H_0^1(\Omega)$ and for all $i=1,\dots, N$. The assumption (15) implies that the norms of functions $g^i_\varepsilon(x)$ are uniformly bounded with respect to $\varepsilon$ in the space $V'$:
$$
\begin{equation}
\|g^i_\varepsilon(x)\|_{-1}\leqslant M_0\quad \forall\, \varepsilon\in (0,1].
\end{equation}
\tag{16}
$$
The matrix $B_\varepsilon^j(x)$ in (13) is diagonal with bounded elements
$$
\begin{equation*}
b^{11}\biggl(x, \frac{x-P^j_\varepsilon}{\varepsilon^{n/(n-2)}}\biggr),\ \dots,\ b^{NN}\biggl(x, \frac{x-P^j_\varepsilon}{\varepsilon^{n/(n-2)}}\biggr),\qquad j\in \Upsilon_\varepsilon,
\end{equation*}
\notag
$$
where $b^{kk}(x,y)\in C(\Omega \times \mathbb{R}^n)$ is a 1-periodic functions in $y$ such that with some constants $b_0$, $B_0$ for all $k=1, \dots, N$. We denote also the vector
$$
\begin{equation*}
\overline B(x,y):=\bigl(b^{11}(x,y), \dots, b^{NN}(x,y)\bigr)^\top,
\end{equation*}
\notag
$$
and the diagonal matrix $B(x,y)$ with elements $b^{11}(x,y), \dots,b^{NN}(x,y)$. We assume that the nonlinear vector-valued function $f(v) \in C(\mathbb{R}^N;\mathbb{R}^N)$ satisfies the following inequalities:
$$
\begin{equation}
\sum_{i=1}^N|f^i(v)|^{p_i/{(p_i-1)}}\leqslant C_0\biggl(\sum_{i=1}^N|v^i|^{p_i}+1\biggr),\qquad 2\leqslant p_1\leqslant \dots \leqslant p_{N-1}\leqslant p_{N},
\end{equation}
\tag{18}
$$
$$
\begin{equation}
\sum_{i=1}^N\gamma_i|v^i|^{p_i}-C\leqslant \sum_{i=1}^N f^i(v)v^i\quad \forall\, v\in \mathbb{R}^N,
\end{equation}
\tag{19}
$$
where $\gamma_i>0$ for all $i=1, \dots, N$. The inequality (18) is connected with a fact that, the functions $f^i(u)$ in a real reaction-diffusion systems are polynomials of, possibly, different degrees. The inequality (19) is called the dissipation condition for a reaction-diffusion system (13). In a simple model case $p_i\equiv p$ for all $i=1, \dots, N$, and the conditions (18) and (19) are reduced to the following inequalities:
$$
\begin{equation}
|f(v)| \leqslant C_0(|v|^{p-1}+1),\quad \gamma |v|^p-C\leqslant f(v)v\qquad \forall\, v\in \mathbb{R}^N.
\end{equation}
\tag{20}
$$
Note that no Lipschitz condition is assumed for the function $f(v)$ with respect to the variable $v$. Remark 2.1. The methods presented here may be used also to study systems with nonlinear terms of the form $\sum_{j=1}^m a_j(x,x/\varepsilon) f_j(u)$ where $a_j$ are matrices whose elements admit homogenization and $f_j(u)$ are vector-valued polynomial in $u$ that satisfy conditions of the form (18) and (19). For the sake of brevity we consider only the case $m=1$ and $a_1(x,x/\varepsilon) =a(x,x/\varepsilon) I$, where $I$ is the identity matrix. Let us consider some examples of functions satisfying the homogenization conditions (14) and (15). Their justification can be found in [36]. Example 2.1. Let the function $a(x,y)\in C(\overline{\Omega} \times \mathbb{R}^n)$ be periodic in each variable $y_k$, $k=1,\dots,n$, with period $1$. Then property (14) clearly holds for the function $a(x, x/\varepsilon)$ and the mean is where $\mathbb{T}^{n}=\mathbb{R}^{n}\ (\operatorname{mod} 1)$ is an $n$-dimensional torus. Let the vector function $g(x,y)\in C(\mathbb{R}^n;H^{-1}(\Omega))$ is also $1$-periodic with values in $H^{-1}(\Omega)$ in each variable $y_k$, $k=1,\dots,n$. Property (15) takes place for $g(x, x/\varepsilon)$ with the mean Example 2.2. Let the functions $a(x,y)$ and $g(x,y)$ be quasiperiodic in the corresponding spaces. For example, for $a(x,y)$ it means that there exists a continuous function which is $1$-periodic in each variable $\xi_{ij}$ and such that where $\{\alpha_{ij}\}_{j=1,\dots,k_i}^{i=1,\dots,n}$ are some rationally independent real numbers. The similar formulas hold for components of the vector-valued function $g(x,y)$. The mean function $\overline{a}(x)$ is obtained by homogenization of the function $A(x,\,\cdot\,)$ over all tori $\mathbb{T}^{k_1}\times \dots \times\mathbb{T}^{k_n}$: The set $\overline{\Omega }\times \mathbb{T}^{k_1}\times \dots \times \mathbb{T}^{k_n}$ is compact. Therefore, the function $a(x,y)$ is uniformly continuous in $x$:
$$
\begin{equation}
|a(x_1,y)-a(x_2,y)|\leqslant \alpha (|x_1-x_2|)\quad\forall\, x_1,x_2\in \overline{\Omega},\quad \forall\, y\in \mathbb{R}^n,
\end{equation}
\tag{23}
$$
where $\alpha (s)\to 0$ $(s\to 0{+})$ and $\alpha (s)$ is independent of $y$. Example 2.3. Consider a function $a(x,y)\in C_b(\overline{\Omega} \times \mathbb{R}^n)$ that satisfies (23). Let $a(x,y)$ be almost periodic in $y$ in the sense of Bohr, that is, there exist quasiperiodic functions $a_N(x,y)\in C_b(\overline{\Omega}\times \mathbb{R}^n)$ (see (21)) that satisfy (23) with the same function $\alpha(s)$ such that Analogously, we can construct examples of vector-valued almost periodic functions $g(x,y)$ with values in $H^{-1}(\Omega)$, for which the function $g(x,x/\varepsilon)$ can be homogenized over $\varepsilon$. Notice that all examples considered admit vector functions
$$
\begin{equation*}
g\biggl(x, \frac x\varepsilon\biggr)=G_0\biggl( x,\frac x\varepsilon \biggr) +\sum_{i=1}^n\partial_{x_i}G_i\biggl( x,\frac x\varepsilon \biggr),
\end{equation*}
\notag
$$
where $G_i(x,y)\in C(\mathbb{R}^n;L_2(\Omega))$ are periodic, quasiperiodic or almost periodic functions with values in the space $L_2(\Omega)$ and have means $\overline G_i(x)\in L_2(\Omega)$, $i=0,1,\dots,n$. Hence, the unbounded growth is possible for the $L_2(\Omega)$-norms of functions
$$
\begin{equation*}
\partial_{x_i}G_i\biggl( x,\frac x\varepsilon \biggr) =G_{ix_i}\biggl( x,\frac x\varepsilon \biggr) +\frac 1\varepsilon\, G_{iy_i}\biggl( x,\frac x\varepsilon \biggr)\quad \text{as}\quad \varepsilon \to 0{+}.
\end{equation*}
\notag
$$
These functions are bounded only in the space $H^{-1}(\Omega)$. Let us put $\mathbf{H}:=[L_2(\Omega)]^N$, $\mathbf{H_\varepsilon}:=[L_2(\Omega_\varepsilon)]^N$, $\mathbf{V}:=[H_0^1(\Omega)]^N$, and let $\mathbf{V}_\varepsilon:=[H^1(\Omega_\varepsilon;\partial\Omega)]^N$ be the set of functions from $[H^1(\Omega_\varepsilon)]^N$ with zero trace on $\partial\Omega$. The norm in these spaces are defined by
$$
\begin{equation*}
\begin{alignedat}{2} \| v\|^2 &:=\int_{\Omega}\sum_{i=1}^N|v^i(x)|^2\, dx, &\qquad \|v\|^2_\varepsilon &:=\int_{\Omega_\varepsilon}\sum_{i=1}^N|v^i(x)|^2\, dx, \\ \| v\|_1^2 &:=\int_{\Omega}\sum_{i=1}^N|\nabla v^i(x)|^2\, dx, &\qquad \|v\|_{1\varepsilon}^2 &:=\int_{\Omega_\varepsilon}\sum_{i=1}^N|\nabla v^i(x)|^2\, dx. \end{alignedat}
\end{equation*}
\notag
$$
Let $\mathbf{V}':=[H^{-1}(\Omega)]^N$ be the dual space of $\mathbf{V}$, while $\mathbf{V}'_\varepsilon$ be the dual space of $\mathbf{V}_\varepsilon$. We set $q_i=p_i/{(p_i-1)}$ for all $i=1,\dots, N$, we put the following vector notations: $\mathbf{p}=(p_1, \dots,p_N)$ and $\mathbf{q}=(q_1, \dots, q_N)$, and we introduce the spaces
$$
\begin{equation*}
\begin{gathered} \, \mathbf{L}_\mathbf{p} :=L_{p_1}(\Omega)\times\dots\times L_{p_N}(\Omega), \qquad \mathbf{L}_{\mathbf{p},\varepsilon}:=L_{p_1}(\Omega_\varepsilon)\times\dots\times L_{p_N}(\Omega_\varepsilon), \\ \begin{aligned} \, \mathbf{L}_{\mathbf{p}}(\mathbb{R}_+;\mathbf{L}_{\mathbf{p}}) &:=L_{p_1}(\mathbb{R}_+;L_{p_1}(\Omega))\times\dots\times L_{p_N}(\mathbb{R}_+;L_{p_N}(\Omega)), \\ \mathbf{L}_{\mathbf{p}}(\mathbb{R}_+;\mathbf{L}_{\mathbf{p},\varepsilon}) &:=L_{p_1}(\mathbb{R}_+;L_{p_1}(\Omega_\varepsilon))\times\dots\times L_{p_N}(\mathbb{R}_+;L_{p_N}(\Omega_\varepsilon)). \end{aligned} \end{gathered}
\end{equation*}
\notag
$$
As in [21] and [48], we study weak solutions of the initial-boundary value problem (13)
$$
\begin{equation*}
u_\varepsilon(x,t)\in \mathbf{L}_\infty^{\mathrm{loc}}(\mathbb{R}_+;\mathbf{H}_\varepsilon) \cap \mathbf{L}_2^{\mathrm{loc}}(\mathbb{R}_+;\mathbf{V}_\varepsilon)\cap \mathbf{L}_{\mathbf{p}}^{\mathrm{loc}}(\mathbb{R}_+;\mathbf{L}_{\mathbf{p},\varepsilon}),
\end{equation*}
\notag
$$
which satisfy the problem (13) in the distribution sense, which means that the following integral identity holds:
$$
\begin{equation}
\begin{aligned} \, &-\int_{Q_\varepsilon} u_\varepsilon \cdot\frac{\partial\psi}{\partial t} \, dx\, dt+ \int_{Q_\varepsilon} \lambda\nabla u_\varepsilon \cdot\nabla \psi\, dx\, dt+ \int_{Q_\varepsilon} a_\varepsilon(x) f(u_\varepsilon)\cdot\psi\, dx\,dt \nonumber \\ &\qquad+\varepsilon^{n/(n-2)}\sum_{j\in\Upsilon_\varepsilon} \int_0^{+\infty} \int_{\partial G^j_\varepsilon} B^j_\varepsilon (x) u_\varepsilon \cdot\psi\, d\omega\, dt =\int_{Q_\varepsilon} g_\varepsilon(x)\cdot\psi\, dx\, dt \end{aligned}
\end{equation}
\tag{24}
$$
for all functions $\psi\in \mathbf{C}_0^\infty(\mathbb{R}_+;\mathbf{V}_\varepsilon\cap \mathbf{L}_{\mathbf{p},\varepsilon})$. Here, $y_1\cdot y_2$ denotes the inner product of vectors $y_1, y_2\in\mathbb{R}^N$. By $d\omega$, we denote an element of $(n-1)$-dimensional volume on the boundaries $\partial G^j_\varepsilon$. If $u_\varepsilon(x,t)\in \mathbf{L}_{\mathbf{p}}(0,M;\mathbf{L}_{\mathbf{p},\varepsilon})$, then (18) implies $f(u(x,t))\in \mathbf{L}_{\mathbf{q}}(0,M;\mathbf{L}_{\mathbf{q},\varepsilon})$. At the same time, if $u_\varepsilon(x,t)\in \mathbf{L}_2(0,M;\mathbf{V}_\varepsilon)$, then $\lambda\Delta u_\varepsilon(x,t)+g_\varepsilon(x) \in \mathbf{L}_2(0,M;\mathbf{V}'_\varepsilon)$. Therefore, for an arbitrary weak solution $u_\varepsilon(x,s)$ of the problem (13) we have
$$
\begin{equation*}
\frac{\partial u_\varepsilon(x,t)}{\partial t}\in \mathbf{L}_{\mathbf{q}}(0,M;\mathbf{L}_{\mathbf{q}, \varepsilon})+\mathbf{L}_2(0,M;\mathbf{V}'_\varepsilon).
\end{equation*}
\notag
$$
Sobolev embedding theorem implies
$$
\begin{equation*}
\mathbf{L}_{\mathbf{q}}(0,M;\mathbf{L}_{\mathbf{q},\varepsilon}) +\mathbf{L}_2(0,M;\mathbf{V}'_\varepsilon)\subset \mathbf{L}_{\mathbf{q}}(0,M;\mathbf{H}^{-\mathbf{r}}_\varepsilon),
\end{equation*}
\notag
$$
where $\mathbf{H}^{-\mathbf{r}}_\varepsilon:=H^{-r_1}(\Omega_\varepsilon)\times\dots\times H^{-r_N}(\Omega_\varepsilon)$, $\mathbf{r}=(r_1,\dots,r_N)$, and exponents $r_i=\max \{1,n(1/{q_i}-1/2)\}$ for $i=1,\dots, N$. Here, $H^{-r}(\Omega_\varepsilon)$ denotes the dual of the Sobolev space $H^{r}(\Omega_\varepsilon)$ with exponent $r>0$ in the perforated domain $\Omega_\varepsilon$. Therefore, for any weak solution $u_\varepsilon(x,t)$ of the problem (13), its time derivative $\partial u_\varepsilon(x,t)/\partial t$ belongs to $\mathbf{L}_{\mathbf{q}}(0,M;\mathbf{H}^{-\mathbf{r}}_\varepsilon)$. Remark 2.2. The existence of a weak solution $u(x,t)$ of the problem (13) for any initial function $U\in \mathbf{H}_\varepsilon$ and fixed $\varepsilon$ is proved in a standard way (see, for example, [20], [48]). This solution is not necessarily unique since the function $f(v)$ satisfies only the conditions (18), (19) and no Lipschitz condition is imposed with respect to $v$. The following lemma can be proved by the same way as Proposition XV.3.1 in [21]. Lemma 2.1. Let $u_{\varepsilon}(x,t)\in \mathbf{L}_2^{\mathrm{loc}}(\mathbb{R}_+;\mathbf{V}_{\varepsilon})\cap \mathbf{L}_{\mathbf{p}}^{\mathrm{loc}}(\mathbb{R}_+;\mathbf{L}_{\mathbf{p},\varepsilon })$ be a weak solution of the problem (13). Then (i) $u_\varepsilon\in \mathbf{C}(\mathbb{R}_+;\mathbf{H}_{\varepsilon})$; (ii) the scalar function $\| u_{\varepsilon}(\,{\cdot}\,,t)\|^2$ is absolutely continuous on $\mathbb{R}_+$ and
$$
\begin{equation}
\begin{aligned} \, &\frac{1}{2}\, \frac{d}{dt}\| u_{\varepsilon}(\,{\cdot}\,,t)\|^2+\int_{\Omega_{\varepsilon}} \lambda \nabla u_{\varepsilon}(x,t)\cdot \nabla u_{\varepsilon}(x,t)\, dx +\int_{\Omega_{\varepsilon}}a_\varepsilon(x)f(u_{\varepsilon}(x,t))\cdot u_{\varepsilon}(x,t)\,dx \notag \\ &\qquad+\varepsilon^{n/(n-2)}\sum_{j\in \Upsilon_{\varepsilon}}\int_{\partial G_{\varepsilon}^j}B_{\varepsilon}^j(x)u_{\varepsilon}(x,t)\cdot u_{\varepsilon}(x,t)\, d\omega =\int_{\Omega_{\varepsilon}}g_\varepsilon(x)\cdot u_\varepsilon (x,t)\, dx \end{aligned}
\end{equation}
\tag{25}
$$
for almost all $t\in \mathbb{R}_+$. The integrals over the boundaries of $G_{\varepsilon}^j$ in (25) are non-negative due to (17), and hence integrating this differential equality in time, we obtain that any weak solution $u(t)$ of the problem (13) satisfies the following inequalities:
$$
\begin{equation}
\| u_{\varepsilon}(t)\|^2\leqslant \| u_{\varepsilon}(0)\|^2e^{-\lambda_1\beta t}+R_1^2,
\end{equation}
\tag{26}
$$
$$
\begin{equation}
\begin{split} &\beta \int_t^{t+1}\| u_{\varepsilon}(s)\|_1^2\, ds+2a_0\sum_{i=1}^{N}\gamma_i\int_t^{t+1} \|u^i_{\varepsilon}(s)\|_{L_{p_i}(\Omega_\varepsilon)}^{p_i}\, ds \\ &\qquad+2B_0\varepsilon^{n/(n-2)}\sum_{j\in \Upsilon_{\varepsilon}}\int_t^{t+1} \|u_{\varepsilon}(s)\|_{\mathbf{L}_2(\partial G_{\varepsilon}^j)}^2\, ds \leqslant \|u_{\varepsilon}(t)\|^2+R_2^2, \end{split}
\end{equation}
\tag{27}
$$
where $\lambda_1$ is the first eigenvalue of the operator $-\Delta $ for the considered boundary-value problem. Positive values $R_1$ and $R_2$ depend on the number $M_0$ (see (16)), and they are independent on $u_{\varepsilon}(0) $ and $\varepsilon$. The proof is given in [21].
§ 3. Trajectory attractor construction for reaction-diffusion system in the perforated domainIn this section, we construct the trajectory attractor for the reaction-diffusion system (13) in the perforated domain for a fixed $\varepsilon$. In what follows, we will omit the subscript $\varepsilon$ in the notation for solutions to system (13) and function spaces, if it does not cause any confusion. Now we apply the scheme described in Section 1 to construct a trajectory attractor of problem (13) that has the form (1) with $E_1=\mathbf{L}_{\mathbf{p}}\cap \mathbf{V}$, $E_0=\mathbf{H}^{-\mathbf{r}}$, $E=\mathbf{H}$, and $A(u)=\lambda\Delta u-a(\,{\cdot}\,)f(u)+g(\,{\cdot}\,)$. To describe the trajectory space $\mathcal{K}_{\varepsilon}^+$ for problem (13) we follow the general scheme of Section 1 and for each segment $[t_1,t_2]\in \mathbb{R}$ define the Banach spaces:
$$
\begin{equation}
\mathcal{F}_{t_1,t_2}:=\mathbf{L}_{\mathbf{p}}(t_1,t_2;\mathbf{L}_p)\cap \mathbf{L}_2(t_1,t_2;\mathbf{V})\cap \mathbf{L}_{\infty }(t_1,t_2; \mathbf{H})\cap \biggl\{ v \biggm| \frac{\partial v}{\partial t}\in \mathbf{L}_{\mathbf{q}}(t_1,t_2;\mathbf{H}^{-\mathbf{r}})\biggr\}
\end{equation}
\tag{28}
$$
with the norms
$$
\begin{equation}
\| v\|_{\mathcal{F}_{t_1,t_2}}:=\| v\|_{\mathbf{L}_{\mathbf{p}}(t_1,t_2; \mathbf{L}_{\mathbf{p}})} +\| v\|_{\mathbf{L}_2(t_1,t_2;\mathbf{V})}+ \|v\|_{\mathbf{L}_{\infty }(0,M;\mathbf{H})}+\biggl\| \frac{\partial v}{\partial t}\biggr\|_{\mathbf{L}_{\mathbf{q}}(t_1,t_2;\mathbf{H}^{-\mathbf{r}})}.
\end{equation}
\tag{29}
$$
It is obvious that the condition (2) holds with the norm (29) and the translation semigroup $\{S(h)\}$ satisfies (3). Putting $\mathcal{D}_{t_1,t_2}=\mathbf{L}_{\mathbf{q}}(t_1,t_2;\mathbf{H}^{-\mathbf{r}})$, we obtain $\mathcal{F}_{t_1,t_2}\subseteq \mathcal{D}_{t_1,t_2}$ and $u(s)\in \mathcal{F}_{t_1,t_2}$ implies $A(u(s))\in \mathcal{D}_{t_1,t_2}$. In what follows, we can consider a weak solution of problem (13) as a solution of the system of equations from the general scheme of Section 1. Having defined the space (4), we obtain
$$
\begin{equation*}
\begin{aligned} \, \mathcal{F}_+^{\mathrm{loc}} &=\mathbf{L}_{\mathbf{p}}^{\mathrm{loc}}(\mathbb{R}_+; \mathbf{L}_{\mathbf{p}})\cap \mathbf{L}_2^{\mathrm{loc}}(\mathbb{R}_+;\mathbf{V})\cap \mathbf{L}_{\infty }^{\mathrm{loc}}(\mathbb{R}_+;\mathbf{H})\,{\cap} \biggl\{ v \biggm| \frac{\partial v}{\partial t}\in \mathbf{L}^{\mathrm{loc}}_{\mathbf{q}} (\mathbb{R}_+;\mathbf{H}^{-\mathbf{r}})\biggr\}, \\ \mathcal{F}_{\varepsilon,+}^{\mathrm{loc}}&=\mathbf{L}_{\mathbf{p}}^{\mathrm{loc}}(\mathbb{R}_+; \mathbf{L}_{\mathbf{p},\varepsilon})\,{\cap}\, \mathbf{L}_2^{\mathrm{loc}}(\mathbb{R}_+;\mathbf{V_\varepsilon})\,{\cap}\, \mathbf{L}_{\infty }^{\mathrm{loc}}(\mathbb{R}_+;\mathbf{H_\varepsilon})\,{\cap} \biggl\{ v \biggm| \frac{\partial v}{\partial t}\,{\in}\, \mathbf{L}^{\mathrm{loc}}_{\mathbf{q}} (\mathbb{R}_+;\mathbf{H_\varepsilon}^{-\mathbf{r}})\biggr\}. \end{aligned}
\end{equation*}
\notag
$$
Let $\mathcal{K}_\varepsilon^+$ be the set of all weak solutions of problem (13). We recall that for any function $U\in \mathbf{H}$ there exists at least one trajectory $u(\,{\cdot}\,)\in \mathcal{K}_{\varepsilon}^+$ such that $u(0)=U(x)$. Consequently, the trajectory space $\mathcal{K}_{\varepsilon}^+$ of problem (13) is not empty and it is sufficiently large. It is clear that $\mathcal{K}_{\varepsilon}^+\subset \mathcal{F}_+^{\mathrm{loc}}$ and the trajectory space $\mathcal{K}_{\varepsilon}^+$ is translation invariant, that is, $u(h+s)\in \mathcal{K}_{\varepsilon}^+$ for any $h\geqslant 0$, whenever $u(s)\in \mathcal{K}_{\varepsilon}^+$. Hence, for any $h\geqslant 0$,
$$
\begin{equation*}
S(h)\mathcal{K}_{\varepsilon}^+\subseteq \mathcal{K}_{\varepsilon}^+.
\end{equation*}
\notag
$$
Further, using the norm of the space $\mathbf{L}_2(t_1,t_2;\mathbf{H})$, we introduce the metrics $\rho_{t_1,t_2}(\,{\cdot}\,,{\cdot}\,)$ in $\mathcal{F}_{t_1,t_2}$ as follows:
$$
\begin{equation*}
\rho_{t_1,t_2}(u,v)=\biggl( \int_{t_1}^{t_2}\| u(s)-v(s)\|^2\, ds\biggr)^{1/2}\quad \forall\, u(\,{\cdot}\,),v(\,{\cdot}\,)\in \mathcal{F}_{t_1,t_2}.
\end{equation*}
\notag
$$
These metrics generate the topology $\Theta_+^{\mathrm{loc}}$ in $\mathcal{F}_+^{\mathrm{loc}}$ (respectively, the topology $\Theta_{\varepsilon,+}^{\mathrm{loc}}$ in $\mathcal{F}_{\varepsilon,+}^{\mathrm{loc}}$). We recall that a sequence $\{v_k\}\subset\mathcal{F}_+^{\mathrm{loc}} $ converges to a function $v\in \mathcal{F}_+^{\mathrm{loc}}$ as $k\to \infty $ in $\Theta_+^{\mathrm{loc}}$, if $\|v_k(\,{\cdot}\,)-v(\,{\cdot}\,)\|_{\mathbf{L}_2(0,M;\mathbf{H})}\to 0$ $(k\to \infty)$ for every $M>0$. The topology $\Theta_+^{\mathrm{loc}}$ is metrizable (see (6)) and the corresponding metric space is complete. We consider this topology in the trajectory space $\mathcal{K}_{\varepsilon}^+$ of problem (13). The translation semigroup $\{S(t)\}$ acting on $\mathcal{K}_{\varepsilon}^+$ is continuous in the topology $\Theta_+^{\mathrm{loc}}$. Following the general scheme of Section 1, we define bounded sets in $\mathcal{K}_{\varepsilon}^+$, using the Banach spaces $\mathcal{F}_+^{\mathrm b}$ (see (7)). It is clear that
$$
\begin{equation}
\mathcal{F}_+^{\mathrm b}=\mathbf{L}_{\mathbf{p}}^{\mathrm b}(\mathbb{R}_+; \mathbf{L}_{\mathbf{p}})\cap \mathbf{L}_2^{\mathrm b}(\mathbb{R}_+;\mathbf{V})\cap \mathbf{L}_{\infty }(\mathbb{R}_+;\mathbf{H})\cap \biggl\{ v \biggm| \frac{\partial v}{\partial t}\in \mathbf{L}^{\mathrm b}_{\mathbf{q}}(\mathbb{R}_+;\mathbf{H}^{-\mathbf{r}})\biggr\}
\end{equation}
\tag{30}
$$
and $\mathcal{F}_+^{\mathrm b}$ is a subspace of $\mathcal{F}_+^{\mathrm{loc}}$. We consider the translation semigroup $\{S(t)\}$ on $\mathcal{K}_{\varepsilon}^+$, $S(t)\colon \mathcal{K}_{\varepsilon}^+\to \mathcal{K}_{\varepsilon}^+$, $t\geqslant 0$. The inequalities (26) and (27) imply the following statement. Proposition 3.1. The trajectory space $\mathcal{K}_{\varepsilon}^+$ belongs to $\mathcal{F}_+^{\mathrm b}$ and, for any trajectory $u(\,{\cdot}\,)\in \mathcal{K}_{\varepsilon}^+$, the following inequalities hold where $\sigma =\beta \lambda_1$, and the values $R_3$, $R_4$ are defined by $R_1$, $R_2$. These values are independent of $u(0)$ and $\varepsilon $. The detailed proof of (31) is given in [21], while (32) follows directly from (25) similarly to (27). From (31) we conclude that the ball
$$
\begin{equation*}
\mathcal{B}_0=\{ u\in \mathcal{F}_+^{\mathrm b}\mid \| u(\,{\cdot}\,)\|_{\mathcal{F}_+^{\mathrm b}}\leqslant 2R_3\}
\end{equation*}
\notag
$$
is an absorbing set of the translation semigroup $\{S(t)\}$ on $\mathcal{K}_{\varepsilon}^+$, that is, for any set $\mathcal{B}\subset \mathcal{K}_{\varepsilon}^+$, bounded in $\mathcal{F}_+^{\mathrm b}$, there is a number $t_1=t_1(\mathcal{B})$, such that $S(t)\mathcal{B}\subseteq \mathcal{B}_0$ for all $t\geqslant t_1$. Consider the set
$$
\begin{equation*}
\mathcal{P}_{\varepsilon}=\mathcal{B}_0\cap \mathcal{K}_{\varepsilon}^+.
\end{equation*}
\notag
$$
The set $\mathcal{P}_{\varepsilon}\subseteq\mathcal{K}_{\varepsilon}^+$ is also absorbing, that is,
$$
\begin{equation*}
S(t)\mathcal{P}_{\varepsilon}\subseteq \mathcal{P}_{\varepsilon}\quad \forall\, t\geqslant 0,
\end{equation*}
\notag
$$
and $\mathcal{P}_{\varepsilon}$ is uniformly (with respect to $\varepsilon $) bounded in $\mathcal{F}_+^{\mathrm b}$. With the help of Lemma 1.1, where we need to set $E_1=\mathbf{V}$, $E_0=\mathbf{H}^{-\mathbf{r}}$, $E=\mathbf{H}$, $p_1=2$, and $p_0=q_{N}$, we obtain the following statement. Proposition 3.2. The set $\mathcal{P}_{\varepsilon}$ is compact in the topology $\Theta_+^{\mathrm{loc}}$ and is uniformly bounded in the norm of $\mathcal{F}_+^{\mathrm b}$. Let $\mathcal{K}_{\varepsilon}$ denote the kernel of problem (13) that consists of all weak solutions $u(s), s\in \mathbb{R}$, which are bounded in the space
$$
\begin{equation*}
\mathcal{F}^{\mathrm b}=\mathbf{L}_{\mathbf{p}}^{\mathrm b}(\mathbb{R}; \mathbf{L}_{\mathbf{p}})\cap \mathbf{L}_2^{\mathrm b}(\mathbb{R};\mathbf{V})\cap \mathbf{L}_{\infty }(\mathbb{R};\mathbf{H})\cap \biggl\{ v \biggm| \frac{\partial v}{\partial t}\in \mathbf{L}^{\mathrm b}_{\mathbf{q}}(\mathbb{R};\mathbf{H}^{-\mathbf{r}})\biggr\}.
\end{equation*}
\notag
$$
We apply Theorems 1.1 and 1.2 using Propositions 3.1 and 3.2. Proposition 3.3. Under the assumptions (18) and (19), the problem (13) has a trajectory attractor $\mathfrak{A}_\varepsilon $ in the topological space $\Theta_+^{\mathrm{loc}}$. The set $\mathfrak{A}_\varepsilon $ is uniformly (with respect to $\varepsilon \in (0,1)$) bounded in $\mathcal{F}_+^{\mathrm b}$ and compact in $\Theta_+^{\mathrm{loc}}$. Moreover, the kernel $\mathcal{K}_\varepsilon $ is non-empty and uniformly (with respect to $\varepsilon \in (0,1)$) bounded in $\mathcal{F}^{\mathrm b}$. Recall that the spaces $\mathcal{F}^{\mathrm b}_+$ and $\Theta^{\mathrm{loc}}_+$ are depended on $\varepsilon$. We note that
$$
\begin{equation}
\mathfrak{A}_\varepsilon \subset \mathcal{B}_0\quad \forall\, \varepsilon \in (0,1).
\end{equation}
\tag{33}
$$
We now prove that the trajectory attractor ${\mathfrak{A}}_{\varepsilon}$ in a “weak” topology $\Theta_+^{\mathrm{loc}}=\mathbf{L}_2^{\mathrm{loc}}(\mathbb{R}_+;\mathbf{H})$ is also a trajectory attractor in the strong topology generated by the spaces $\mathcal{F}_{t_1,t_2}$. We denote by $\Theta_+^{s,\mathrm{loc}}$ the topology in $\mathcal{F}_+^{\mathrm{loc}}$, generated by the convergence in the metric spaces $\mathcal{F}_{t_1,t_2}$. Thus, by definition, a sequence $\{v_k\}\subset \mathcal{F}_+^{\mathrm{loc}}$ converges to $v\in \mathcal{F}_+^{\mathrm{loc}}$ as $k\to \infty $ in $\Theta_+^{s,\mathrm{loc}}$, if $\|v_k(\,{\cdot}\,)-v(\,{\cdot}\,)\|_{\mathcal{F}_{t_1,t_2}}\to 0$ for any $(t_1,t_2)\subset\mathbb{R}_+$ (see (29)). Introduced topology is clearly metrizable. Theorem 3.1. The trajectory attractor ${\mathfrak{A}}_{\varepsilon}$ is compact in the topology $\Theta_+^{s,\mathrm{loc}}$ and attracts bounded sets of trajectories from $\mathcal{K}_{\varepsilon}^+$ in this topology, that is, ${\mathfrak{A}}_{\varepsilon}$ is a trajectory attractor in the strong topology $\Theta_+^{s,\mathrm{loc}}$. Proof. We fix $\varepsilon >0$. Since the set $\mathcal{P}_{\varepsilon}$ is absorbing, it is sufficient to prove that the set $S(1)\mathcal{P}_{\varepsilon}$ is compact in the strong topology of the space $\mathbf{L}_{\mathbf{p}}(0,M;\mathbf{L}_{\mathbf{p}}(\Omega))\cap \mathbf{L}_2(0,M;\mathbf{V})$ for every $M>0$. We note that $\mathbf{L}_{\mathbf{p}}(0,M;\mathbf{L}_{\mathbf{p}}(\Omega)) =\mathbf{L}_{\mathbf{p}}(\Omega\times [0,M])$.
We have to prove that any sequence $\{u_k(t)\}\subset \mathcal{P}_{\varepsilon}$ has a subsequence that converges strongly in the space $\mathbf{L}_{\mathbf{p}}(\Omega\times [1,M])\cap \mathbf{L}_2(1,M;\mathbf{V})$ for every $M>0$. The set $\mathcal{P}_{\varepsilon}$ is bounded in the space $\mathcal{F}_+^{\mathrm b}$. Therefore, $\{u_k(t)\}$ is bounded in the spaces $\mathbf{L}_{\mathbf{p}}(\Omega\times [0,M])$ and $\mathbf{L}_2(0,M;\mathbf{V})$. Passing to a subsequence that we denote again $\{u_k(t)\}$, we can assume that $u_k(\,{\cdot}\,)\rightharpoondown \widehat{u}(\,{\cdot}\,)$ as $k\to \infty $ weakly in spaces $\mathbf{L}_{\mathbf{p}}(\Omega\times [0,M])$ and $\mathbf{L}_2(0,M;\mathbf{V})$, where $\widehat{u}(t)$ is a solution of the system (13) that belongs to $\mathcal{P}_{\varepsilon}$. The Lions–Magenes lemma (see, for example, [49], [50]) implies that $u_k(t)\rightharpoondown \widehat{u}(t)$ weakly in $\mathbf{H}$ for all $t\in [0,M]$. Moreover, it follows from the standard embedding theorems that $u_k(\,{\cdot}\,)\to \widehat{u}(\,{\cdot}\,)$ strongly in the space $\mathbf{L}_2(\Omega\times [0,M])$ and $u_k(x,t)\to \widehat{u}(x,t)$ for almost all $(x,t)\in \Omega\times [0,M]$. We note that the inequality (32) implies that the sequence $\{u_k(t)\}$ is bounded in the space $\mathbf{L}_2(\partial G_{\varepsilon}^j\times [0,M])$ for every $j\in\Upsilon_{\varepsilon}$. Therefore, passing once again to a subsequence, we can assume that $u_k(\,{\cdot}\,)\rightharpoondown \widehat{u}(\,{\cdot}\,)$ as $k\to \infty$ weakly in the space $\mathbf{L}_2(\partial G_{\varepsilon}^j\times [0,M])$ for every $j\in\Upsilon_{\varepsilon}$. We recall the following fact from functional analysis: if a sequence $\chi_k\rightharpoondown \widehat{\chi}$ weakly in a Banach space $X$, then
$$
\begin{equation*}
\| \widehat{\chi}\|_{X}\leqslant \liminf_{k\to \infty}\| \chi_k\|_{X}
\end{equation*}
\notag
$$
(see, for example, [51]). Hence, for a weakly convergent subsequence of trajectories $\{u_k(\,{\cdot}\,)\}$ we obtain the following limit relations:
$$
\begin{equation}
\| \widehat{u}(M)\| \leqslant \liminf_{k\to \infty} \| u_k(M)\|,
\end{equation}
\tag{34}
$$
$$
\begin{equation}
\int_0^M\int_{\Omega}s\lambda\nabla \widehat{u}\cdot \nabla \widehat{u}\,dx\, ds \leqslant \liminf_{k\to \infty}\int_0^M\int_{\Omega}s\lambda\nabla u_k\cdot \nabla u_k\,dx\, ds,
\end{equation}
\tag{35}
$$
$$
\begin{equation}
\int_0^M\int_{\Omega}sa_{\varepsilon}(x)|\widehat{u}^{\,i}|^{p_i}\,dx\, ds \leqslant \liminf_{k\to \infty}\int_0^M\int_{\Omega}sa_{\varepsilon }(x)|u_k^i|^{p_i}\,dx\, ds,\qquad i=1,\dots,N,
\end{equation}
\tag{36}
$$
$$
\begin{equation}
\int_0^M\int_{\partial G_{\varepsilon}^j}sB_{\varepsilon}^j(x)\widehat{u}\cdot \widehat{u}\, d\omega\, ds \leqslant\liminf_{k\to \infty}\int_0^M\int_{\partial G_{\varepsilon}^j}sB_{\varepsilon}^j(x)u_k\cdot u_k\,d\omega\, ds, \qquad j\in\Upsilon_{\varepsilon},
\end{equation}
\tag{37}
$$
where for brevity we denote ${u_k=u_k(x,s)}$ and ${\widehat{u}=\widehat{u}(x,s)}$. The norms in (35)–(37) correspond to weighted spaces $\mathbf{L}_{2,s}(0,M;\mathbf{V})$, $\mathbf{L}_{\mathbf{p},sa_{\varepsilon}(x)}(\Omega\times [0,M])$ and $\mathbf{L}_{2,sB_{\varepsilon}^j(x)}(\partial G_{\varepsilon}^j\times [0,M])$ with weights $s$, $sa_{\varepsilon}(x)$, and $sB_{\varepsilon}^j(x)$, respectively. Besides, the quadratic form $\lambda y\cdot y$ with $y\in \mathbb{R}^{N}$ is equivalent to the standard norm of a vector $y$ in $\mathbb{R}^{N}$, since the matrix $\lambda$ has a positive symmetric part. Therefore, the quadratic form $\int_{\Omega}\lambda\nabla v(x)\cdot \nabla v(x)\, dx$ is equivalent to the norm of a function $v(\,{\cdot}\,)$ in the space $\mathbf{V}$.
We note that weak convergence $u_k(\,{\cdot}\,)\rightharpoondown \widehat{u}(\,{\cdot}\,)$ holds also in the weighted spaces $\mathbf{L}_{2,s}(0,M;\mathbf{V})$, $\mathbf{L}_{\mathbf{p},sa_{\varepsilon}(x)}(\Omega\times [0,M])$ and $\mathbf{L}_{2,sB_{\varepsilon}^j(x)}(\partial G_{\varepsilon}^j\times [0,M])$. Consider the continuous scalar function
$$
\begin{equation*}
F(v)=\sum_{i=1}^{N}f^i(v)v^i-\sum_{i=1}^{N}\gamma_i|v^i|^{p_i},\qquad v\in \mathbb{R}^{N}.
\end{equation*}
\notag
$$
Then $sa_{\varepsilon}(x) F(u_k(x,s))\to sa_{\varepsilon}(x) F(\widehat{u}(x,s))$ as $k\to \infty $ for almost all $(x,t)\in \Omega\times [0,M]$, since the function $F(v)$ is continuous. We claim that
$$
\begin{equation}
\int_0^M\int_{\Omega}sa_{\varepsilon}(x) F(\widehat{u}(x,t))\, dx\, ds\leqslant \liminf_{k\to \infty}\int_0^M\int_{\Omega}sa_{\varepsilon }(x) F(u_k(x,t))\, dx\, ds.
\end{equation}
\tag{38}
$$
The proof of this inequality uses the inequalities $F(v)+C_1\geqslant 0$, $a_{\varepsilon}(\,{\cdot}\,)\geqslant 0$ (cf. (19) and (17)), the convergence of the sequence $\{u_k(x,s)\}$ for almost all $(x,s)\in \Omega\times [0,M]$, and the Fatou lemma on the estimate from above of the integral of a limit function in terms of the lower limit of integrals of a convergent sequence of non-negative functions (see, for example, [51]).
Recall that the weak solutions $u_k(\,{\cdot}\,)$ and $\widehat{u}(\,{\cdot}\,)$ of the system (13) satisfy the differential identity (25). Multiplying this identity by $t$, integrating the result over $[0,M]$, and using the definition of the function $F(\,{\cdot}\,)$ we obtain the following equalities:
$$
\begin{equation}
\frac{1}{2}\| u_k(M)\|^2+\int_0^M\int_{\Omega}s\lambda\nabla u_k\cdot \nabla u_k\,dx\, ds +\sum_{i=1}^{N}\gamma_i\int_0^M\int_{\Omega}sa_{\varepsilon}(x)|u_k^i|^{p_i}\,dx\,ds \nonumber
\end{equation}
\notag
$$
$$
\begin{equation}
\qquad+\int_0^M\int_{\Omega}sa_{\varepsilon}(x) F(u_k)\,dx\, ds+ \varepsilon^{n/(n-2)}\sum_{j\in\Upsilon_{\varepsilon}}\int_0^M\int_{\partial G_{\varepsilon}^j}sB_{\varepsilon}^j(x)u_k\cdot u_k\,d\omega\, ds \nonumber
\end{equation}
\notag
$$
$$
\begin{equation}
=\frac{1}{2}\int_0^M\int_{\Omega}|u_k|^2\, dx\, ds +\int_0^M\int_{\Omega}g_{\varepsilon}(x)\cdot u_k\,dx\, ds,
\end{equation}
\tag{39}
$$
$$
\begin{equation}
\frac{1}{2}\| \widehat{u}(M)\|^2+\int_0^M\int_{\Omega}s\lambda\nabla \widehat{u} \cdot \nabla \widehat{u}\,dx\, ds +\sum_{i=1}^{N}\gamma_i\int_0^M\int_{\Omega}sa_{\varepsilon}(x) |\widehat{u}^{\,i}|^{p_i}\,dx\, ds \nonumber
\end{equation}
\notag
$$
$$
\begin{equation}
\qquad+\int_0^M\int_{\Omega}sa_{\varepsilon}(x) F(\widehat{u})\,dx\, ds +\varepsilon^{n/(n-2)}\sum_{j\in\Upsilon_{\varepsilon}}\int_0^M\int_{\partial G_{\varepsilon}^j}sB_{\varepsilon}^j(x)\widehat{u}\cdot \widehat{u}\,d\omega\, ds \nonumber
\end{equation}
\notag
$$
$$
\begin{equation}
=\frac{1}{2}\int_0^M\int_{\Omega}|\widehat{u}|^2\,dx\, ds +\int_0^M\int_{\Omega}g_{\varepsilon}(x)\cdot \widehat{u}\,dx\, ds.
\end{equation}
\tag{40}
$$
Recall that $u_k(\,{\cdot}\,)\to \widehat{u}(\,{\cdot}\,)$ strongly in the space $\mathbf{L}_2(\Omega\times [0,M])$. Therefore, the right-hand side of (39) tends to that of (40). Hence, the left-hand side of (39) also converges to the left-hand side of (40). Then, it follows from (34)–(38) that each of five real sequences in the sum in the left-hand side of (39) has a limit as $k\to\infty $, which coincides with the corresponding quantity in the left-hand side of (40). In particular, we have
$$
\begin{equation*}
\begin{aligned} \, \lim_{k\to \infty }\int_0^M\int_{\Omega}s\lambda\nabla u_k\cdot \nabla u_k\, dx\, ds &=\int_0^M\int_{\Omega}s\lambda\nabla \widehat{u}\cdot \nabla \widehat{u}\,dx \,ds, \\ \lim_{k\to \infty }\int_0^M\int_{\Omega}sa_{\varepsilon}(x)|u_k^i|^{p_i}\, dx\, ds &= \int_0^M\int_{\Omega}sa_{\varepsilon}(x)|\widehat{u}^{\,i}|^{p_i}\,dx\, ds,\qquad i=1,\dots,N. \end{aligned}
\end{equation*}
\notag
$$
It is known that, in a uniformly convex Banach space $X$, the weak convergence $\chi_k\rightharpoondown \widehat{\chi}$ of elements and the convergence of their norms $\| \chi_k\|_{X}\to \| \widehat{\chi}\|_{X}$ implies the strong convergence $\| \chi_k-\widehat{\chi}\|_{X}\to 0$ as $k\to \infty $ (this assertion follows from the Mazur theorem, see [51]). The weight spaces $\mathbf{L}_{2,s}(0,M;\mathbf{V})$ and $\mathbf{L}_{\mathbf{p},sa_{\varepsilon}(x)}(\Omega\times [0,M])$ are uniformly convex. Therefore, the weak convergence of the sequences of functions $u_k^i(\,{\cdot}\,)$ to $\widehat{u}(\,{\cdot}\,)$ and the convergence of their norms in the space $\mathbf{L}_{2,s}(0,M;\mathbf{V})\cap \mathbf{L}_{\mathbf{p},sa_{\varepsilon}(x)}(\Omega\times [0,M])$ implies the strong convergence $u_k(\,{\cdot}\,)\to \widehat{u}(\,{\cdot}\,)$ in the space $\mathbf{L}_{2,s}(1,M;\mathbf{V})\cap \mathbf{L}_{\mathbf{p},sa_{\varepsilon}(x)}(\Omega\times [1,M])$, which is, obviously, equivalent to the convergence in $\mathbf{L}_2(1,M;\mathbf{V})\cap \mathbf{L}_{\mathbf{p}}(\Omega\times [1,M])$ (without weights). We have proved the compactness of the set $S(1)\mathcal{P}_{\varepsilon}$ in the strong topology of the space $\mathbf{L}_{\mathbf{p}}^{\mathrm{loc}}(\mathbb{R}_+;\mathbf{L}_{\mathbf{p}}(\Omega))\cap \mathbf{L}_2^{\mathrm{loc}}(\mathbb{R}_+;\mathbf{V})$. The compactness of the corresponding set of derivatives $\partial_tu(\,{\cdot}\,)$ in the strong topology of the space $\mathcal{L}^{\mathrm{loc}}(\mathbb{R}_+) :=\mathbf{L}_2^{\mathrm{loc}}(\mathbb{R}_+;\mathbf{V}') +\mathbf{L}_{\mathbf{q}}^{\mathrm{loc}}(\mathbb{R}_+;\mathbf{L}_{\mathbf{q}}(\Omega))$ follows directly from the equation (13) and from the continuity of the Nemytskii operator $u\,{\mapsto}\,f(u)$, which, by virtue of (18), acts from $\mathbf{L}_{\mathbf{p}}(\Omega\,{\times}\, [0,M])$ to $\mathbf{L}_{\mathbf{q}}(\Omega\,{\times}\, [0,M])$ (see [52]), and, hence, $a_{\varepsilon}(\,{\cdot}\,)f(u_k(x,s))\to a_{\varepsilon}(\,{\cdot}\,) f(\widehat{u}(\,{\cdot}\,))$ strongly in $\mathbf{L}_{\mathbf{q}}^{\mathrm{loc}}(\mathbb{R}_+;\mathbf{L}_{\mathbf{q}}(\Omega))$. At the same time, it is clear that $\lambda\nabla u_k(\,{\cdot}\,)\to $ $\lambda\nabla \widehat{u}(\,{\cdot}\,)$ strongly in $\mathbf{L}_2^{\mathrm{loc}}(\mathbb{R}_+;\mathbf{V}')$. Therefore, from the equation (13) we conclude that $\partial_tu_k(\,{\cdot}\,)\to \partial_t\widehat{u}(\,{\cdot}\,)$ strongly in $\mathcal{L}^{\mathrm{loc}}(\mathbb{R}_+)$. It is remains to note that the set $\mathcal{P}_{\varepsilon}$ belongs to the space $C^{\mathrm{loc}}(\mathbb{R}_+;\mathbf{H})$ (this fact follows from the energy identity (25)), and the set $S(1)\mathcal{P}_{\varepsilon}$ is compact in $C^{\mathrm{loc}}(\mathbb{R}_+;\mathbf{H})$. The last assertion follows from the continuity of the embedding
$$
\begin{equation*}
\mathbf{L}_2(0,M;\mathbf{V})\cap \mathbf{L}_{\mathbf{p}}(0,M;\mathbf{L}_{\mathbf{p}}(\Omega)) \cap \{ \partial_tv\in \mathcal{L}(0,M)\} \subset C([0,M];\mathbf{H}),
\end{equation*}
\notag
$$
which was proved, for example, in [21]. This completes the proof of Theorem 3.1. § 4. Homogenization of attractors for reaction-diffusion equations in the perforated domainIn this section, we study the limit behaviour of trajectory attractors ${\mathfrak{A}}_{\varepsilon}$ for the reaction-diffusion system (13) as $\varepsilon \to 0{+}$ and their convergence to the trajectory attractor of the corresponding homogenized system. The limit system contains some additional “strange term” (potential). To define this term, we consider the following problem:
$$
\begin{equation}
\begin{alignedat}{2} &-\Delta_y v=0, &\qquad &y\in \mathbb{R}^n\setminus G_0, \\ &\frac{\partial v}{\partial \nu_y} + B(x, y) v =\overline{B}(x, y), &\qquad &y\in \partial G_0, \\ &v\to 0, &\qquad &|y|\to \infty, \end{alignedat}
\end{equation}
\tag{41}
$$
where the matrix $B(x,y)$ and the vector $\overline B(x,y)$ are defined above. In this problem, the variable $x$ plays the role of a slow parameter. We define the limit potential by the formula
$$
\begin{equation}
V^{kk}(x)=\int_{\partial G_0}\frac{\partial}{\partial \nu_y}v^k(x,y)\,d\omega_y, \qquad k=1, \dots, N.
\end{equation}
\tag{42}
$$
The homogenized (limit) problem for the considered reaction-diffusion system has the form
$$
\begin{equation}
\begin{alignedat}{2} &\frac{\partial u}{\partial t} =\lambda\Delta u-\overline{a}(x)f(u)-V(x)u +\overline{g}(x), &\qquad &x\in \Omega, \\ &u =0, &\qquad &x\in \partial\Omega, \\ &u= U(x), &\qquad &t=0, \end{alignedat}
\end{equation}
\tag{43}
$$
where $V(x)$ is a diagonal matrix with elements $V^{kk}(x)$, $k=1,\dots, N$. The mean functions $\overline{a}(x)$ and $\overline{g}(x)$ were defined in (14) and (15). The following statement is similar to Lemma 2.1. Lemma 4.1. Let $u(x,t)\in \mathbf{L}_2^{\mathrm{loc}}(\mathbb{R}_+;\mathbf{V})\cap \mathbf{L}_{\mathbf{p}}^{\mathrm{loc}}(\mathbb{R}_+;\mathbf{L}_{\mathbf{p}})$ be a weak solution of (43). Then (i) $u\in \mathbf{C}(\mathbb{R}_+;\mathbf{H})$; (ii) the function $\| u(\,{\cdot}\,,t)\|^2$ is absolutely continuous on $\mathbb{R}_+$ and, moreover,
$$
\begin{equation*}
\begin{aligned} \, &\frac{1}{2}\frac{d}{dt}\| u(\,{\cdot}\,,t)\|^2 +\int_{\Omega}\lambda\nabla u(x,t)\cdot \nabla u(x,t)\, dx +\int_{\Omega}\overline a(x)f(u(x,t))\cdot u(x,t)\, dx \\ &\qquad+\int_{\Omega}V(x)u(x,t)\cdot u(x,t)\,dx=\int_{\Omega}\overline g(x)\cdot u(x,t)\, dx. \end{aligned}
\end{equation*}
\notag
$$
Propositions 3.1–3.3, as well as Theorem 3.1 hold for the limit system in the corresponding spaces $\mathcal{F}^{\mathrm{loc}}_+$ and $\Theta^{s,\mathrm{loc}}_+$ in the domain $\Omega$ without perforation. Problem (43) has a trajectory attractor $\overline{\mathfrak{A}}$ in the trajectory space $\overline{\mathcal{K}}^{\,+}$ corresponding to this problem and, moreover, where $\overline{\mathcal{K}}$ is the kernel of problem (43) in the space $\mathcal{F}^{\mathrm b}$ (in the domain $\Omega$ without perforation).Let us state the homogenization theorem for the reaction-diffusion system in the perforated domain with rapidly oscillating terms in the weak topology. Theorem 4.1. The following limit relation holds in the topological space $\Theta_+^{\mathrm{loc}}$ Furthermore, Remark 4.1. The functions from the sets $\mathfrak{A}_\varepsilon$ and $\mathcal{K}_\varepsilon$ are defined on perforated domains $\Omega_{\varepsilon}$. However, all these functions can be extended inside the cavities in such a way that the norms of the extended functions in the spaces $\mathbf{H}$, $\mathbf{V}$, and $\mathbf{L}_{\mathbf{p}}$ (defined without perforations) coincide with the corresponding norms in the perforated spaces $\mathbf{H}_{\varepsilon}$, $\mathbf{V}_{\varepsilon}$ and $\mathbf{L}_{\mathbf{p},\varepsilon}$. Therefore, in Theorem 4.1, all distances are measured in the spaces without perforation, taking into account the extension inside the cavities. Proof of Theorem 4.1. It is clear that (45) implies (44). Therefore, it suffices to prove (45), that is, to demonstrate that for any neighborhood $\mathcal{O}(\overline{\mathcal{K}})$ in $\Theta^{\mathrm{loc}}$ there is a number $\varepsilon_1=\varepsilon_1(\mathcal{O})>0$ such that
$$
\begin{equation}
\mathcal{K}_\varepsilon\subset \mathcal{O}(\overline{\mathcal{K}})\quad \text{for all}\quad \varepsilon<\varepsilon_1.
\end{equation}
\tag{46}
$$
Assume that (46) fails. Then there exists a neighborhood $\mathcal{O}'(\overline{\mathcal{K}})$ in $\Theta^{\mathrm{loc}}$, and sequences $\varepsilon_k\to 0{+}$ $(k\to \infty )$ and $u_{\varepsilon_k}(\,{\cdot}\,)=u_{\varepsilon_k}(s)\in \mathcal{K}_{\varepsilon_k}$ such that
$$
\begin{equation}
u_{\varepsilon_k}\notin \mathcal{O}'(\overline{\mathcal{K}})\quad\text{for all}\quad k\in\mathbb{N}.
\end{equation}
\tag{47}
$$
The functions $u_{\varepsilon_k}(s)$, $s\in \mathbb{R}$, satisfy the equations
$$
\begin{equation}
\begin{alignedat}{2} &\frac{\partial u_{\varepsilon_k}}{\partial t} =\lambda \Delta u_{\varepsilon_k} -a_{\varepsilon_k}(x)f(u_{\varepsilon_k}) +g_{\varepsilon_k}(x), &\qquad x&\in \Omega_{\varepsilon_k}, \\ &\frac{\partial u_{\varepsilon_k}}{\partial \nu} +\varepsilon_k^{n/(n-2)} B^j_{\varepsilon_k}(x) u_{\varepsilon_k} =0, &\qquad x&\in \partial G^j_{\varepsilon_k},\, j\in \Upsilon_{\varepsilon_k}, \\ &u_{\varepsilon_k} =0, &\qquad x&\in \partial\Omega, \end{alignedat}
\end{equation}
\tag{48}
$$
on the entire time axis $t\in \mathbb{R}$.
To derive an $\varepsilon$-uniform estimate, we use the following lemmas (see [53], Ch. III, Sec. 5, and [54], respectively). Lemma 4.2. Let be a bilinear form on $\mathbf{V}_\varepsilon$ and let $q(x)\geqslant0$ and $r(x)\geqslant 0$ ($q\not\equiv0$ or $r\not\equiv0$). Then the bilinear form $W(f,g)$ defines an inner product on $\mathbf{V}_\varepsilon$, which is equivalent to the inner product Lemma 4.3. The coercitivity of the problem (43) implies the coercitivity of the initial problem (13). It is proved in Proposition 3.3 that the kernels $\mathcal{K}_{\varepsilon}$ are uniformly bounded (with respect to $\varepsilon$) in the space $\mathcal{F}^{\mathrm b}$ (see (33)). Consequently, the sequence $\{u_{\varepsilon_k}(s)\}$ is bounded in $\mathcal{F}^{\mathrm b}$, that is,
$$
\begin{equation}
\begin{aligned} \, \| u_{\varepsilon_k}\|_{\mathcal{F}^{\mathrm b}} &=\sup_{t\in \mathbb{R}}\| u_{\varepsilon_k}(t)\| +\sup_{t\in \mathbb{R}}\biggl(\int_t^{t+1}\| u_{\varepsilon_k}(s)\|_1^2\, ds\biggr)^{1/2} +\sup_{t\in \mathbb{R}}\|u_{\varepsilon_k}(s)\|_{\mathbf{L}_{\mathbf{p}}(t, t+1; \mathbf{L}_{\mathbf{p}})} \nonumber \\ &\qquad +\sup_{t\in \mathbb{R}}\biggl\| \frac{\partial u_{\varepsilon_k}}{\partial t}(s)\biggr\|_{\mathbf{L}_{\mathbf{q}}(t, t+1;\mathbf{H}^{-\mathbf{r}})}\leqslant C\quad \forall\, k\in \mathbb{N}. \end{aligned}
\end{equation}
\tag{50}
$$
Therefore, there is a subsequence $\{u_{\varepsilon_k'}(s)\}\subset \{u_{\varepsilon_k}(s)\}$ such that
$$
\begin{equation}
u_{\varepsilon_k}(s)\to \overline u(s)\quad\text{as}\quad k\to \infty \text{ in } \Theta^{\mathrm{loc}},
\end{equation}
\tag{51}
$$
where $\overline u(s)\in \mathcal{F}^{\mathrm b}$ and $\overline u(s)$ satisfy (50) with the same constant $C$. From (50) we conclude that $u_{\varepsilon_k}(s)\rightharpoonup \overline u(s)$ $(k\to\infty )$ weakly in $\mathbf{L}_2^{\mathrm{loc}}(\mathbb{R};\mathbf{V})$, weakly in $\mathbf{L}_{\mathbf{p}}^{\mathrm{loc}}(\mathbb{R};\mathbf{L}_{\mathbf{p}})$, $*$-weakly in $\mathbf{L}_{\infty }^{\mathrm{loc}}(\mathbb{R}_+;\mathbf{H})$, and $\partial u_{\varepsilon_k}(s)/\partial t\rightharpoonup \partial \overline u(s)/\partial t$ $(k\to \infty )$ weakly in $\mathbf{L}_{\mathbf{q},w}^{\mathrm{loc}}(\mathbb{R};\mathbf{H}^{-\mathbf{r}})$. We claim that $\overline u(s)\in\overline{\mathcal{K}}$. We have already proved that $\| \overline u\|_{\mathcal{F}^{\mathrm b}}\leqslant C$. It remains to check that $\overline u(s)$ is a weak solution of (43). Using (50) and (15), we find that
$$
\begin{equation}
\frac{\partial u_{\varepsilon_k}}{\partial t}-\lambda\Delta u_{\varepsilon_k} -g_{\varepsilon_k}(x) \to \frac{\partial \overline u}{\partial t}-\lambda\Delta \overline u- \overline{g}(x) \quad \text{as}\quad k\to \infty
\end{equation}
\tag{52}
$$
in the space $D'(\mathbb{R};\mathbf{H}^{-\mathbf{r}}_\varepsilon)$, since the derivative operators are continuous in the space of distributions. Let us prove that
$$
\begin{equation}
a\biggl( x,\frac x{\varepsilon_k}\biggr) f(u_{\varepsilon_k})\rightharpoondown \overline a(x) f(\overline u\,)\quad \text{as}\quad k\to \infty
\end{equation}
\tag{53}
$$
weakly in $\mathbf{L}_{\mathbf{q},w}^{\mathrm{loc}}(\mathbb{R};\mathbf{L}_{\mathbf{q}})$. We fix an arbitrary number $M>0$. The sequence $\{u_{\varepsilon_k}(s)\}$ is bounded in $\mathbf{L}_{\mathbf{p}}(-M,M; \mathbf{L}_{\mathbf{p},\varepsilon}) $ (see (50)). Then, due to (18), the sequence $\{f(u_{\varepsilon_k}(s))\}$ is bounded in $\mathbf{L}_{\mathbf{q}}(-M,M; \mathbf{L}_{\mathbf{q},\varepsilon})$. Since $\{u_{\varepsilon_k}(s)\}$ is bounded in $\mathbf{L}_2(-M,M; \mathbf{V}_\varepsilon)$ and $\{\partial_tu_{\varepsilon_k}(s)\}$ is bounded in $\mathbf{L}_{\mathbf{q}}(-M,M; \mathbf{H}^{-\mathbf{r}}_\varepsilon)$, we can assume that $u_{\varepsilon_k}(s)\to \overline u(s)$ as $k\to \infty $ strongly in $\mathbf{L}_2(-M,M; \mathbf{L}_2) =\mathbf{L}_2(\Omega \times\,]{-}M,M[\,)$ and, therefore,
$$
\begin{equation*}
u_{\varepsilon_k}(x,s)\to \overline u(x,s)\quad\text{ as }k\to \infty\text{ for almost all } (x,s)\in \Omega \times\, ]{-}M,M[.
\end{equation*}
\notag
$$
Since the function $f(v)$ is continuous in $v\in \mathbb R$, we conclude that
$$
\begin{equation}
f(u_{\varepsilon_k}(x,s))\to f(\overline u(x,s))\quad\text{as }k\to \infty \text{ for almost all } (x,s)\in \Omega \times\, ]{-}M,M[.
\end{equation}
\tag{54}
$$
We have
$$
\begin{equation}
\begin{aligned} \, &a\biggl( x,\frac x{\varepsilon_k}\biggr) f(u_{\varepsilon_k})-\overline a(x) f(\overline u\,) \nonumber \\ &\qquad= a\biggl( x,\frac x{\varepsilon_k}\biggr) \bigl( f(u_{\varepsilon_k})-f(\overline u\,)\bigr)+\biggl( a\biggl( x,\frac x{\varepsilon_k}\biggr) -\overline a(x) \biggr) f(\overline u\,). \end{aligned}
\end{equation}
\tag{55}
$$
Let us show that both items in the right-hand side of (55) tend to zero as $k\to \infty $ weakly in $\mathbf{L}_{\mathbf{q}}(-M,M; \mathbf{L}_{\mathbf{q}}) =\mathbf{L}_{\mathbf{q}}(\Omega\times\, ]{-}M,M[)$. First, the sequence $a(x,x/\varepsilon_k) (f(u_{\varepsilon_k})-f(\overline u\,))$ tends to zero as $k\to \infty $ for almost all $(x,s)\in \Omega\, \times\, ]{-}M,M[$ (see (54)) and it is bounded in $\mathbf{L}_{\mathbf{q}}(\Omega\times\, ]{-}M,M[\,)$ (see (17)). Applying [50], Ch. 1, Sec. 1, Lemma 1.3, we conclude that
$$
\begin{equation*}
a\biggl( x,\frac x{\varepsilon_k}\biggr) \bigl( f(u_{\varepsilon_k})-f(\overline u\,)\bigr) \rightharpoondown 0\quad\text{as}\quad k\to \infty
\end{equation*}
\notag
$$
weakly in $\mathbf{L}_{\mathbf{q}}(\Omega \times ]{-}M,M[)$. Second, the sequence $(a(x,x/\varepsilon_k)- \overline a(x)) f(\overline u\,)$ also tends to zero as $k\to \infty$ weakly in $\mathbf{L}_{\mathbf{q}}(\Omega \times\, ]{-}M,M[\,)$, because, due to the condition, $a(x,x/\varepsilon_k) \rightharpoondown \overline a(x)$ as $k\to \infty $ $*$-weakly in $\mathbf{L}_{\infty,*w}(-M,M; \mathbf{L}_2)$ and $f(\overline u\,)\in \mathbf{L}_{\mathbf{q}}(\Omega \times\, ]{-}M,M[)$. Thus, (53) is established. Following [55] and [56], we prove Lemma 4.4. Lemma 4.4. For any function $\varphi \in \mathbf{H}_\varepsilon$ and for all $t$, the following inequality holds: Proof. The inequality (56) is proved by the same scheme as Lemma 2, inequality (21) in [56]. We multiply the equation in (41) by the function $\varphi \in \mathbf{H}_\varepsilon$ and integrate by part in the domain $\Omega_\varepsilon$. We subtract from and add to the obtained equality the term $\int_{\Omega}V(x)\overline \varphi\, dx$. Then, moving the difference
$$
\begin{equation*}
\varepsilon^{n/(n-2)}\sum_{j\in\Upsilon_\varepsilon} \int_{\partial G^j_\varepsilon} B^j_\varepsilon(x) \varphi\, d\omega- \int_{\Omega}V(x)\overline \varphi\, dx
\end{equation*}
\notag
$$
to the left-hand side of the obtained equality, and estimating it by the modulo, we obtain (56).
To prove (57), first of all, by substituting $u_\varepsilon$ as a test function in (24), one can obtain the uniform boundedness:
$$
\begin{equation*}
\|\nabla u_\varepsilon\|_{\mathbf{H}_\varepsilon}\leqslant K, \qquad \sum_{j\in\Upsilon_\varepsilon} \int_{\partial G^j_\varepsilon} B^j_\varepsilon u_\varepsilon \psi\, d\omega \leqslant K\varepsilon^{-n/(n-2)},
\end{equation*}
\notag
$$
where the constant $K$ is independent of $\varepsilon$.
Consider the family of the extension operators
$$
\begin{equation*}
P_\varepsilon \colon \mathbf{V}_\varepsilon\to\mathbf{V}
\end{equation*}
\notag
$$
such that $P_\varepsilon v = v$ almost everywhere in $\Omega_\varepsilon$ and
$$
\begin{equation*}
\|\nabla P_\varepsilon v\|_{\mathbf{H}} \leqslant \|\nabla v\|_{\mathbf{H}_\varepsilon}\quad \text{for any function } v\in \mathbf{V}_\varepsilon
\end{equation*}
\notag
$$
(the detailed construction of such operators can be found in [11]).
Having in mind the previous inequality and the estimate (50), we conclude that the sequence $\widetilde u_\varepsilon = P_\varepsilon u_\varepsilon$ is bounded in $\mathbf{V}$. Therefore, it converges weakly in $\mathbf{V}$. Then there is a function $u\in\mathbf{V}$ such that
$$
\begin{equation*}
\widetilde u_\varepsilon\rightharpoonup u \quad \text{in }\mathbf{V}\text{ as } \varepsilon\to0.
\end{equation*}
\notag
$$
In what follows, we write $u_\varepsilon$ instead $\widetilde u_\varepsilon$.
We denote $T^j_r=\{ x\in \mathbb{R}^n \colon |x-P_\varepsilon^j|\leqslant r\}$. Consider the following temporary function $v_\varepsilon^j$ that satisfies the problem
$$
\begin{equation}
\begin{alignedat}{2} &\Delta v_\varepsilon^j=0, &\qquad x&\in T^j_{\varepsilon/4}\setminus \overline{G^j_\varepsilon}, \\ &\frac{\partial v_\varepsilon^j}{\partial \nu}+\varepsilon^{n/(n-2)}B_\varepsilon^j(x) v_\varepsilon^j=\varepsilon^{n/(n-2)}\overline{B}_\varepsilon^j(x), &\qquad x&\in \partial G_\varepsilon^j, \\ &v_\varepsilon^j=0, &\qquad x&\in \partial T^j_{\varepsilon/4}. \end{alignedat}
\end{equation}
\tag{58}
$$
It is easy to show that
$$
\begin{equation*}
\varepsilon^{n/(n-2)}\sum_{j\in \Upsilon_\varepsilon}\int_{\partial G_\varepsilon^j} B_\varepsilon^j(x) u_\varepsilon \phi\, d\omega=- \sum_{j\in \Upsilon_\varepsilon} \int_{\partial T^j_{\varepsilon/4}} \frac{\partial v_\varepsilon^j}{\partial\nu}\, u_\varepsilon \phi\, d\omega.
\end{equation*}
\notag
$$
Thus, we have established that
$$
\begin{equation}
\mathcal{V}_\varepsilon(x)= \begin{cases} v_\varepsilon^j(x), &x\in T^j_{\varepsilon/4}\setminus \overline{G^j_\varepsilon},\ j\in \Upsilon_\varepsilon, \\ 0, & x\in \mathbb{R}^n\setminus \overline{T^j_{\varepsilon/4}}. \end{cases}
\end{equation}
\tag{59}
$$
It is proved in [55] that
$$
\begin{equation*}
\|\mathcal{V}_\varepsilon\|^2_{\mathbf{V}_\varepsilon}\leqslant K\varepsilon^2
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\widetilde {\mathcal{V}}_\varepsilon\rightharpoonup 0 \quad\text{weakly in }\mathbf{V}, \qquad \widetilde {\mathcal{V}}_\varepsilon\to 0 \quad\text{strongly in }\mathbf{H}\quad\text{as}\quad \varepsilon\to0,
\end{equation*}
\notag
$$
where $\widetilde {\mathcal{V}}_\varepsilon=P_\varepsilon \mathcal{V}_\varepsilon$.
Using Lemmas 4.1, 4.2 in [55] we obtain that
$$
\begin{equation}
\biggl|\sum_{j\in \Upsilon_\varepsilon}\int_{\partial T^j_{\varepsilon/4}} \frac{\partial v_\varepsilon^j}{\partial\nu}\, h_\varepsilon\, d\omega+\int_{\Omega}V(x) h\, dx\biggr|\to 0
\end{equation}
\tag{60}
$$
as $\varepsilon\to0$ for functions $h_\varepsilon$, $h\in \mathbf{V}$, such that $h_\varepsilon\rightharpoonup h$ in $\mathbf{V}$.
Finally, (60) implies the convergence (57). Lemma 4.4 is proved. Using (52), (53), and (57) and passing to the limit in the equation of problem (48) as $k\to \infty$ in the space $D'(\mathbb{R}_+; \mathbf{H}^{-\mathbf{r}})$, we conclude that the function $\overline u(x,s)$ satisfies the equations
$$
\begin{equation}
\begin{alignedat}{2} &\frac{\partial \overline u}{\partial t} =\lambda \Delta \overline u -\overline{a}(x)f(\overline u\,)-V(x)\overline u +\overline{g}(x), &\qquad x&\in \Omega, \\ &\overline u =0, &\qquad x&\in \partial\Omega. \end{alignedat}
\end{equation}
\tag{61}
$$
Consequently, $\overline u\in \overline{\mathcal{K}}$. We have proved above that $u_{\varepsilon_k}(s)\to \overline u(s)$ as $k\to\infty$ in $\Theta^{\mathrm{loc}}$. The assumption $u_{\varepsilon_k}(s)\notin \mathcal{O}'(\overline{\mathcal{K}})$ yields $\overline u\notin \mathcal{O}'(\overline{\mathcal{K}})$ and, moreover, $\overline u\notin \overline{\mathcal{K}}$. We arrive to a contradiction. Theorem 4.1 is proved. We can strengthen the convergence (44) using compactness of inclusions (11) and (12), Corollary 4.1. For every $0<\delta \leqslant 1$ and for all $M>0$ To prove (62) and (63), we repeat the proof of Theorem 4.1 with the topology $\Theta^{\mathrm{loc}}$ replaced by the topology $\mathbf{L}_2^{\mathrm{loc}}(\mathbb{R}_+;\mathbf{H}^{1-\delta })$ or $\mathbf{C}^{\mathrm{loc}}(\mathbb{R}_+;\mathbf{H}^{-\delta })$. A natural question arise: is it possible to take $\delta=0$ in the limit relations (62) and (63)? The answer is affirmative under some additional conditions. For simplicity, we assume that the coefficient $a_{\varepsilon}(x)$ is independent of $\varepsilon$, that is, the function $a(x,y)=a(x)$ is independent of the fast variable $y$. Besides, for the function $g_\varepsilon(x)$, we have to assume a more strong homogenization condition instead of (15): for all $\varepsilon>0$ the functions $g^i_\varepsilon(x)=g^i(x,x/\varepsilon)\in L_2(\Omega)$ and have the means $\overline{g}^{\,i}(x)$ in the space $L_2(\Omega)$ as $\varepsilon \to 0{+}$, that is,
$$
\begin{equation}
\int_{\Omega}g^i\biggl(x,\frac{x}{\varepsilon}\biggr)\varphi(x)\, dx\to \int_{\Omega}\overline{g}^{\,i}(x)\varphi(x)\, dx, \qquad \varepsilon \to 0{+},
\end{equation}
\tag{64}
$$
for any function $\varphi\in L_2(\Omega)$ and for all $i=1,\dots, N$. For the proof, we need the following statement that is analogous to Lemma 4.4. Lemma 4.5. Under the assumptions of Lemma 4.4, the following limit relation holds Remark 4.2. We note that, in the proof of the main Theorem 4.2, the convergence (65) can be replaced by a weaker inequality Proof of Lemma 4.5. We estimate the difference, adding an subtracting the corresponding terms. We have
$$
\begin{equation}
\begin{aligned} \, &\biggl|\varepsilon_k^{n/(n-2)}\sum_{j\in\Upsilon_{\varepsilon_k}} \int_0^M\int_{\partial G_{\varepsilon_k }^j}sB_{\varepsilon_k}^j(x)u_{\varepsilon_k}(x,s)\cdot u_{\varepsilon_k}(x,s) \,d\omega\, ds \nonumber \\ &\qquad\qquad-\int_0^M\int_{\Omega}sV(x) \overline{u}(x,s)\cdot \overline{u}(x,s)\,dx\, ds \biggr| \nonumber \\ &\qquad\leqslant\biggl|\varepsilon_k^{n/(n-2)}\sum_{j\in\Upsilon_{\varepsilon_k}} \int_0^M\int_{\partial G_{\varepsilon_k }^j}sB_{\varepsilon_k}^j(x)u_{\varepsilon_k}(x,s)\cdot u_{\varepsilon_k}(x,s)\,d\omega\, ds \nonumber \\ &\qquad\qquad-\varepsilon_k^{n/(n-2)}\sum_{j\in\Upsilon_{\varepsilon_k}}\int_0^M \int_{\partial G_{\varepsilon_k }^j}sB_{\varepsilon_k}^j(x)\overline{u}(x,s)\cdot u_{\varepsilon_k}(x,s)\,d\omega\, ds\biggr| \nonumber \\ &\qquad+\biggl|\varepsilon_k^{n/(n-2)}\sum_{j\in\Upsilon_{\varepsilon_k}}\int_0^M \int_{\partial G_{\varepsilon_k }^j}sB_{\varepsilon_k}^j(x)\overline{u}(x,s)\cdot u_{\varepsilon_k}(x,s)\,d\omega\, ds \nonumber \\ &\qquad\qquad-\int_0^M\int_{\Omega}sV(x) \overline{u}(x,s)\cdot u_{\varepsilon_k}(x,s)\,dx\, ds \biggr| \nonumber \\ &\qquad+\biggl|\int_0^M\int_{\Omega}sV(x) \overline{u}(x,s)\cdot u_{\varepsilon_k}(x,s)\,dx\, ds \nonumber \\ &\qquad\qquad- \int_0^M\int_{\Omega}sV(x) \overline{u}(x,s)\cdot \overline{u}(x,s)\,dx\, ds \biggr|= I_1+ I_2+ I_3. \end{aligned}
\end{equation}
\tag{67}
$$
Applying the Cauchy–Bunyakovsky inequality to the terms $I_1$ and $I_3$, and having the strong convergence $u_{\varepsilon_k}(x,s)\to \overline{u}(x,s)$ in $\mathbf{L}_2(\Omega\times [0,M])$, we show that $I_1\to 0$ and $I_3\to0$ as $\varepsilon_k\to 0{+}$. The rest term $I_2$ approaches zero by Lemma 4.4. Lemma 4.5 is proved. We now formulate the main theorem on convergence of trajectory attractors of the system (13) in the strong topology $\Theta_+^{s,\mathrm{loc}}$, in which the trajectory attractors have been constructed for a fixed $\varepsilon$ (Theorem 3.1). Theorem 4.2. Assume that the coefficient $a=a(x)$ is independent of $\varepsilon$ and the function $g_{\varepsilon}(x)$ satisfies condition (64). Then the following convergence holds in the strong topology $\Theta_+^{s,\mathrm{loc}}$: Besides, Proof. Repeating the reasoning from the proof of Theorem 4.1, we construct a bounded in $\mathcal{F}^{\mathrm b}$ sequence $\{u_{\varepsilon_k}(s),\, s\in \mathbb{R}\}$ of complete trajectories of the system (13), which converges in the topology $\Theta^{\mathrm{loc}}$ as $\varepsilon_k\to 0{+}$ to the function $\overline{u}(s)$ that is a complete bounded trajectory of the limit (homogenized) system (43).
We claim that $u_{\varepsilon_k}(s)$ converges to $\overline{u}(s)$ in the strong topology $\Theta_+^{s,\mathrm{loc}}$. To establish this, we use the method of energy equalities from the proof of Theorem 3.1. It is sufficient to check that the sequence $\{u_{\varepsilon_k}(s)\}$ has a subsequence which converges strongly to $\overline{u}(s)$ in the space $\mathbf{L}_{\mathbf{p} }(\Omega\times [-M+1,M])\cap \mathbf{L}_2(-M+1,M;\mathbf{V})$ for every $M> 0$. For any fixed $M$, shifting the time back on $s=-M+s'$, we can assume that the functions $\{u_{\varepsilon_k}(s')\}$ and $\overline{u}(s')$ are defined on the interval $[0,M']$, $M'=2M$, and we are looking a subsequence that converges strongly in $\mathbf{L}_{\mathbf{p}}(\Omega\times [1,M'])\cap \mathbf{L}_2(1,M';\mathbf{V})$. For brevity, we omit the primes in $s'$ and $M'$. Since $\{u_{\varepsilon_k}(s)\}$ are bounded in the spaces $\mathbf{L}_{\mathbf{p}}(\Omega\times [0,M])$ and $\mathbf{L}_2(0,M;\mathbf{V})$, we can assume that $u_{\varepsilon_k}(\,{\cdot}\,)\rightharpoondown \overline{u}(\,{\cdot}\,)$ as $\varepsilon_k\to 0{+}$ weakly in the spaces $\mathbf{L}_{\mathbf{p}}(\Omega\times [0,M])$ and $\mathbf{L}_2(0,M;\mathbf{V})$. We can also assume that $u_{\varepsilon_k}(M)\rightharpoondown \overline{u}(M)$ as $\varepsilon_k\to 0{+}$ weakly in $\mathbf{H}$. Similarly to (34)–(36), we have
$$
\begin{equation}
\| \overline{u}(M)\| \leqslant \liminf_{k\to \infty}\| u_{\varepsilon_k}(M)\|,
\end{equation}
\tag{70}
$$
$$
\begin{equation}
\int_0^M\int_{\Omega}s\lambda\nabla \overline{u}\cdot \nabla \overline{u}\, dx\, ds \leqslant \liminf_{k\to \infty}\int_0^M\int_{\Omega}s\lambda\nabla u_{\varepsilon_k}\cdot \nabla u_{\varepsilon_k}\, dx\, ds,
\end{equation}
\tag{71}
$$
$$
\begin{equation}
\int_0^M\int_{\Omega}sa(x)|\overline{u}^{\,i}|^{p_i}\, dx\, ds \leqslant \lim\inf_{k\to \infty} \int_0^M\int_{\Omega}sa(x)|u_{\varepsilon_k}^i|^{p_i}\,dx\, ds,\qquad i=1,\dots,N,
\end{equation}
\tag{72}
$$
where, for brevity, we denote $u_{\varepsilon_k}=u_{\varepsilon_k}(x,s)$ and $\overline{u}=\overline{u}(x,s)$. Moreover, due to Lemma 4.5, the relation (66) holds.
Similarly to (38), we obtain
$$
\begin{equation}
\int_0^M\int_{\Omega}sa(x) F(\overline{u}(x,t))\, dx\, ds\leqslant \liminf_{k\to \infty} \int_0^M\int_{\Omega}sa(x) F(u_{\varepsilon_k}(x,t))\, dx\, ds
\end{equation}
\tag{73}
$$
(recall that we consider the case when the function $a(x)$ is independent on $\varepsilon$).
We now apply the energy equalities for the functions $u_{\varepsilon_k}(s) $ and $\overline{u}(\,{\cdot}\,)$, and obtain, similarly to (39) and (40), the following equalities:
$$
\begin{equation}
\begin{aligned} \, &\frac{1}{2}\| u_{\varepsilon_k}(M)\|^2+\int_0^M\int_{\Omega}s\lambda\nabla u_{\varepsilon_k}\cdot \nabla u_{\varepsilon_k}\, dx\, ds +\sum_{i=1}^{N}\gamma_i\int_0^M\int_{\Omega}sa(x) |u_{\varepsilon_k}^i|^{p_i}\, dx\, ds \notag \\ &\qquad+\int_0^M\int_{\Omega}sa(x)F(u_{\varepsilon_k})\, dx\, ds + \varepsilon^{n/(n-2)}\sum_{j\in\Upsilon_{\varepsilon}}\int_0^M \int_{\partial G_{\varepsilon_k}^j}sB_{\varepsilon_k}^j(x)u_{\varepsilon_k }\cdot u_{\varepsilon_k }\,d\omega \, ds \notag \\ &=\frac{1}{2} \int_0^M\int_{\Omega}|u_{\varepsilon_k}|^2\, dx\, ds +\int_0^M\int_{\Omega}g_{\varepsilon_k}(x)\cdot u_{\varepsilon_k}\, dx\, ds, \end{aligned}
\end{equation}
\tag{74}
$$
$$
\begin{equation}
\begin{aligned} \, &\frac{1}{2}\| \overline{u}(M)\|^2+\int_0^M\int_{\Omega}s\lambda\nabla \overline{u} \cdot \nabla \overline{u}\, dx\, ds +\sum_{i=1}^{N}\gamma_i\int_0^M\int_{\Omega}sa(x) |\overline{u}^{\,i}|^{p_i}\, dx\, ds \notag \\ &\qquad+\int_0^M\int_{\Omega}sa(x) F(\overline{u})\, dx\, ds +\int_0^M\int_{\Omega}sV(x) \overline{u}(x,s)\cdot \overline{u}(x,s)\,dx\, ds \notag \\ &=\frac{1}{2} \int_0^M\int_{\Omega}|\overline{u}|^2\, dx\, ds +\int_0^M\int_{\Omega}\overline{g}(x)\cdot \overline{u}\, dx\, ds. \end{aligned}
\end{equation}
\tag{75}
$$
Consider the difference
$$
\begin{equation}
\begin{aligned} \, &\biggl| \int_0^M\int_{\Omega}g_{\varepsilon_k}(x)\cdot u_{\varepsilon_k}\, dx\, ds -\int_0^M\int_{\Omega}\overline{g}(x)\cdot \overline{u}\, dx\, ds\biggr| \notag \\ &\qquad=\biggl| \int_0^M\int_{\Omega}g_{\varepsilon_k}(x)\cdot (u_{\varepsilon_k}-\overline{u}) \, dx\, ds +\int_0^M\int_{\Omega}\bigl(g_{\varepsilon_k}(x) -\overline{g}(x)\bigr) \cdot \overline{u}\, dx\, ds\biggr| \notag \\ &\qquad\leqslant \| g_{\varepsilon_k}(\,{\cdot}\,)\|_{\mathbf{L}_2}\| u_{\varepsilon_k}-\overline{u}\|_{\mathbf{L}_2}+\biggl| \int_0^M\int_{\Omega}\bigl( g_{\varepsilon_k}(x)-\overline{g}(x)\bigr) \cdot \overline{u}\, dx\, ds\biggr|. \end{aligned}
\end{equation}
\tag{76}
$$
Recall that $u_{\varepsilon_k}(\,{\cdot}\,)\to \overline{u}(\,{\cdot}\,)$ strongly in the space $\mathbf{L}_2(\Omega\times [0,M])$ and $g_{\varepsilon_k}(\,{\cdot}\,)\rightharpoondown \overline{g}(\,{\cdot}\,)$ weakly in $\mathbf{L}_2(\Omega\times [0,M])$ (see (64)) and, hence, $g_{\varepsilon_k}(\,{\cdot}\,)$ is uniformly bounded in $\mathbf{L}_2(\Omega\times [0,M])$. Therefore, both items in (76) tend to zero and, hence,
$$
\begin{equation}
\int_0^M\int_{\Omega}g_{\varepsilon_k}(x)\cdot u_{\varepsilon_k}\, dx\, ds\to \int_0^M\int_{\Omega}\overline{g}(x)\cdot \overline{u}\, dx\, ds\quad \text{as} \quad \varepsilon_k\to 0{+}.
\end{equation}
\tag{77}
$$
Thus, the right-hand side of the equation (74) tends to the right-hand side of the equation (75). Then the left-hand side of the equation (74) also tends to the left-hand side of the equation (75). Combining this observations with the inequalities (70)–(73) and (66), we conclude that
$$
\begin{equation*}
\begin{gathered} \, \lim_{k\to \infty }\| u_{\varepsilon_k}(M)\|^2 =\|\overline{u}(M)\|^2, \\ \lim_{k\to \infty }\int_0^M\int_{\Omega}s\lambda\nabla u_{\varepsilon_k}\cdot \nabla u_{\varepsilon_k}\, dx\, ds =\int_0^M\int_{\Omega}s\lambda\nabla \overline{u}\cdot \nabla \overline{u}\, dx\, ds, \\ \lim_{k\to \infty }\int_0^M\int_{\Omega}sa(x)|u_{\varepsilon_k}^i|^{p_i}\, dx\, ds =\int_0^M\int_{\Omega}sa(x)|\overline{u}^{\,i}|^{p_i}\, dx\, ds,\qquad i=1,\dots,N, \\ \lim_{k\to \infty }\int_0^M\int_{\Omega}sa(x)F(u_{\varepsilon_k})\, dx\, ds =\int_0^M\int_{\Omega}sa(x) F(\overline{u})\, dx\, ds, \\ \begin{aligned} \, &\lim_{k\to \infty}\varepsilon^{n/(n-2)}\sum_{j\in\Upsilon_{\varepsilon}}\int_0^M \int_{\partial G_{\varepsilon_k}^j}sB_{\varepsilon_k}^j(x)u_{\varepsilon_k }\cdot u_{\varepsilon_k }\,d\omega \, ds \\ &\qquad =\int_0^M\int_{\Omega}sV(x) \overline{u}(x,s)\cdot \overline{u}(x,s)\,dx\, ds. \end{aligned} \end{gathered}
\end{equation*}
\notag
$$
To complete the proof, we use the reasonings as in the end of the proof of Theorem 3.1 and obtain that $u_{\varepsilon_k}(\,{\cdot}\,)\to \overline{u}(\,{\cdot}\,)$ strongly in the space $\mathbf{L}_{\mathbf{p}}(\Omega\times [0,M])\cap \mathbf{L}_2(0,M;\mathbf{V})\cap C([0,M];\mathbf{H})$ and $\partial_tu_{\varepsilon_k}(\,{\cdot}\,)\to \partial_t\overline{u}(\,{\cdot}\,)$ strongly in the space $\mathcal{L(}0,M)$ as $\varepsilon_k\to 0{+}$. We have proved (69) and, consequently (68). Theorem 4.2 is proved. Remark 4.3. Theorem 4.2 holds also in the general case, when the coefficient $a_{\varepsilon}(x)$ depends on $\varepsilon$ and satisfies the homogenization condition (14). Finally we consider the reaction-diffusion systems for which the uniqueness theorem of the Cauchy problem takes place. It is sufficient to assume that the nonlinear term $f(u)$ in the system (13) satisfies the condition
$$
\begin{equation}
(f(v_1)-f(v_2),\, v_1-v_2) \geqslant -C|v_1-v_2|^2\quad\forall\, v_1,v_2\in \mathbb{R}^{N}
\end{equation}
\tag{78}
$$
(see [21], [48]). It was proved in [48] that (78) implies that the systems (13) and (43) generate dynamical semigroups in $\mathbf{H}$, which have the global attractors $\mathcal{A}_{\varepsilon}$ and $\overline{\mathcal{A}}$, that are bounded in the space $\mathbf{V}=\mathbf{H}_0^1(\Omega)$ (see also [20], [22]). Moreover, the following identities holds:
$$
\begin{equation*}
\mathcal{A}_{\varepsilon}=\{u(0)\mid u\in \mathfrak{A}_{\varepsilon}\},\qquad \overline{\mathcal{A}}=\{u(0)\mid u\in \overline{\mathfrak{A}}\}.
\end{equation*}
\notag
$$
In this case, the convergence (63) implies the following corollary 4.2. Corollary 4.2. Under the assumptions of Theorem 4.2 the following limit relation holds: 
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