Application of Fatou’s Lemma for Strong Homogenization of Attractors to Reaction–Diffusion Systems with Rapidly Oscillating Coefficients in Orthotropic Media with Periodic Obstacles

Abstract

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.

1. Introduction

Modern technologies are connected with the production of micro-inhomogeneous materials including reinforced, porous, skeleton, fractal structures and other composites. In this connection the appearance of initial-boundary value problems with rapidly oscillating terms or in domains with complicated micro-inhomogeneous structures is very natural. This leads to almost insoluble numerical problems. To resolve these difficulties, specialists use homogenization and asymptotic methods (see, for instance, [1,2,3,4,5,6,7,8,9,10], which also contain detailed references for the subject). The investigation of fractals is an independently complicated problem (see recent papers [11,12]).

In this paper, we study the limit behaviour of trajectory attractors to the initial-boundary value problem for a general reaction–diffusion equations in periodically perforated domains (media with periodic obstacles), depending on the small parameter $\epsilon$ that approaches zero. The parameter $\epsilon$ describes the perforation step and the cavities diameter (obstacles), while the value ${\epsilon }^{-1}$ is proportional to the rate of oscillations in the cavity (obstacles) boundary conditions and the coefficients of equations. At the same time, unlike the paper [13], we manage to prove the strong convergence of attractors under more general conditions for the oscillating terms of the equations.

We study the strong convergence and limit behaviour of trajectory attractors as the small parameter $\epsilon$ tends to zero. To analyse this phenomenon, we utilize the homogenization methods (e.g., [1,2,6,7,9,14,15], as well as subtle analysis of trajectory dynamic systems.

We note several works on the homogenization of attractors for reaction–diffusion equations that appear recently ([13,16,17]). The homogenization of attractors of scalar evolution equations with dissipation in a domain with periodic perforation has been studied in [17]. The paper [18] is devoted to the homogenization of Ginzburg–Landau equations in perforated domains.

The attractors appear in dissipative dynamic systems and reflect the limit behaviour of trajectories (solutions) as time tends to infinity. Informally, the attractor is the smallest set of phase spaces of a dissipative dynamic system that attracts all trajectories starting from bounded domains of initial data. Roughly speaking the attractor characterizes the whole dynamic system and is especially useful in the investigations of the final behaviour of the solutions of non-linear evolution partial dissipative equations ([19,20,21] and references therein). Using the attractors, it is also useful to study the global perturbation solutions for evolution equations.

In the paper, we are interested in asymptotic behaviour of trajectory and global attractors of a reaction–diffusion system with rapidly oscillating terms in a domain with small obstacles that describe the behaviour of orthotropic media.

The theory of trajectory attractors for dissipative partial differential equations is sufficiently completed and it can be found in [20,22]. For equations with the absence of the uniqueness to the corresponding initial-value problem (e.g., for the inhomogeneous 3D Navier–Stokes system in a bounded domain, for which the uniqueness is not yet proven, or for the general reaction–diffusion systems considered in this paper, for which it does not hold), the developed approach is effective in the study of the long-term behaviour of the solutions.

The pioneering papers [23,24,25] on averaging the attractors of evolution equations with rapidly oscillating terms used the classical Bogolyubov averaging principle [26]. The averaging of global attractors for equations with oscillating parameters was considered in [20,27,28,29,30] for parabolic-type equations and in [31,32,33,34,35] for hyperbolic-type equations. Similar averaging problems for autonomous and non-autonomous 2D Navier–Stokes equations was studied in [20,33,36]. The authors of papers [37,38] dealt with partial differential equations containing singular oscillating terms with growing amplitudes.

The main theorem of the paper stated that the trajectory attractors ${\mathfrak{A}}_{\epsilon }$ of a general dissipative reaction–diffusion system containing rapidly oscillating terms and acting in a domain with small obstacles converges as $\epsilon \to 0$ in strong topology to the trajectory attractor $\overline{\mathfrak{A}}$ of a homogenized system in an appropriate functional space. The mentioned strong topology corresponds to the maximal possible topology in which the solutions of the system are constructed.

In Section 2, we define the geometric structure of the perforated domain, formulate the problem and describe necessary functional spaces. Section 3 is devoted to the construction of trajectory attractors for the reaction–diffusion systems in strong topology for a fixed parameter $\epsilon$. In Section 4, we study the homogenization of attractors for autonomous a reaction–diffusion system with rapidly oscillating terms in a perforated domain as well as demonstrate the appearance of a potential in the homogenized system (see pioneering works in [4,7] on the appearance of a “strange term” (potential)).

2. Statement of the Problem

Assume that $\mathsf{\Omega }$ is a bounded domain in ${\mathbb{R}}^n$, $n\ge 3,$ with piece-wise $C^1$-smooth boundary $\partial \mathsf{\Omega }$ and $0\in \mathsf{\Omega }$. Let $G_0$ be a domain belonging to ${(-1/2,1/2)}^n$ such that ${\overline{G}}_0$ is a compact set diffeomorphic to a ball.

For multi-indexes $j\in {\mathbb{Z}}^n$, we define points and sets

$$P_{\epsilon }^j=\epsilon j,G_{\epsilon }^j=P_{\epsilon }^j+{\epsilon }^{n/(n-2)}{G}_0.$$

Denoted by ${\tilde{\mathsf{\Omega }}}_{\epsilon }$ the domain $\{x\in \mathsf{\Omega }:\rho (x,\partial \mathsf{\Omega })>\sqrt{n}\epsilon \}\subset \mathsf{\Omega }$ and by ${\mathrm{Y}}_{\epsilon }$ the set of permissible multi-indices

$${\mathrm{Y}}_{\epsilon }=\left\{j\in {\mathbb{Z}}^n:G_{\epsilon }^j\cap {\overline{\tilde{\mathsf{\Omega }}}}_{\epsilon }\ne ⌀\right\}.$$

Note that $|{\mathrm{Y}}_{\epsilon }|\cong d{\epsilon }^{-n}$, is some constant $d>0$. Consider the following domain with periodic perforation:

$${\mathsf{\Omega }}_{\epsilon }=\mathsf{\Omega }\backslash {\overline{G}}_{\epsilon },\mathrm{where}{G}_{\epsilon }=\bigcup _{j\in {\mathrm{Y}}_{\epsilon }}{G}_{\epsilon }^j.$$

Consider also the cylindrical domains

$$Q_{\epsilon }={\mathsf{\Omega }}_{\epsilon }\times (0,+\infty ),Q=\mathsf{\Omega }\times (0,+\infty ).$$

We study the initial-boundary value problem for the reaction–diffusion equations in the perforated domain (domain with obstacles) ${\mathsf{\Omega }}_{\epsilon }$:

$$\begin{array}{cc}\hfill & \frac{\partial u_{\epsilon }}{\partial t}=\lambda \mathrm{\Delta }{u}_{\epsilon }-a\left(x,\frac{x}{\epsilon }\right)f\left(u_{\epsilon }\right)+g\left(x,\frac{x}{\epsilon }\right),x\in {\mathsf{\Omega }}_{\epsilon },t\ge 0,\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill & \frac{\partial u_{\epsilon }}{\partial \nu }+{\epsilon }^{n/(n-2)}{B}_{\epsilon }^j\left(x\right)u_{\epsilon }=0,x\in \partial G_{\epsilon }^j,t\ge 0,j\in {\mathrm{Y}}_{\epsilon },\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill & u_{\epsilon }(x,t)=0,x\in \partial \mathsf{\Omega },t\ge 0,\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill & u_{\epsilon }(x,0)=U\left(x\right),x\in {\mathsf{\Omega }}_{\epsilon },\hfill \end{array}$$

(4)

where $u_{\epsilon }(x,t)={(u_{\epsilon }^1,\cdots ,u_{\epsilon }^N)}^{\top }$ is the unknown vector function, $f={(f^1,\cdots ,f^N)}^{\top }$ is the known non-linear function, $g={(g^1,\cdots ,g^N)}^{\top }$ is the known vector function, and $\lambda$ is the $N\times N$-matrix with constant coefficients such that $(\lambda +{\lambda }^{\top })/2\ge \beta I$ (here $\beta >0$ and I are the identity matrix of order N), $\nu$ is the outward normal vector to the boundaries $G_{\epsilon }^j$.

Let the non-linear vector-valued function $f\left(v\right)\in C({\mathbb{R}}^N;{\mathbb{R}}^N)$ satisfy the inequalities

$$\begin{array}{c}\sum _{i=1}^N{\left|f^i\left(v\right)\right|}^{p_i/(p_i-1)}\le C_0\left(\sum _{i=1}^N{\left|v^i\right|}^{p_i}+1\right),2\le p_1\le \cdots \le p_{N-1}\le p_N,\end{array}$$

(5)

$$\begin{array}{c}\sum _{i=1}^N{\gamma }_i{\left|v^i\right|}^{p_i}-C\le \sum _{i=1}^N{f}^i\left(v\right)v^i\forall v\in {\mathbb{R}}^N,\end{array}$$

(6)

where ${\gamma }_i>0$ for all $i=1,\cdots ,N$. Note that in real reaction–diffusion systems, functions $f^i\left(u\right)$ are polynomial, possibly, having different degrees and the Inequality (5) reflects this fact. The Estimate (6) is called the dissipation condition for the reaction–diffusion system in (1). In a simple model case $p_i\equiv p$ for all $i=1,\cdots ,N$, and the Conditions (5) and (6) are simplified as follows:

$$\left|f\left(v\right)\right|\le C_0{\left(\right|v|}^{p-1}+{1),\gamma |v|}^p-C\le f\left(v\right)v\forall v\in {\mathbb{R}}^N.$$

(7)

We assume that no Lipschitz condition for the non-linear vector function $f\left(v\right)$ with respect to variable v is valid.

The scalar function $a(x,y)\in C(\overline{\mathsf{\Omega }}\times {\mathbb{R}}^n)$ satisfies the inequalities

$$0<a_0\le a(x,y)\le A_0$$

with constants $a_0$ and $A_0$. We assume that the function $a_{\epsilon }\left(x\right)=a(x,x/\epsilon )$ has the mean $\overline{a}\left(x\right)$ as $\epsilon \to 0+$ in the space $L_{\infty ,*w}\left(\mathsf{\Omega }\right)$, that is,

$${\int }_{\mathsf{\Omega }}a\left(x,\frac{x}{\epsilon }\right)\phi \left(x\right)dx\to {\int }_{\mathsf{\Omega }}\overline{a}\left(x\right)\phi \left(x\right)dx(\epsilon \to 0+),\forall \phi \in L_1\left(\mathsf{\Omega }\right).$$

(8)

Suppose that the vector function $g(x,y)$ satisfies the following: for every $\epsilon >0$ the components $g_{\epsilon }^i\left(x\right)=g^i(x,x/\epsilon )\in L_2\left(\mathsf{\Omega }\right)$ and have the mean ${\overline{g}}^i\left(x\right)$ in the space $L_2\left(\mathsf{\Omega }\right)$ as $\epsilon \to 0+$, that is,

$${\int }_{\mathsf{\Omega }}{g}^i\left(x,\frac{x}{\epsilon }\right)\phi \left(x\right)dx\to {\int }_{\mathsf{\Omega }}{\overline{g}}^i\left(x\right)\phi \left(x\right)dx(\epsilon \to 0)+,\forall \phi \in L_2\left(\mathsf{\Omega }\right),i=1,\cdots ,N.$$

(9)

For example, $a(x,y)=a(x,y_1,\dots ,y_n)$ and $g(x,y)=g(x,y_1,\dots ,y_n)$ are 1-periodic function in each variable $y_i,i=1,\dots ,n.$

The Assumption (9) implies that the norms of function $g_{\epsilon }^i\left(x\right)$ are uniformly bounded with respect to $\epsilon$ in the space $L_2\left(\mathsf{\Omega }\right)$:

$$\parallel g_{\epsilon }^i{\left(x\right)\parallel }_{L_2\left(\mathsf{\Omega }\right)}\le M_0\forall \epsilon \in (0,1].$$

(10)

In the boundary Conditions (2), the matrix $B_{\epsilon }^j\left(x\right)$ is diagonal with elements

$$b^{11}\left(x,\frac{x-P_{\epsilon }^j}{{\epsilon }^{n/(n-2)}}\right),\cdots ,b^{NN}\left(x,\frac{x-P_{\epsilon }^j}{{\epsilon }^{n/(n-2)}}\right),j\in {\mathrm{Y}}_{\epsilon },$$

(11)

where $b^{kk}(x,y)\in C(\mathsf{\Omega }\times {\mathbb{R}}^n)$ is an 1-periodic in y functions satisfying

$$0<b_0\le b^{kk}(x,y)\le B_0$$

(12)

with constants $b_0$ and $B_0$ for all $k=1,\cdots ,N$.

Remark 1.

We can also study reaction–diffusion systems with more general non-linear terms of the form ${\sum }_{j=1}^m{a}_j(x,x/\epsilon )f_j\left(u\right)$ where $a_j$ are matrices with positive coefficients whose elements have means as $\epsilon \to 0+$ in the space $L_{\infty ,*w}\left(\mathsf{\Omega }\right)$ and $f_j\left(u\right)$ are vector-valued polynomials in u that satisfy Conditions (5) and (6). For simplicity, we only consider the case $m=1$ and $a_1(x,x/\epsilon )=a(x,x/\epsilon )I$, where I is the identity matrix.

Interesting examples of functions which satisfy the homogenization Conditions (8) and (9) can be found in [13], including almost periodic functions (see, for example, [39]).

We denote spaces $\mathbf{H}:={\left[L_2\left(\mathsf{\Omega }\right)\right]}^N$, ${\mathbf{H}}_{\epsilon }:={\left[L_2\left({\mathsf{\Omega }}_{\epsilon }\right)\right]}^N$, $\mathbf{V}:={\left[H_0^1\left(\mathsf{\Omega }\right)\right]}^N$, and ${\mathbf{V}}_{\epsilon }:={\left[H^1({\mathsf{\Omega }}_{\epsilon };\partial \mathsf{\Omega })\right]}^N$ as the space of functions from ${\left[H^1\left({\mathsf{\Omega }}_{\epsilon }\right)\right]}^N$ with zero trace on $\partial \mathsf{\Omega }$ with the respective norms

$$\begin{array}{cccc}\hfill {\parallel v\parallel }^2& :={\int }_{\mathsf{\Omega }}\sum _{i=1}^N{\left|v^i\left(x\right)\right|}^{2}dx,\hfill & \hfill {\parallel v\parallel }_{\epsilon }^2& :={\int }_{{\mathsf{\Omega }}_{\epsilon }}\sum _{i=1}^N{\left|v^i\left(x\right)\right|}^{2}dx,\hfill \\ \hfill {\parallel v\parallel }_1^2& :={\int }_{\mathsf{\Omega }}\sum _{i=1}^N{|\nabla v^i\left(x\right)|}^{2}dx,\hfill & \hfill {\parallel v\parallel }_{1\epsilon }^2& :={\int }_{{\mathsf{\Omega }}_{\epsilon }}\sum _{i=1}^N{|\nabla v^i\left(x\right)|}^{2}dx.\hfill \end{array}$$

We also denote ${\mathbf{V}}^{\prime }$ and ${\mathbf{V}}_{\epsilon }^{\prime }$ as the dual spaces of $\mathbf{V}$ and ${\mathbf{V}}_{\epsilon }$, respectively.

Thus, with index $\epsilon$, we denote spaces in the perforated domain $H^1\left({\mathsf{\Omega }}_{\epsilon }\right)$ and, without index $\epsilon ,$ we denote the spaces in the domain $\mathsf{\Omega }$ without perforations.

We set $q_i=p_i/(p_i-1)$ for all $i=1,\cdots ,N$. We use the following vector notations: $\mathbf{p}=(p_1,\cdots ,p_N)$ and $\mathbf{q}=(q_1,\cdots ,q_N)$, and introduce the spaces

$$\begin{array}{c}{\mathbf{L}}_{\mathbf{p}}:=L_{p_1}\left(\mathsf{\Omega }\right)\times \cdots \times L_{p_N}\left(\mathsf{\Omega }\right),{\mathbf{L}}_{\mathbf{p},\epsilon }:=L_{p_1}\left({\mathsf{\Omega }}_{\epsilon }\right)\times \cdots \times L_{p_N}\left({\mathsf{\Omega }}_{\epsilon }\right),\\ \begin{array}{cc}\hfill {\mathbf{L}}_{\mathbf{p}}({\mathbb{R}}_{+};{\mathbf{L}}_{\mathbf{p}})& :=L_{p_1}({\mathbb{R}}_{+};L_{p_1}\left(\mathsf{\Omega }\right))\times \cdots \times L_{p_N}({\mathbb{R}}_{+};L_{p_N}\left(\mathsf{\Omega }\right)),\hfill \\ \hfill {\mathbf{L}}_{\mathbf{p}}({\mathbb{R}}_{+};{\mathbf{L}}_{\mathbf{p},\epsilon })& :=L_{p_1}({\mathbb{R}}_{+};L_{p_1}\left({\mathsf{\Omega }}_{\epsilon }\right))\times \cdots \times L_{p_N}({\mathbb{R}}_{+};L_{p_N}\left({\mathsf{\Omega }}_{\epsilon }\right)).\hfill \end{array}\end{array}$$

As in [20,40], we study the weak solutions to the initial-boundary value Problem (1)–(4). The function

$$u_{\epsilon }(x,t)\in {\mathbf{L}}_{\infty }^{\mathrm{loc}}({\mathbb{R}}_{+};{\mathbf{H}}_{\epsilon })\cap {\mathbf{L}}_2^{\mathrm{loc}}({\mathbb{R}}_{+};{\mathbf{V}}_{\epsilon })\cap {\mathbf{L}}_{\mathbf{p}}^{\mathrm{loc}}({\mathbb{R}}_{+};{\mathbf{L}}_{\mathbf{p},\epsilon }),$$

satisfies Problem (1)–(4) in the distribution sense, that is, the integral identity

$$\begin{array}{cc}\hfill & -{\int }_{Q_{\epsilon }}{u}_{\epsilon }·\frac{\partial \psi }{\partial t}dxdt+{\int }_{Q_{\epsilon }}\lambda \nabla u_{\epsilon }·\nabla \psi dxdt+{\int }_{Q_{\epsilon }}{a}_{\epsilon }\left(x\right)f\left(u_{\epsilon }\right)·\psi dxdt+\hfill \\ \hfill & {\epsilon }^{n/(n-2)}\sum _{j\in {\mathrm{Y}}_{\epsilon }}{\int }_0^{+\infty }{\int }_{\partial G_{\epsilon }^j}{B}_{\epsilon }^j\left(x\right)u_{\epsilon }·\psi d\omega dt={\int }_{Q_{\epsilon }}{g}_{\epsilon }\left(x\right)·\psi dxdt\hfill \end{array}$$

(13)

holds for all test functions $\psi \in {\mathbf{C}}_0^{\infty }({\mathbb{R}}_{+};{\mathbf{V}}_{\epsilon }\cap {\mathbf{L}}_{\mathbf{p},\epsilon })$. Here, $w_1·w_2$ denotes the inner product of vectors $w_1,w_2\in {\mathbb{R}}^N$. By $d\omega$, we denote an element of $(n-1)$-dimensional volume on the boundaries $\partial G_{\epsilon }^j$.

If $u_{\epsilon }(x,t)\in {\mathbf{L}}_{\mathbf{p}}(0,M;{\mathbf{L}}_{\mathbf{p},\epsilon })$ then Condition (5) implies that the function $f\left(u(x,t)\right)\in {\mathbf{L}}_{\mathbf{q}}(0,M;{\mathbf{L}}_{\mathbf{q},\epsilon })$. At the same time, if $u_{\epsilon }(x,t)\in {\mathbf{L}}_2(0,M;{\mathbf{V}}_{\epsilon })$ then $\lambda \mathrm{\Delta }{u}_{\epsilon }(x,t)+g_{\epsilon }\left(x\right)\in {\mathbf{L}}_2(0,M;{\mathbf{V}}_{\epsilon }^{\prime })$. Hence, for an arbitrary $u_{\epsilon }(x,s)$ (weak solution to Problem (1)–(4)) we have

$$\frac{\partial u_{\epsilon }(x,t)}{\partial t}\in {\mathbf{L}}_{\mathbf{q}}(0,M;{\mathbf{L}}_{\mathbf{q},\epsilon })+{\mathbf{L}}_2(0,M;{\mathbf{V}}_{\epsilon }^{\prime }).$$

We have (by means of the Sobolev embedding theorem)

$${\mathbf{L}}_{\mathbf{q}}(0,M;{\mathbf{L}}_{\mathbf{q},\epsilon })+{\mathbf{L}}_2(0,M;{\mathbf{V}}_{\epsilon }^{\prime })\subset {\mathbf{L}}_{\mathbf{q}}(0,M;{\mathbf{H}}_{\epsilon }^{-\mathbf{r}}),$$

where the space ${\mathbf{H}}_{\epsilon }^{-\mathbf{r}}:=H^{-r_1}\left({\mathsf{\Omega }}_{\epsilon }\right)\times \cdots \times H^{-r_N}\left({\mathsf{\Omega }}_{\epsilon }\right)$, $\mathbf{r}=(r_1,\cdots ,r_N)$, and exponents $r_i=max\{1,n(1/q_i-1/2)\}$ for $i=1,\cdots ,N$. The space $H^{-r}\left({\mathsf{\Omega }}_{\epsilon }\right)$ is the dual of the Sobolev space $H^r\left({\mathsf{\Omega }}_{\epsilon }\right)$ with exponent $r>0$ in the perforated domain ${\mathsf{\Omega }}_{\epsilon }$. We also use the space ${\mathbf{H}}^{-\mathbf{r}}$ in domain $\mathsf{\Omega }$ without perforation.

Hence, for any solution $u_{\epsilon }(x,t)$ of Problem (1)–(4), its time derivative $\partial u_{\epsilon }(x,t)/\partial t\in {\mathbf{L}}_{\mathbf{q}}$$(0,M;{\mathbf{H}}_{\epsilon }^{-\mathbf{r}})$.

Proposition 1.

For a fixed $\epsilon >0$ and for any initial function $U\in {\mathbf{H}}_{\epsilon }$ the Problem (1)–(4) has a weak solution $u_{\epsilon }(x,t)$ such that $u_{\epsilon }(x,0)=U\left(x\right)$ and $u_{\epsilon }(x,t)\in C_b({\mathbb{R}}_{+};{\mathbf{H}}_{\epsilon }^{-\mathbf{r}})\cap C_w({\mathbb{R}}_{+};{\mathbf{H}}_{\epsilon }).$

The proof of this proposition is standard (see, for example, [19,40]). This solution is not unique since the function $f\left(v\right)$ only satisfies the Conditions (5) and (6) and no Lipschitz condition is imposed with respect to v.

The next key proposition is proved in the same manner as Proposition XV.3.1 in [20].

Proposition 2.

Assume that $u_{\epsilon }(x,t)\in {\mathbf{L}}_2^{\mathrm{loc}}({\mathbb{R}}_{+};{\mathbf{V}}_{\epsilon })\cap {\mathbf{L}}_{\mathbf{p}}^{\mathrm{loc}}({\mathbb{R}}_{+};{\mathbf{L}}_{\mathbf{p},\epsilon })$ is an arbitrary weak solution of Problem (1)–(4). Then

(i)

$u_{\epsilon }\in \mathbf{C}({\mathbb{R}}_{+};{\mathbf{H}}_{\epsilon })$;

(ii)

the scalar function $\parallel u_{\epsilon }{(·,t)\parallel }^2$ is absolutely continuous on ${\mathbb{R}}_{+}$ and

$$\begin{array}{cc}\hfill & \frac{1}{2}\frac{d}{dt}{\parallel u_{\epsilon }(·,t)\parallel }^2+{\int }_{{\mathsf{\Omega }}_{\epsilon }}\lambda \nabla u_{\epsilon }(x,t)·\nabla u_{\epsilon }(x,t)dx+{\int }_{{\mathsf{\Omega }}_{\epsilon }}{a}_{\epsilon }\left(x\right)f\left(u_{\epsilon }(x,t)\right)·u_{\epsilon }(x,t)dx+\hfill \\ \hfill & {\epsilon }^{n/(n-2)}\sum _{j\in {\mathrm{Y}}_{\epsilon }}{\int }_{\partial G_{\epsilon }^j}{B}_{\epsilon }^j\left(x\right)u_{\epsilon }(x,t)·u_{\epsilon }(x,t)d\omega ={\int }_{{\mathsf{\Omega }}_{\epsilon }}{g}_{\epsilon }\left(x\right)·u_{\epsilon }(x,t)dx\hfill \end{array}$$

(14)

for almost all $t\in {\mathbb{R}}_{+}$.

Note that in the Equality (14), the integrals over the boundaries of $G_{\epsilon }^j$ are non-negative due to the Condition (12). Integrating this differential equality in time and using the dissipation Condition (6), we obtain

Proposition 3.

The following estimates:

$$\begin{array}{c}\parallel u_{\epsilon }{\left(t\right)\parallel }^2\le {\parallel u_{\epsilon }\left(0\right)\parallel }^2{e}^{-{\lambda }_1\beta t}+R_1^2,\end{array}$$

(15)

$$\begin{array}{c}\begin{array}{cc}& \beta {\int }_t^{t+1}\parallel u_{\epsilon }{\left(s\right)\parallel }_1^{2}ds+2a_0\sum _{i=1}^N{\gamma }_i{\int }_t^{t+1}{\parallel u_{\epsilon }^i\left(s\right)\parallel }_{L_{p_i}\left({\mathsf{\Omega }}_{\epsilon }\right)}^{p_i}ds\hfill \\ & +2B_0{\epsilon }^{n/(n-2)}\sum _{j\in {\mathrm{Y}}_{\epsilon }}{\int }_t^{t+1}\parallel u_{\epsilon }{\left(s\right)\parallel }_{{\mathbf{L}}_2(\partial G_{\epsilon }^j)}^{2}ds\le {\parallel u_{\epsilon }\left(t\right)\parallel }^2+R_2^2,\hfill \end{array}\end{array}$$

(16)

are valid for any weak solution $u\left(t\right)$ to Problem (1)–(4), where ${\lambda }_1$ is the first eigenvalue of the operator $-\mathrm{\Delta }$ for the considered boundary-value problem. Positive values $R_1$ and $R_2$ depend on the number $M_0$ (see (10))) and are independent of $u_{\epsilon }\left(0\right)$ and ε.

The proof of this Proposition is given in [20].

3. Trajectory Attractor of Reaction–Diffusion System in a Domain with Obstacles

Here, we define the trajectory attractor for the general reaction–diffusion Equations (1)–(4) in a perforated domain for a fixed $\epsilon$.

We omit the subscript $\epsilon$ in the denotations of function spaces provided that no confusion arises.

To describe the trajectory space ${\mathcal{K}}_{\epsilon }^{+}$ (the trajectory space to Equations (1)–(4)), for every segment $[0,M]\subset {\mathbb{R}}_{+}$, we define the Banach spaces

$${\mathcal{F}}_{0,M}:={\mathbf{L}}_{\mathbf{p}}(0,M;{\mathbf{L}}_p)\cap {\mathbf{L}}_2(0,M;\mathbf{V})\cap {\mathbf{L}}_{\infty }(0,M;\mathbf{H})\cap \left\{v|\frac{\partial v}{\partial t}\in {\mathbf{L}}_{\mathbf{q}}(0,M;{\mathbf{H}}^{-\mathbf{r}})\right\}$$

(17)

with norms

$${\parallel v\parallel }_{{\mathcal{F}}_{0,M}}:={\parallel v\parallel }_{{\mathbf{L}}_{\mathbf{p}}(0,M;{\mathbf{L}}_{\mathbf{p}})}+{\parallel v\parallel }_{{\mathbf{L}}_2(0,M;\mathbf{V})}+{\parallel v\parallel }_{{\mathbf{L}}_{\infty }(0,M;\mathbf{H})}+{∥\frac{\partial v}{\partial t}∥}_{{\mathbf{L}}_{\mathbf{q}}(0,M;{\mathbf{H}}^{-\mathbf{r}})}.$$

(18)

We also define the spaces

$$\begin{array}{cc}\hfill {\mathcal{F}}_{+}^{\mathrm{loc}}& ={\mathbf{L}}_{\mathbf{p}}^{\mathrm{loc}}({\mathbb{R}}_{+};{\mathbf{L}}_{\mathbf{p}})\cap {\mathbf{L}}_2^{\mathrm{loc}}({\mathbb{R}}_{+};\mathbf{V})\cap {\mathbf{L}}_{\infty }^{\mathrm{loc}}({\mathbb{R}}_{+};\mathbf{H})\cap \left\{v|\frac{\partial v}{\partial t}\in {\mathbf{L}}_{\mathbf{q}}^{\mathrm{loc}}({\mathbb{R}}_{+};{\mathbf{H}}^{-\mathbf{r}})\right\},\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill {\mathcal{F}}_{\epsilon ,+}^{\mathrm{loc}}& ={\mathbf{L}}_{\mathbf{p}}^{\mathrm{loc}}({\mathbb{R}}_{+};{\mathbf{L}}_{\mathbf{p},\epsilon })\cap {\mathbf{L}}_2^{\mathrm{loc}}({\mathbb{R}}_{+};{\mathbf{V}}_{\epsilon })\cap {\mathbf{L}}_{\infty }^{\mathrm{loc}}({\mathbb{R}}_{+};{\mathbf{H}}_{\epsilon })\cap \left\{v|\frac{\partial v}{\partial t}\in {\mathbf{L}}_{\mathbf{q}}^{\mathrm{loc}}({\mathbb{R}}_{+};{\mathbf{H}}_{\epsilon }^{-\mathbf{r}})\right\}.\hfill \end{array}$$

(20)

We denote by ${\mathcal{K}}_{\epsilon }^{+}$ the set of all weak solutions to the Problem (1)–(4). We recall that for any $U\in \mathbf{H}$ there exists at least one trajectory $u(·)\in {\mathcal{K}}_{\epsilon }^{+}$ such that $u\left(0\right)=U$. Therefore, the trajectory space ${\mathcal{K}}_{\epsilon }^{+}$ is sufficiently large. It is easy to verify that ${\mathcal{K}}_{\epsilon }^{+}\subset {\mathcal{F}}_{+}^{\mathrm{loc}}.$

Consider the translation operators $S\left(h\right),h\ge 0$, acting on functions $f\in {\mathcal{F}}_{+}^{loc}$ by the

$$S\left(h\right)f\left(t\right)=f(h+t).$$

The mappings $\left\{S\right(h),h\ge 0\}$ forms a semigroup in ${\mathcal{F}}_{+}^{loc}$: $S(h_1+h_2)=S\left(h_1\right)S\circ \left(h_1\right)$ for all $h_1,h_2\ge 0$ and $S\left(0\right)=Id$ is the identity operator.

The trajectory space ${\mathcal{K}}_{\epsilon }^{+}$ is translation invariant, that is, if $u\left(t\right)\in {\mathcal{K}}_{\epsilon }^{+}$, then $u(h+t)\in {\mathcal{K}}_{\epsilon }^{+}$ for all $h\ge 0$. Hence,

$$S\left(h\right){\mathcal{K}}_{\epsilon }^{+}\subseteq {\mathcal{K}}_{\epsilon }^{+}\forall h\ge 0.$$

We define the metrics ${\rho }_{0,M}(·,·)$ on the spaces ${\mathcal{F}}_{0,M}$, using the norms in these spaces

$${\rho }_{0,M}(u,v)={\parallel u(·)-v(·)\parallel }_{{\mathcal{F}}_{0,M}},\forall u(·),v(·)\in {\mathcal{F}}_{0,M}.$$

These metrics generate the strong topology ${\mathsf{\Theta }}_{+}^{loc}$ in the space ${\mathcal{F}}_{+}^{loc}$. By definition, a sequence $\left\{v_k\right\}\subset {\mathcal{F}}_{+}^{loc}$ converges to $v\in {\mathcal{F}}_{+}^{loc}$ as $k\to \infty$ in ${\mathsf{\Theta }}_{+}^{loc}$, if for any $M>0$

$$\begin{array}{cc}\hfill & {\parallel u(·)-v(·)\parallel }_{{\mathcal{F}}_{0,M}}=\parallel v_k{(·)-v(·)\parallel }_{{\mathbf{L}}_{\mathbf{p}}(0,M;{\mathbf{L}}_{\mathbf{p}})}+{\parallel v_k(·)-v(·)\parallel }_{L_2(0,M;\mathbf{V})}+\hfill \\ \hfill & \parallel v_k{(·)-v(·)\parallel }_{L_{\infty }(0,M;\mathbf{H})}+{\parallel {\partial }_t{v}_k(·)-{\partial }_{t}v(·)\parallel }_{{\mathbf{L}}_{\mathbf{q}}\left(0,M;{\mathbf{H}}^{-\mathbf{r}}\right)}\to 0(k\to \infty ).\hfill \end{array}$$

The topology ${\mathsf{\Theta }}_{+}^{\mathrm{loc}}$ is metrizable, for example, using the following Frechét metrics

$${\rho }_{+}(f_1,f_2):=\sum _{M\in \mathbb{N}}{2}^{-M}\frac{{\rho }_{0,M}(f_1,f_2)}{1+{\rho }_{0,M}(f_1,f_2)},$$

(21)

and the corresponding metric space is complete.

Consider this topology on ${\mathcal{K}}_{\epsilon }^{+}$ of Problem (1)–(4).

The translation semigroup $\left\{S\right(h\left)\right\}$ acting on ${\mathcal{K}}_{\epsilon }^{+}$ is continuous in the topology ${\mathsf{\Theta }}_{+}^{loc}$. This directly follows from the definition of the considered local convergence topologies.

Now we define the bounded sets in the trajectory space ${\mathcal{K}}_{\epsilon }^{+}$. For this we aim to use the (uniform) norm in the Banach space

$${\mathcal{F}}_{+}^b={\mathbf{L}}_{\mathbf{p}}^b({\mathbb{R}}_{+};{\mathbf{L}}_{\mathbf{p}})\cap L_2^b({\mathbb{R}}_{+};\mathbf{V})\cap L_{\infty }({\mathbb{R}}_{+};\mathbf{H})\cap \left\{\frac{\partial v}{\partial t}\in {\mathbf{L}}_{\mathbf{q}}^b({\mathbb{R}}_{+};{\mathbf{H}}^{-\mathbf{r}})\right\}.$$

Note that ${\mathcal{F}}_{+}^b$ is a subspace of ${\mathcal{F}}_{+}^{loc}.$ Recall that the (uniform) norm in the space $L_q^b({\mathbb{R}}_{+};\mathbf{X})$, where X is a Banach space, is defined by

$${\parallel w\parallel }_{L_q^b({\mathbb{R}}_{+};\mathbf{X})}:={\left(\underset{h\ge 0}{sup}{\int }_h^{h+1}{\parallel w\left(s\right)\parallel }_{\mathbf{X}}^{q}ds\right)}^{1/q}.$$

Consider now the translation semigroup $\left\{S\right(h\left)\right\}$ on ${\mathcal{K}}_{\epsilon }^{+},$

$$S\left(h\right):{\mathcal{K}}_{\epsilon }^{+}\to {\mathcal{K}}_{\epsilon }^{+},t\ge 0.$$

Definition 1.

The set $\mathfrak{A}\subseteq {\mathcal{K}}^{+}$ is called the trajectory attractor of the translation semigroup $\left\{S\right(h\left)\right\}$ on ${\mathcal{K}}^{+}$ in the topology ${\mathsf{\Theta }}_{+}^{loc}$, if

(i)

$\mathfrak{A}$ is bounded in ${\mathcal{F}}_{+}^b$ and compact in ${\mathsf{\Theta }}_{+}^{loc}$;

(ii)

the set $\mathfrak{A}$ is strictly invariant:

$$S\left(h\right)\mathfrak{A}=\mathfrak{A}\forall h\ge 0;$$

(iii)

$\mathfrak{A}$ is an attracting set for $\left\{S\right(h\left)\right\}$ on ${\mathcal{K}}^{+}$ in the topology ${\mathsf{\Theta }}_{+}^{loc}$, that is, for any $M>0$.

$${\mathrm{dist}}_{{\mathcal{F}}_{0,M}}({\mathsf{\Pi }}_{0,M}S\left(h\right)\mathcal{B},{\mathsf{\Pi }}_{0,M}\mathfrak{A})\to 0(h\to +\infty ).$$

Here,

$${\mathrm{dist}}_{\mathcal{M}}(A,B)=\underset{a\in A}{sup}\underset{b\in B}{inf}{\mathrm{dist}}_{\mathcal{M}}(a,b)$$

denotes the Hausdorff semi-distance from set A to set B of a metric space $\mathcal{M}.$

The Inequalities (15) and (16) imply the following assertion.

Proposition 4.

The trajectory space ${\mathcal{K}}_{\epsilon }^{+}$ belongs to ${\mathcal{F}}_{+}^{\mathrm{b}}$, and for any trajectory $u(·)\in {\mathcal{K}}_{\epsilon }^{+}$ the following inequalities hold

$$\begin{array}{c}{\parallel S\left(h\right)u(·)\parallel }_{{\mathcal{F}}_{+}^{\mathrm{b}}}^2\le C_2{\parallel u\left(0\right)\parallel }^2{e}^{-\sigma h}+R_3^2,\end{array}$$

(22)

$$\begin{array}{c}{\epsilon }^{n/(n-2)}\sum _{j\in {\mathrm{Y}}_{\epsilon }}{\int }_h^{h+1}{\int }_{\partial G_{\epsilon }^j}{B}_{\epsilon }^j\left(x\right)u(x,s)·u(x,t)d\omega ds\le C_3{\parallel u\left(0\right)\parallel }^2{e}^{-\sigma h}+R_4^2\forall h\ge 0,\end{array}$$

(23)

where $\sigma =\beta {\lambda }_1$, and the values $R_3$ and $R_4$ are defined by $R_1$ and $R_2$, respectively. These values are independent of $u\left(0\right)$ and ε.

The proof of the Estimate (22) and Inequality (23) follows directly from Conditions (15) and (16), which can be obtained from Equality(14) by integrating this identity with respect to t, keeping in mind the Gronwall inequality and Inequality (6) (see proof in [20]). We conclude from Inequality (22) that the ball

$${\mathcal{B}}_0=\{u\in {\mathcal{F}}_{+}^{\mathrm{b}}\mid {\parallel u(·)\parallel }_{{\mathcal{F}}_{+}^{\mathrm{b}}}\le 2R_3\}$$

is an absorbing set of the translation semigroup $\left\{S\right(h\left)\right\}$ on ${\mathcal{K}}_{\epsilon }^{+}$, that is, for any set $\mathcal{B}\subset {\mathcal{K}}_{\epsilon }^{+}$, bounded in ${\mathcal{F}}_{+}^{\mathrm{b}}$, there is a number $t_1=t_1\left(\mathcal{B}\right)$, such that $S\left(t\right)\mathcal{B}\subseteq {\mathcal{B}}_0$ for all $t\ge t_1$. Consider the set

$${\mathcal{P}}_{\epsilon }={\mathcal{B}}_0\cap {\mathcal{K}}_{\epsilon }^{+}.$$

The set ${\mathcal{P}}_{\epsilon }\subseteq {\mathcal{K}}_{\epsilon }^{+}$ is also absorbing, that is,

$$S\left(h\right){\mathcal{P}}_{\epsilon }\subseteq {\mathcal{P}}_{\epsilon }\forall h\ge 0,$$

(24)

and ${\mathcal{P}}_{\epsilon }$ is uniformly bounded in space ${\mathcal{F}}_{+}^{\mathrm{b}}$, with respect to $\epsilon$.

To describe the structure of the trajectory attractor we use the notion of complete trajectories (weak solutions) $u\left(t\right),t\in \mathbb{R},$ of the Problem (1)–(4), e.g., weak solutions to this system that are defined on the entire t-axis.

Let ${\mathcal{K}}_{\epsilon }$ denote the kernel of Problem (1)–(4) that consists of all weak solutions $u\left(t\right),t\in \mathbb{R}$ (complete trajectories), bounded in space

$${\mathcal{F}}^{\mathrm{b}}={\mathbf{L}}_{\mathbf{p}}^{\mathrm{b}}(\mathbb{R};{\mathbf{L}}_{\mathbf{p}})\cap {\mathbf{L}}_2^{\mathrm{b}}(\mathbb{R};\mathbf{V})\cap {\mathbf{L}}_{\infty }(\mathbb{R};\mathbf{H})\cap \left\{v|\frac{\partial v}{\partial t}\in {\mathbf{L}}_{\mathbf{q}}^{\mathrm{b}}(\mathbb{R};{\mathbf{H}}^{-\mathbf{r}})\right\}.$$

We define the norm in space ${\mathcal{F}}^{\mathrm{b}}$ in a similar way as in space ${\mathcal{F}}_{+}^{\mathrm{b}}$.

In space ${\mathcal{F}}_{+}^{loc},$ we also consider the local “weak” topology ${\mathsf{\Theta }}_{+}^{\mathrm{loc},\mathrm{w}}$ generated by the metrics

$${\rho }_{0,M}^w(u,v)={\parallel u\left(t\right)-v\left(t\right)\parallel }_{L_2(0,M;\mathbf{H})}$$

by the formula similar to Equation (21).

The following assertion is valid.

Proposition 5.

The set ${\mathcal{P}}_{\epsilon }$ is compact in the “weak” topology ${\mathsf{\Theta }}_{+}^{\mathrm{loc},\mathrm{w}}$ and is uniformly bounded in the norm of ${\mathcal{F}}_{+}^{\mathrm{b}}$.

Proof.

To prove that a ball from the space ${\mathcal{F}}_{+}^{\mathrm{b}}$ is compact in ${\mathsf{\Theta }}_{+}^{\mathrm{loc}}$, we use the following Lemma 1. Let $E_0$ and $E_1$ be Banach spaces such that $E_1\subset E_0$. Consider the Banach spaces

$$\begin{array}{cc}\hfill W_{p_1,p_0}(0,M;E_1,E_0)& =\{\psi \left(s\right),s\in 0,M\mid \psi (·)\in L_{p_1}(0,M;E_1),\hfill \\ \hfill & {\psi }^{\prime }(·)\in L_{p_0}(0,M;E_0)\},\hfill \\ \hfill W_{\infty ,p_0}(0,M;E_1,E_0)& =\{\psi \left(s\right),s\in 0,M\mid \psi (·)\in L_{\infty }(0,M;E_1),\hfill \\ \hfill & {\psi }^{\prime }(·)\in L_{p_0}(0,M;E_0)\}\hfill \end{array}$$

where $p_1\ge 1$ and $p_0>1$, with norms

$$\begin{array}{cc}\hfill {\parallel \psi \parallel }_{W_{p_1,p_0}}& :={\left({\int }_0^M{\parallel \psi \left(s\right)\parallel }_{E_1}^{p_1}ds\right)}^{1/p_1}+{\left({\int }_0^M{\parallel {\psi }^{\prime }\left(s\right)\parallel }_{E_0}^{p_0}ds\right)}^{1/p_0},\hfill \\ \hfill {\parallel \psi \parallel }_{W_{\infty ,p_0}}& :={ess\; sup\{\parallel \psi \left(s\right)\parallel }_{E_1}\mid s\in [0,M]\}+{\left({\int }_0^M{\parallel {\psi }^{\prime }\left(s\right)\parallel }_{E_0}^{p_0}ds\right)}^{1/p_0}.\hfill \end{array}$$

Lemma 1

(Oben–Lions–Simon, [41]). Assume that $E_1⋐E\subset E_0$. Then the following embeddings are compact:

$$\begin{array}{cc}\hfill W_{p_1,p_0}(0,T;E_1,E_0)& ⋐L_{p_1}(0,T;E),\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill W_{\infty ,p_0}(0,T;E_1,E_0)& ⋐C\left(\right[0,T];E).\hfill \end{array}$$

(26)

Then, using the Inequality (23), we complete the proof of the second statement of this proposition. □

Similar to Definition 1 we define a trajectory attractor for the translation semigroup $\left\{S\right(h\left)\right\}$ on ${\mathcal{K}}^{+}$ in the weak topology ${\mathsf{\Theta }}_{+}^{loc,w}$ replacing the topology ${\mathsf{\Theta }}_{+}^{loc}$ with ${\mathsf{\Theta }}_{+}^{loc,w}$.

Proposition 6.

The Problem (1)–(4) has the trajectory attractors ${\mathfrak{A}}_{\epsilon }$ in the “weak” topology ${\mathsf{\Theta }}_{+}^{loc,w}$. The set ${\mathfrak{A}}_{\epsilon }$ is uniformly (with respect to $\epsilon \in (0,1)$) bounded in ${\mathcal{F}}_{+}^b$ and compact in ${\mathsf{\Theta }}_{+}^{loc,w}$. Moreover,

$${\mathfrak{A}}_{\epsilon }={\mathsf{\Pi }}_{+}{\mathcal{K}}_{\epsilon },$$

the kernel ${\mathcal{K}}_{\epsilon }$ is non-empty and uniformly (with respect to $\epsilon \in (0,1)$) bounded in ${\mathcal{F}}^b$. Recall that the spaces ${\mathcal{F}}_{+}^{\mathrm{b}}$ and ${\mathsf{\Theta }}_{+}^{\mathrm{loc},\mathrm{w}}$ depend on ε.

The proof can be found in [13].

We are now going to verify that the trajectory attractor ${\mathfrak{A}}_{\epsilon }$ in the “weak” topology ${\mathsf{\Theta }}_{+}^{\mathrm{loc},\mathrm{w}}={\mathbf{L}}_2^{\mathrm{loc}}({\mathbb{R}}_{+};\mathbf{H})$, constructed in Proposition 6, is also a trajectory attractor in the strong topology ${\mathsf{\Theta }}_{+}^{\mathrm{loc},\mathrm{w}}$ generated by the spaces ${\mathcal{F}}_{0,M}$ (Equations (17) and (18)). For this aim we use the method of energy identities from [42,43].

Theorem 1.

The trajectory attractor ${\mathfrak{A}}_{\epsilon }$ is compact in the strong topology ${\mathsf{\Theta }}_{+}^{\mathrm{loc}}$ and attracts bounded sets of trajectories from ${\mathcal{K}}_{\epsilon }^{+}$ in this topology, that is, ${\mathfrak{A}}_{\epsilon }$ is a trajectory attractor in the topology ${\mathsf{\Theta }}_{+}^{\mathrm{loc}}$.

Proof.

We fix $\epsilon >0$. Since the set ${\mathcal{P}}_{\epsilon }$ is absorbing, it is sufficient to prove that the set $S\left(1\right){\mathcal{P}}_{\epsilon }$ is compact in the strong topology of the space ${\mathbf{L}}_{\mathbf{p}}(0,M;{\mathbf{L}}_{\mathbf{p}}\left(\mathsf{\Omega }\right))\cap {\mathbf{L}}_2(0,M;\mathbf{V})$ for every $M>0$. Thus, it is easy to see that ${\mathbf{L}}_{\mathbf{p}}(0,M;{\mathbf{L}}_{\mathbf{p}}\left(\mathsf{\Omega }\right))={\mathbf{L}}_{\mathbf{p}}(\mathsf{\Omega }\times [0,M])$.

We show that any sequence $\left\{u_k\left(t\right)\right\}\subset {\mathcal{P}}_{\epsilon }$ is strongly pre-compact in the space ${\mathbf{L}}_{\mathbf{p}}(\mathsf{\Omega }\times [1,M])\cap {\mathbf{L}}_2(1,M;\mathbf{V})$ for every $M>0$.

The set ${\mathcal{P}}_{\epsilon }$ is bounded in the space ${\mathcal{F}}_{+}^{\mathrm{b}}$. Hence, $\left\{u_k\left(t\right)\right\}$ is bounded in the spaces ${\mathbf{L}}_{\mathbf{p}}(\mathsf{\Omega }\times [0,M])$ and ${\mathbf{L}}_2(0,M;\mathbf{V})$. Taking a subsequence labelled again $\left\{u_k\left(t\right)\right\}$, we can assume that $u_k(·)\rightharpoondown \widehat{u}(·)$ as $k\to \infty$ weakly in spaces ${\mathbf{L}}_{\mathbf{p}}(\mathsf{\Omega }\times [0,M])$ and ${\mathbf{L}}_2(0,M;\mathbf{V})$, where $\widehat{u}\left(t\right)$ is a solution to Problem (1)–(4) that belongs to ${\mathcal{P}}_{\epsilon }$. The Lions–Magenes lemma (e.g., [44,45]) implies that $u_k\left(t\right)\rightharpoondown \widehat{u}\left(t\right)$ strongly in the space ${\mathbf{L}}_2(\mathsf{\Omega }\times [0,M])$ and (passing again to a subsequence) $u_k(x,t)\to \widehat{u}(x,t)$ for almost all $(x,t)\in \mathsf{\Omega }\times [0,M]$.

Note that from the Inequality (23) it follows that the sequence $\left\{u_k\left(t\right)\right\}$ is bounded in the space ${\mathbf{L}}_2(\partial G_{\epsilon }^j\times [0,M])$ for every $j\in {\mathrm{Y}}_{\epsilon }$. Therefore, passing to a subsequence, we can suppose that $u_k(·)\rightharpoondown \widehat{u}(·)$ as $k\to \infty$ weakly in the space ${\mathbf{L}}_2(\partial G_{\epsilon }^j\times [0,M])$ for every $j\in {\mathrm{Y}}_{\epsilon }$.

We recall the following assertion from the functional analysis: if a sequence ${\chi }_k\rightharpoondown \widehat{\chi }$ weakly in a Banach space X, then

$$\parallel \widehat{\chi }{\parallel }_X\le \underset{k\to \infty }{lim\; inf}{\parallel {\chi }_k\parallel }_X$$

(see, for example, [46]). Therefore, for a weakly convergent subsequence of trajectories $\left\{u_k(·)\right\}$ (labelled the same) we obtain the following limit relations:

$$\begin{array}{cc}\hfill \parallel \widehat{u}\left(M\right)\parallel & \le \underset{k\to \infty }{lim\; inf}\parallel u_k\left(M\right)\parallel ,\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill {\int }_0^M{\int }_{\mathsf{\Omega }}t\lambda \nabla \widehat{u}·\nabla \widehat{u}dxdt& \le \underset{k\to \infty }{lim\; inf}{\int }_0^M{\int }_{\mathsf{\Omega }}t\lambda \nabla u_k·\nabla u_{k}dxdt,\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill {\int }_0^M{\int }_{\mathsf{\Omega }}t{a}_{\epsilon }\left(x\right){\left|{\widehat{u}}^i\right|}^{p_i}dxdt& \le \underset{k\to \infty }{lim\; inf}{\int }_0^M{\int }_{\mathsf{\Omega }}t{a}_{\epsilon }\left(x\right){\left|u_k^i\right|}^{p_i}dxdt,i=1,\cdots ,N,\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill {\int }_0^M{\int }_{\partial G_{\epsilon }^j}t{B}_{\epsilon }^j\left(x\right)\widehat{u}·\widehat{u}d\omega dt& \le \underset{k\to \infty }{lim\; inf}{\int }_0^M{\int }_{\partial G_{\epsilon }^j}t{B}_{\epsilon }^j\left(x\right)u_k·u_{k}d\omega dt,j\in {\mathrm{Y}}_{\epsilon },\hfill \end{array}$$

(30)

where we denote $u_k=u_k(x,t)$ and $\widehat{u}=\widehat{u}(x,t)$. The norms in Relations (28)–(30) correspond to weighted spaces ${\mathbf{L}}_{2,t}(0,M;\mathbf{V})$, ${\mathbf{L}}_{\mathbf{p},t{a}_{\epsilon }\left(x\right)}(\mathsf{\Omega }\times [0,M])$ and ${\mathbf{L}}_{2,t{B}_{\epsilon }^j\left(x\right)}(\partial G_{\epsilon }^j\times [0,M])$ with weights t, $t{a}_{\epsilon }\left(x\right)$, and $t{B}_{\epsilon }^j\left(x\right)$, respectively. Furthermore, the quadratic form $\lambda y·y$ 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. Hence, ${\int }_{\mathsf{\Omega }}\lambda \nabla v\left(x\right)·\nabla v\left(x\right)dx$ is equivalent to the norm of a function $v(·)$ in the space $\mathbf{V}$.

We note that weak convergence $u_k(·)\rightharpoondown \widehat{u}(·)$ also holds in the weighted spaces ${\mathbf{L}}_{2,t}(0,M;\mathbf{V})$, ${\mathbf{L}}_{\mathbf{p},t{a}_{\epsilon }\left(x\right)}(\mathsf{\Omega }\times [0,M])$ and ${\mathbf{L}}_{2,t{B}_{\epsilon }^j\left(x\right)}(\partial G_{\epsilon }^j\times [0,M])$.

Let us consider the continuous scalar function

$$F\left(v\right)=\sum _{i=1}^N{f}^i\left(v\right)v^i-\sum _{i=1}^N{\gamma }_i{\left|v^i\right|}^{p_i},v\in {\mathbb{R}}^N.$$

(31)

Then $s{a}_{\epsilon }\left(x\right)F\left(u_k(x,s)\right)\to s{a}_{\epsilon }\left(x\right)F\left(\widehat{u}(x,s)\right)$ as $k\to \infty$ for almost all $(x,t)\in \mathsf{\Omega }\times [0,M]$, because of the continuity of the function $F\left(v\right)$. We have

$${\int }_0^M{\int }_{\mathsf{\Omega }}t{a}_{\epsilon }\left(x\right)F\left(\widehat{u}(x,t)\right)dxdt\le \underset{k\to \infty }{lim\; inf}{\int }_0^M{\int }_{\mathsf{\Omega }}t{a}_{\epsilon }\left(x\right)F\left(u_k(x,t)\right)dxdt.$$

(32)

Here, we used the estimates $F\left(v\right)+C\ge 0$, $a_{\epsilon }(·)\ge 0$ (cf. (Conditions (6) and (12)), the properties of $\left\{u_k(x,t)\right\}$ (the convergence of it for almost all $(x,t)\in \mathsf{\Omega }\times [0,M]$), and the Fatou lemma on the estimate above of the integral over the limit function by the $liminf$ of the integrals over a sequence of non-negative convergent functions (e.g., [46]).

Now let us recall that $u_k(·)$ and $\widehat{u}(·)$ satisfy the differential identity Equality (14). Then, we multiply this identity by t and integrate it over $[0,M]$. By means of the definition of the function $F(·)$ we obtain

$$\begin{array}{cc}\hfill & \frac{1}{2}\parallel u_k{\left(M\right)\parallel }^2+{\int }_0^M{\int }_{\mathsf{\Omega }}t\lambda \nabla u_k·\nabla u_{k}dxdt+\sum _{i=1}^N{\gamma }_i{\int }_0^M{\int }_{\mathsf{\Omega }}t{a}_{\epsilon }\left(x\right){\left|u_k^i\right|}^{p_i}dxdt\hfill \\ \hfill & +{\int }_0^M{\int }_{\mathsf{\Omega }}t{a}_{\epsilon }\left(x\right)F\left(u_k\right)dxdt+{\epsilon }^{n/(n-2)}\sum _{j\in {\mathrm{Y}}_{\epsilon }}{\int }_0^M{\int }_{\partial G_{\epsilon }^j}t{B}_{\epsilon }^j\left(x\right)u_k·u_{k}d\omega dt=\hfill \\ \hfill & \frac{1}{2}{\int }_0^M{\int }_{\mathsf{\Omega }}{\left|u_k\right|}^{2}dxdt+{\int }_0^M{\int }_{\mathsf{\Omega }}{g}_{\epsilon }\left(x\right)·u_{k}dxdt,\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill & \frac{1}{2}\parallel \widehat{u}{\left(M\right)\parallel }^2+{\int }_0^M{\int }_{\mathsf{\Omega }}t\lambda \nabla \widehat{u}·\nabla \widehat{u}dxdt+\sum _{i=1}^N{\gamma }_i{\int }_0^M{\int }_{\mathsf{\Omega }}t{a}_{\epsilon }\left(x\right){\left|{\widehat{u}}^i\right|}^{p_i}dxdt+\hfill \\ \hfill & {\int }_0^M{\int }_{\mathsf{\Omega }}t{a}_{\epsilon }\left(x\right)F\left(\widehat{u}\right)dxdt+{\epsilon }^{n/(n-2)}\sum _{j\in {\mathrm{Y}}_{\epsilon }}{\int }_0^M{\int }_{\partial G_{\epsilon }^j}t{B}_{\epsilon }^j\left(x\right)\widehat{u}·\widehat{u}d\omega dt=\hfill \\ \hfill & \frac{1}{2}{\int }_0^M{\int }_{\mathsf{\Omega }}{\left|\widehat{u}\right|}^{2}dxdt+{\int }_0^M{\int }_{\mathsf{\Omega }}{g}_{\epsilon }\left(x\right)·\widehat{u}dxdt.\hfill \end{array}$$

(34)

Recall that $u_k(·)\to \widehat{u}(·)$ strongly in the space ${\mathbf{L}}_2(\mathsf{\Omega }\times [0,M])$. Consequently, the right-hand side of the Equality (33) tends to that of Equality (34). Therefore, the left-hand side of Equality (33) also converges to the left-hand side of Equality (34).

Then, by means of Equations (27)–(32), we conclude that each of the five real sequences in the sum in the left-hand side of the Equality (33) has a limit as $k\to \infty$, which coincides with the corresponding quantity in the left-hand side of Equality (34). In particular, we obtain the next relations

$$\begin{array}{cc}\hfill \underset{k\to \infty }{lim}{\int }_0^M{\int }_{\mathsf{\Omega }}t\lambda \nabla u_k·\nabla u_{k}dxdt& ={\int }_0^M{\int }_{\mathsf{\Omega }}t\lambda \nabla \widehat{u}·\nabla \widehat{u}dxdt,\hfill \\ \hfill \underset{k\to \infty }{lim}{\int }_0^M{\int }_{\mathsf{\Omega }}t{a}_{\epsilon }\left(x\right){\left|u_k^i\right|}^{p_i}dxdt& ={\int }_0^M{\int }_{\mathsf{\Omega }}t{a}_{\epsilon }\left(x\right){\left|{\widehat{u}}^i\right|}^{p_i}dxdt,i=1,\cdots ,N.\hfill \end{array}$$

Using the Mazur theorem (see [46]), we conclude that from the weak convergence ${\chi }_k\rightharpoondown \widehat{\chi }$ of elements from a uniformly convex Banach space X and the convergence of their norms $\parallel {\chi }_k{\parallel }_X\to {\parallel \widehat{\chi }\parallel }_X$ we obtain the strong convergence $\parallel {\chi }_k-\widehat{\chi }{\parallel }_X\to 0$ as $k\to \infty$. The weighted spaces ${\mathbf{L}}_{2,t}(0,M;\mathbf{V})$ and ${\mathbf{L}}_{\mathbf{p},t{a}_{\epsilon }\left(x\right)}(\mathsf{\Omega }\times [0,M])$ are uniformly convex. Hence, the weak convergence of $u_k^i(·)$ to $\widehat{u}(·)$ and the convergence of their norms in the space ${\mathbf{L}}_{2,t}(0,M;\mathbf{V})\cap {\mathbf{L}}_{\mathbf{p},t{a}_{\epsilon }\left(x\right)}(\mathsf{\Omega }\times [0,M])$ implies strong convergence $u_k(·)\to \widehat{u}(·)$ in the space ${\mathbf{L}}_{2,t}(1,M;\mathbf{V})\cap {\mathbf{L}}_{\mathbf{p},t{a}_{\epsilon }\left(x\right)}(\mathsf{\Omega }\times [1,M])$, which is, obviously, equivalent to the strong convergence in ${\mathbf{L}}_2(1,M;\mathbf{V})\cap {\mathbf{L}}_{\mathbf{p}}(\mathsf{\Omega }\times [1,M])$ (without weights).

Clearly, we showed the compactness of the set $S\left(1\right){\mathcal{P}}_{\epsilon }$ in the strong topology of the space ${\mathbf{L}}_{\mathbf{p}}^{\mathrm{loc}}({\mathbb{R}}_{+};{\mathbf{L}}_{\mathbf{p}}\left(\mathsf{\Omega }\right))\cap {\mathbf{L}}_2^{\mathrm{loc}}({\mathbb{R}}_{+};\mathbf{V})$.

Using the continuity of the Nemytskii operator $u\mapsto f\left(u\right)$ for Equation (1), which, bearing in mind Equation (5), acts from ${\mathbf{L}}_{\mathbf{p}}(\mathsf{\Omega }\times [0,M])$ to ${\mathbf{L}}_{\mathbf{q}}(\mathsf{\Omega }\times [0,M])$ (see [47]) and, consequently, strong convergence of $a_{\epsilon }(·)f\left(u_k(·)\right)\to a_{\epsilon }(·)f\left(\widehat{u}(·)\right)$ in ${\mathbf{L}}_{\mathbf{q}}^{\mathrm{loc}}({\mathbb{R}}_{+};{\mathbf{L}}_{\mathbf{q}}\left(\mathsf{\Omega }\right))$ we obtain the compactness of the set of derivatives ${\partial }_{t}u(·)$ in the strong topology of the space ${\mathcal{L}}^{\mathrm{loc}}\left({\mathbb{R}}_{+}\right):={\mathbf{L}}_2^{\mathrm{loc}}({\mathbb{R}}_{+};{\mathbf{V}}^{\prime })+{\mathbf{L}}_{\mathbf{q}}^{\mathrm{loc}}({\mathbb{R}}_{+};{\mathbf{L}}_{\mathbf{q}}\left(\mathsf{\Omega }\right))$. On the other hand, $\lambda \nabla u_k(·)\to$$\lambda \nabla \widehat{u}(·)$ strongly in ${\mathbf{L}}_2^{\mathrm{loc}}({\mathbb{R}}_{+};{\mathbf{V}}^{\prime })$. Hence, ${\partial }_t{u}_k(·)\to {\partial }_t\widehat{u}(·)$ strongly in ${\mathcal{L}}^{\mathrm{loc}}\left({\mathbb{R}}_{+}\right)$.

It remains to say that keeping in mind the differential identity Equation (14), the set ${\mathcal{P}}_{\epsilon }$ belongs to the space $C^{\mathrm{loc}}({\mathbb{R}}_{+};\mathbf{H})$ and the set $S\left(1\right){\mathcal{P}}_{\epsilon }$ is compact in $C^{\mathrm{loc}}({\mathbb{R}}_{+};\mathbf{H})$, due to the continuity of the embedding (see [20])

$${\mathbf{L}}_2(0,M;\mathbf{V})\cap {\mathbf{L}}_{\mathbf{p}}(0,M;{\mathbf{L}}_{\mathbf{p}}\left(\mathsf{\Omega }\right))\cap \{{\partial }_{t}v\in \mathcal{L}(0,M)\}\subset C([0,M];\mathbf{H}).$$

This completes the proof of Theorem 1. □

4. Strong Homogenization of Trajectory Attractors for Reaction–Diffusion Systems in Domains with Obstacles

In this section, we study the limit behaviour of trajectory attractors ${\mathfrak{A}}_{\epsilon }$ for the reaction–diffusion system (1)–(4) as $\epsilon \to 0+$ and their convergence to the trajectory attractor of the corresponding homogenized system.

The limit system contains some additional “strange terms” (potential). To define this term, we consider the following problem with separated unknown functions:

$$\begin{array}{cc}\hfill & {\mathrm{\Delta }}_y{v}^k=0,y\in {\mathbb{R}}^n\backslash G_0,\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill & \frac{\partial v^k}{\partial {\nu }_y}+b^{kk}(x,y)v=b^{kk}(x,y),y\in \partial G_0,\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill & v^k\to 0,\left|y^k\right|\to \infty ,k=1,2,\dots ,N.\hfill \end{array}$$

(37)

where $v^k=v^k(x,y)$ and the variable x plays the role of a slow parameter. The limit potential $V\left(x\right)$ is a diagonal $N\times N$-matrix and we define the diagonal elements $V^{kk}\left(x\right)$ by the following formula

$$V^{kk}\left(x\right)={\int }_{\partial G_0}\frac{\partial }{\partial {\nu }_y}{v}^k(x,y)d{\omega }_y,k=1,\cdots ,N.$$

(38)

The homogenized (limit) problem for the considered reaction–diffusion system has the form

$$\begin{array}{cc}\hfill & \frac{\partial u}{\partial t}=\lambda \mathrm{\Delta }u-\overline{a}\left(x\right)f\left(u\right)-V\left(x\right)u+\overline{g}\left(x\right),x\in \mathsf{\Omega },t\ge 0,\hfill \end{array}$$

(39)

$$\begin{array}{cc}\hfill & u(x,t)=0,x\in \partial \mathsf{\Omega },t\ge 0,\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill & u(x,0)=U\left(x\right),x\in \mathsf{\Omega },\hfill \end{array}$$

(41)

where $V\left(x\right)$ is a diagonal matrix with elements defined in Equation (38). The mean functions $\overline{a}\left(x\right)$ and $\overline{g}\left(x\right)$ were defined in Equations (8) and (9).

As usual, we study weak solutions $u(x,t)$ to the Problem (39)–(41) that we define using the integral identity similar to Equation (13). We have also

Proposition 7.

For any initial function $U\in \mathbf{H}$ the Problem (39)–(41) has a weak solution $u(x,t)$ such that $u(x,0)=U\left(x\right)$ and $u(x,t)\in C_b({\mathbb{R}}_{+};{\mathbf{H}}^{-\mathbf{r}})\cap C_w({\mathbb{R}}_{+};\mathbf{H}).$

The following assertions are similar to Propositions 2 and 3.

Proposition 8.

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 the Problem (39)–(41). Then

(i)

$u\in \mathbf{C}({\mathbb{R}}_{+};\mathbf{H})$;

(ii)

the function ${\parallel u(·,t)\parallel }^2$ is absolutely continuous on ${\mathbb{R}}_{+}$ and, moreover,

$$\begin{array}{cc}\hfill & \frac{1}{2}\frac{d}{dt}{\parallel u(·,t)\parallel }^2+{\int }_{\mathsf{\Omega }}\lambda \nabla u(x,t)·\nabla u(x,t)dx+{\int }_{\mathsf{\Omega }}\overline{a}\left(x\right)f\left(u(x,t)\right)·u(x,t)dx+\hfill \\ \hfill & {\int }_{\mathsf{\Omega }}V\left(x\right)u(x,t)·u(x,t)dx={\int }_{\mathsf{\Omega }}\overline{g}\left(x\right)·u(x,t)dx.\hfill \end{array}$$

Proposition 9.

Any weak solution $u\left(t\right)$ of the Problem (39)–(41) satisfies the following inequalities:

$$\begin{array}{c}{\parallel u\left(t\right)\parallel }^2\le {\parallel u\left(0\right)\parallel }^2{e}^{-{\lambda }_1{\beta }_{0}t}+R_3^2,\\ {\beta }_0{\int }_t^{t+1}{\parallel u\left(s\right)\parallel }_1^{2}ds+2a_0\sum _{i=1}^N{\gamma }_i{\int }_t^{t+1}\parallel u^i{\left(s\right)\parallel }_{L_{p_i}\left({\mathsf{\Omega }}_{\epsilon }\right)}^{p_i}ds\le {\parallel u\left(t\right)\parallel }^2+R_4^2\end{array}$$

where ${\lambda }_1$ is the first eigenvalue of the Laplace operator $-\mathrm{\Delta }$ in Ω with Dirichlet boundary conditions. Positive values $R_3$ and $R_4$ depend on the number $M_0$ (Equation (10)) and are independent of $u_{\epsilon }\left(0\right)$ and ε.

The proof is given in [20].

Analogues of Propositions 5 and 6 and Theorem 1 hold for the limit system in the corresponding spaces ${\mathcal{F}}_{+}^{\mathrm{loc}}$ and ${\mathsf{\Theta }}_{+}^{s,\mathrm{loc}}$ in the domain $\mathsf{\Omega }$ without perforation. Therefore, the Problem (39)–(41) has a trajectory attractor $\overline{\mathfrak{A}}$ in the trajectory space ${\overline{\mathcal{K}}}^{+}$ corresponding to this problem and, moreover,

$$\overline{\mathfrak{A}}={\mathsf{\Pi }}_{+}\overline{\mathcal{K}},$$

where $\overline{\mathcal{K}}$ is the kernel of Problem (39)–(41) in the space ${\mathcal{F}}^{\mathrm{b}}$ (in the domain $\mathsf{\Omega }$ without perforation).

To begin with we formulate a homogenization theorem for the reaction–diffusion system with rapidly oscillating terms in a domain with obstacles, in the weak topology.

Theorem 2.

Let the functions $a_{\epsilon }\left(x\right)$ and $g_{\epsilon }\left(x\right)$ have means, that is, Properties (8) and (9) are valid. Then the next relation holds in the “weak” topological space ${\mathsf{\Theta }}_{+}^{\mathrm{loc},\mathrm{w}}$

$${\mathfrak{A}}_{\epsilon }\to \overline{\mathfrak{A}}as\epsilon \to 0+.$$

(42)

Moreover,

$${\mathcal{K}}_{\epsilon }\to \overline{\mathcal{K}}as\epsilon \to 0+in{\mathsf{\Theta }}^{\mathrm{loc},\mathrm{w}}.$$

(43)

Remark 2.

For all the functions from the sets ${\mathfrak{A}}_{\epsilon }$ and ${\mathcal{K}}_{\epsilon }$, defined on perforated domains ${\mathsf{\Omega }}_{\epsilon }$, we can extend inside the cavities keeping the norms of the extended functions in $\mathbf{H}$, $\mathbf{V}$ and ${\mathbf{L}}_{\mathbf{p}}$ to coincide with the corresponding norms in the perforated spaces ${\mathbf{H}}_{\epsilon }$, ${\mathbf{V}}_{\epsilon }$ and ${\mathbf{L}}_{\mathbf{p},\epsilon }$. Hence, in Theorem 2 all distances are measured in the spaces without perforation taking into account the extension inside the cavities.

The proof of Theorem 2 is based on the following proposition and can be found in [13].

Proposition 10.

For any function $\phi \in {\mathbf{H}}_{\epsilon }$ and for all t, the following inequality holds:

$$\left|{\epsilon }^{n/(n-2)}\sum _{j\in {\mathrm{Y}}_{\epsilon }}{\int }_{\partial G_{\epsilon }^j}{B}_{\epsilon }^j\left(x\right)\phi d\omega -{\int }_{\mathsf{\Omega }}V\left(x\right)\overline{\phi }dx\right|\le M\epsilon {\parallel \phi \parallel }_{{\mathbf{H}}_{\epsilon }},$$

(44)

and for any function $\psi \in {\mathcal{F}}^{\mathrm{b}}$ as $\epsilon \to 0$ the following limit relation is valid:

$${\epsilon }^{n/(n-2)}\sum _{j\in {\mathrm{Y}}_{\epsilon }}{\int }_{\partial G_{\epsilon }^j}{B}_{\epsilon }^j\left(x\right)u_{\epsilon }\psi d\omega \to {\int }_{\mathsf{\Omega }}V\left(x\right)\overline{u}\psi dx,$$

(45)

here $V\left(x\right)$ is defined by the Formula (38) and the constant M is independent of ε.

Proof.

The Inequality (44) is proven by using the same scheme as in [48] (Lemma 2, Inequality (21)). We multiply the equation in Problem (35) by the function $\phi \in {\mathbf{H}}_{\epsilon }$ and part integrate it in the domain ${\mathsf{\Omega }}_{\epsilon }$. We subtract from and add to the obtained equality the term ${\int }_{\mathsf{\Omega }}V\left(x\right)\overline{\phi }dx$. Then, we place the difference

$${\epsilon }^{n/(n-2)}\sum _{j\in {\mathrm{Y}}_{\epsilon }}{\int }_{\partial G_{\epsilon }^j}{B}_{\epsilon }^j\left(x\right)\phi d\omega -{\int }_{\mathsf{\Omega }}V\left(x\right)\overline{\phi }dx$$

to the left-hand side of the equality and estimate it by the modulo and obtain the Estimate (44).

To prove the Convergence (45), first of all, substituting $u_{\epsilon }$ as a test function to Equation (13), we establish that the following quantities are uniformly bounded:

$$\parallel \nabla u_{\epsilon }{\parallel }_{{\mathbf{H}}_{\epsilon }}\le K,\sum _{j\in {\mathrm{Y}}_{\epsilon }}{\int }_{\partial G_{\epsilon }^j}{B}_{\epsilon }^j{u}_{\epsilon }\psi d\omega \le K{\epsilon }^{-n/(n-2)},$$

where the constant K is independent of $\epsilon$.

Consider the family of the extension operators

$$P_{\epsilon }:{\mathbf{V}}_{\epsilon }\to \mathbf{V}$$

such that $P_{\epsilon }vs.=v$ almost everywhere in ${\mathsf{\Omega }}_{\epsilon }$ and

$$\parallel \nabla P_{\epsilon }{v\parallel }_{\mathbf{H}}\le {\parallel \nabla v\parallel }_{{\mathbf{H}}_{\epsilon }}\mathrm{for}\mathrm{any}\mathrm{function}v\in {\mathbf{V}}_{\epsilon }$$

(the detailed construction of such operators can be found in [8]).

Having in mind the previous inequality and the boundedness $\parallel u_{{\epsilon }_k}{\parallel }_{{\mathcal{F}}^{\mathrm{b}}}\le C$, we conclude that the sequence ${\tilde{u}}_{\epsilon }=P_{\epsilon }{u}_{\epsilon }$ is bounded in $\mathbf{V}$. Therefore, it converges weakly in $\mathbf{V}$. Then there is a function $u\in \mathbf{V}$ such that

$${\tilde{u}}_{\epsilon }\rightharpoonup u\mathrm{in}\mathbf{V}\mathrm{as}\epsilon \to 0.$$

In what follows, we write $u_{\epsilon }$ instead ${\tilde{u}}_{\epsilon }$.

We denote $T_r^j=\{x\in {\mathbb{R}}^n:|x-P_{\epsilon }^j|\le r\}$. Consider the following temporary function $v_{\epsilon }^j$ that satisfies the problem

$$\begin{array}{cccc}\hfill & \mathrm{\Delta }{v}_{\epsilon }^j=0,\hfill & \hfill x& \in T_{\epsilon /4}^j\backslash \overline{G_{\epsilon }^j},\hfill \\ \hfill & \frac{\partial v_{\epsilon }^j}{\partial \nu }+{\epsilon }^{n/(n-2)}{B}_{\epsilon }^j\left(x\right)v_{\epsilon }^j={\epsilon }^{n/(n-2)}{\overline{B}}_{\epsilon }^j\left(x\right),\hfill & \hfill x& \in \partial G_{\epsilon }^j,\hfill \\ \hfill & v_{\epsilon }^j=0,\hfill & \hfill x& \in \partial T_{\epsilon /4}^j.\hfill \end{array}$$

(46)

It is easy to show that

$${\epsilon }^{n/(n-2)}\sum _{j\in {\mathrm{Y}}_{\epsilon }}{\int }_{\partial G_{\epsilon }^j}{B}_{\epsilon }^j\left(x\right)u_{\epsilon }\varphi d\omega =-\sum _{j\in {\mathrm{Y}}_{\epsilon }}{\int }_{\partial T_{\epsilon /4}^j}\frac{\partial v_{\epsilon }^j}{\partial \nu }{u}_{\epsilon }\varphi d\omega .$$

Thus, we have established that

$${\mathcal{V}}_{\epsilon }\left(x\right)=\left\{\begin{array}{cc}{v}_{\epsilon }^j\left(x\right),\hfill & x\in T_{\epsilon /4}^j\backslash \overline{G_{\epsilon }^j},j\in {\mathrm{Y}}_{\epsilon },\hfill \\ 0,\hfill & x\in {\mathbb{R}}^n\backslash \overline{T_{\epsilon /4}^j}.\hfill \end{array}\right.$$

(47)

It is proven in [15] that

$$\parallel {\mathcal{V}}_{\epsilon }{\parallel }_{{\mathbf{V}}_{\epsilon }}^2\le K{\epsilon }^2$$

and

$${\tilde{\mathcal{V}}}_{\epsilon }\rightharpoonup 0\mathrm{weakly}\mathrm{in}\mathbf{V},{\tilde{\mathcal{V}}}_{\epsilon }\to 0\mathrm{strongly}\mathrm{in}\mathbf{H}\mathrm{as}\epsilon \to 0,$$

where ${\tilde{\mathcal{V}}}_{\epsilon }=P_{\epsilon }{\mathcal{V}}_{\epsilon }$.

Using [15] (Lemmas 4.1, 4.2) we obtain that

$$\left|\sum _{j\in {\mathrm{Y}}_{\epsilon }}{\int }_{\partial T_{\epsilon /4}^j}\frac{\partial v_{\epsilon }^j}{\partial \nu }{h}_{\epsilon }d\omega -{\int }_{\mathsf{\Omega }}V\left(x\right)hdx\right|\to 0$$

(48)

as $\epsilon \to 0$ for functions $h_{\epsilon }$, $h\in \mathbf{V}$, such that $h_{\epsilon }\rightharpoonup h$ in $\mathbf{V}$.

Finally, Equation (48) implies the Convergence (45). Thus, Proposition 10 is proved. □

We now formulate the main theorem of the paper on convergence of trajectory attractors of the Problem (1)–(4) in the strong topology ${\mathsf{\Theta }}_{+}^{\mathrm{loc}}$, in which the trajectory attractors have been constructed for a fixed $\epsilon$ (Theorem 1).

Theorem 3.

Assume that the functions $a_{\epsilon }\left(x\right)$ and $g_{\epsilon }\left(x\right)$ satisfy the conditions of Theorem 2. Then the convergence

$${\mathfrak{A}}_{\epsilon }\to \overline{\mathfrak{A}}as\epsilon \to 0+$$

(49)

is valid in the strong topology ${\mathsf{\Theta }}_{+}^{\mathrm{loc}}$. Furthermore,

$${\mathcal{K}}_{\epsilon }\to \overline{\mathcal{K}}as\epsilon \to 0+in{\mathsf{\Theta }}^{\mathrm{loc}}.$$

(50)

A particular case of Theorem 3 was proven in [13] assuming that the coefficient $a_{\epsilon }\left(x\right)$ is independent of $\epsilon .$ Now we prove the general case.

We also need an assertion that generalizes the classical Fatou’s lemma.

Let $(X,\mathsf{\Sigma })$ be a measurable space with a sequence of measures ${\mu }_m$ on the sigma-algebra $\mathsf{\Sigma }$ in X.

Definition 2.

A sequence of measures $\left\{{\mu }_m\right\}$ on X converges to a measure $\mu ,$ if

$${\mu }_m\left(E\right)\to \mu \left(E\right)(m\to \infty ),\forall E\in \mathsf{\Sigma }.$$

Lemma 2 (Generalized Fatou’s Lemma).

Let $\left\{f_m\left(x\right)\right\}$ be a sequence of measurable functions on $(X,\mathsf{\Sigma })$ such that $f_m\left(x\right)\ge 0$ and $f_m\left(x\right)\to f\left(x\right)$ as $m\to \infty$ for all $x\in X$ and let a sequence of measures $\left\{{\mu }_m\right\}$ converges to a measure $\mu .$ Then

$${\int }_{X}f\left(x\right)d\mu \le \underset{m\to \infty }{liminf}{\int }_X{f}_m\left(x\right)d{\mu }_m.$$

For the proof see [49].

Proof of Theorem 3.

It is clear that Convergence (50) implies Convergence (49). Therefore, it suffices to prove Convergence (50), that is, to establish that for any neighbourhood $\mathcal{O}\left(\overline{\mathcal{K}}\right)$ in ${\mathsf{\Theta }}^{\mathrm{loc}}$ there is a number ${\epsilon }_1={\epsilon }_1\left(\mathcal{O}\right)>0$ such that

$${\mathcal{K}}_{\epsilon }\subset \mathcal{O}\left(\overline{\mathcal{K}}\right)\mathrm{for}\mathrm{all}\epsilon <{\epsilon }_1.$$

(51)

Recall that we are working taking into account Remark 2.

Assume that Convergence (51) fails. Then there exists a neighbourhood ${\mathcal{O}}^{\prime }\left(\overline{\mathcal{K}}\right)$ in ${\mathsf{\Theta }}^{\mathrm{loc}}$, sequences ${\epsilon }_k\to 0+$ $(k\to \infty )$ and $u_{{\epsilon }_k}(·)=u_{{\epsilon }_k}\left(t\right)\in {\mathcal{K}}_{{\epsilon }_k}$ such that

$$u_{{\epsilon }_k}\notin {\mathcal{O}}^{\prime }\left(\overline{\mathcal{K}}\right)\mathrm{for}\mathrm{all}k\in \mathbb{N}.$$

(52)

On the other hand, since Theorem 2 holds, we can assume that the sequence $\left\{u_{{\epsilon }_k}(·)\right\}$ is convergent in the weak topology ${\mathsf{\Theta }}^{\mathrm{loc},\mathrm{w}}:$

$$u_{{\epsilon }_k}(·)\to \overline{u}(·)ask\to \infty \mathrm{in}{\mathsf{\Theta }}^{\mathrm{loc},\mathrm{w}},$$

(53)

where $\overline{u}(·)\in \overline{\mathcal{K}},$ that is, $\overline{u}(·)$ is a complete solution of the limit Problem (39)–(41) and $\overline{u}(·)\in {\mathcal{F}}^{\mathrm{b}}.$ Convergence (53) means that

$$u_{{\epsilon }_k}(·)\to \overline{u}(·)ask\to \infty \mathrm{strongly}\mathrm{in}{\mathbf{L}}_2^{\mathrm{loc}}({\mathbb{R}}_{+};\mathbf{H}),$$

(54)

and, in addition, we can assume that

$$u_{{\epsilon }_k}(x,t)\to \overline{u}(x,t)\mathrm{for}\mathrm{almost}\mathrm{all}(x,t)\in \mathsf{\Omega }\times {\mathbb{R}}_{+}.$$

(55)

Recall that the kernels ${\mathcal{K}}_{\epsilon },\epsilon \in (0,1],$ are uniformly (with respect to $\epsilon$) bounded in ${\mathcal{F}}^{\mathrm{b}}$ (Proposition 6). Therefore, for some constant $C>0$ we have

$$\begin{array}{cc}\hfill \parallel u_{{\epsilon }_k}{\parallel }_{{\mathcal{F}}^{\mathrm{b}}}& =\underset{t\in \mathbb{R}}{sup}\parallel u_{{\epsilon }_k}\left(t\right)\parallel +\underset{t\in \mathbb{R}}{sup}{\left({\int }_t^{t+1}{\parallel u_{{\epsilon }_k}\left(s\right)\parallel }_1^{2}ds\right)}^{1/2}+\underset{t\in \mathbb{R}}{sup}{\parallel u_{{\epsilon }_k}\left(s\right)\parallel }_{{\mathbf{L}}_{\mathbf{p}}(t,t+1;{\mathbf{L}}_{\mathbf{p}})}+\hfill \\ \hfill & \underset{t\in \mathbb{R}}{sup}{∥\frac{\partial u_{{\epsilon }_k}}{\partial t}\left(s\right)∥}_{{\mathbf{L}}_{\mathbf{q}}(t,t+1;{\mathbf{H}}^{-\mathbf{r}})}\le C\forall k\in \mathbb{N}.\hfill \end{array}$$

(56)

Therefore, passing to a subsequence $\{u_{{\epsilon }_k^{\prime }}\left(t\right)\}\subset \{u_{{\epsilon }_k}\left(t\right)\}$ labelled the same as $\left\{u_{{\epsilon }_k}\left(t\right)\right\},$ we can assume that

$$u_{{\epsilon }_k}\left(t\right)\rightharpoonup \overline{u}\left(t\right)(k\to \infty )$$

(57)

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

$$\partial u_{{\epsilon }_k}\left(t\right)/\partial t\rightharpoonup \partial \overline{u}\left(t\right)/\partial t(k\to \infty )\mathrm{weakly}\mathrm{in}{\mathbf{L}}_{\mathbf{q},w}^{\mathrm{loc}}(\mathbb{R};{\mathbf{H}}^{-\mathbf{r}}).$$

(58)

Now, let us verify that $u_{{\epsilon }_k}\left(t\right)\to \overline{u}\left(t\right)$ in the strong topology ${\mathsf{\Theta }}_{+}^{\mathrm{loc}}$. To prove this fact, we use the method of energy equalities from the proof of Theorem 1. It is sufficient to check that a subsequence of $\left\{u_{{\epsilon }_k}\left(t\right)\right\}$ converges strongly to $\overline{u}\left(t\right)$ in the space ${\mathbf{L}}_{\mathbf{p}}(\mathsf{\Omega }\times [-M+1,M])\cap {\mathbf{L}}_2(-M+1,M;\mathbf{V})$ for every $M>0$. For any fixed M, shifting the time back on $t=-M+t^{\prime }$, we can suppose the functions $\left\{u_{{\epsilon }_k}\left(t^{\prime }\right)\right\}$ and $\overline{u}\left(t^{\prime }\right)$ to define on the interval $[0,M^{\prime }]$, $M^{\prime }=2M$. We now have subsequence that converges strongly in ${\mathbf{L}}_{\mathbf{p}}(\mathsf{\Omega }\times [1,M^{\prime }])\cap {\mathbf{L}}_2(1,M^{\prime };\mathbf{V})$. We omit the primes in $t^{\prime }$ and $M^{\prime }$.

Since $\left\{u_{{\epsilon }_k}\left(t\right)\right\}$ are bounded in the spaces ${\mathbf{L}}_{\mathbf{p}}(\mathsf{\Omega }\times [0,M])$ and ${\mathbf{L}}_2(0,M;\mathbf{V})$, we can assume that $u_{{\epsilon }_k}(·)\rightharpoondown \overline{u}(·)$ as ${\epsilon }_k\to 0+$ weakly in the spaces ${\mathbf{L}}_{\mathbf{p}}(\mathsf{\Omega }\times [0,M])$ and ${\mathbf{L}}_2(0,M;\mathbf{V})$. We suppose that $u_{{\epsilon }_k}\left(M\right)\rightharpoondown \overline{u}\left(M\right)$ as ${\epsilon }_k\to 0+$ weakly in $\mathbf{H}$.

Similarly to Relations (27)–(29), we have

$$\begin{array}{cc}\hfill \parallel \overline{u}\left(M\right)\parallel & \le \underset{k\to \infty }{lim\; inf}\parallel u_{{\epsilon }_k}\left(M\right)\parallel ,\hfill \end{array}$$

(59)

$$\begin{array}{cc}\hfill {\int }_0^M{\int }_{\mathsf{\Omega }}t\lambda \nabla \overline{u}·\nabla \overline{u}dxdt& \le \underset{k\to \infty }{lim\; inf}{\int }_0^M{\int }_{\mathsf{\Omega }}t\lambda \nabla u_{{\epsilon }_k}·\nabla u_{{\epsilon }_k}dxdt,\hfill \end{array}$$

(60)

$$\begin{array}{cc}\hfill {\int }_0^M{\int }_{\mathsf{\Omega }}ta\left(x\right){\left|{\overline{u}}^i\right|}^{p_i}dxdt& \le lim\underset{k\to \infty }{inf}{\int }_0^M{\int }_{\mathsf{\Omega }}ta\left(x\right){\left|u_{{\epsilon }_k}^i\right|}^{p_i}dxdt,i=1,\cdots ,N,\hfill \end{array}$$

(61)

where we denote $u_{{\epsilon }_k}=u_{{\epsilon }_k}(x,t)$ and $\overline{u}=\overline{u}(x,t)$, for brevity.

We claim that

$$\begin{array}{cc}\hfill & {\int }_0^M{\int }_{\mathsf{\Omega }}tV\left(x\right)\overline{u}(x,t)·\overline{u}(x,t)dxdt\le \hfill \\ \hfill & \underset{k\to \infty }{lim\; inf}{\epsilon }_k^{n/(n-2)}\sum _{j\in {\mathrm{Y}}_{{\epsilon }_k}}{\int }_0^M{\int }_{\partial G_{{\epsilon }_k}^j}t{B}_{{\epsilon }_k}^j\left(x\right)u_{{\epsilon }_k}(x,t)·u_{{\epsilon }_k}(x,t)d\omega dt.\hfill \end{array}$$

(62)

To establish the Inequality (62) we show that

$$\begin{array}{cc}\hfill & {\epsilon }_k^{n/(n-2)}\sum _{j\in {\mathrm{Y}}_{{\epsilon }_k}}{\int }_0^M{\int }_{\partial G_{{\epsilon }_k}^j}t{B}_{{\epsilon }_k}^j\left(x\right)u_{{\epsilon }_k}(x,t)·u_{{\epsilon }_k}(x,t)d\omega dt\to \hfill \\ \hfill & {\int }_0^M{\int }_{\mathsf{\Omega }}tV\left(x\right)\overline{u}(x,t)·\overline{u}(x,t)dxdtas{\epsilon }_k\to 0+.\hfill \end{array}$$

(63)

Indeed, we estimate the following difference:

$$\begin{array}{cc}\hfill & |{\epsilon }_k^{n/(n-2)}\sum _{j\in {\mathrm{Y}}_{{\epsilon }_k}}{\int }_0^M{\int }_{\partial G_{{\epsilon }_k}^j}t{B}_{{\epsilon }_k}^j\left(x\right)u_{{\epsilon }_k}(x,t)·u_{{\epsilon }_k}(x,t)d\omega dt-\hfill \\ \hfill & {\int }_0^M{\int }_{\mathsf{\Omega }}tV\left(x\right)\overline{u}(x,t)·\overline{u}(x,t)dxdt|\le \hfill \\ \hfill & |{\epsilon }_k^{n/(n-2)}\sum _{j\in {\mathrm{Y}}_{{\epsilon }_k}}{\int }_0^M{\int }_{\partial G_{{\epsilon }_k}^j}t{B}_{{\epsilon }_k}^j\left(x\right)u_{{\epsilon }_k}(x,t)·u_{{\epsilon }_k}(x,t)d\omega dt-\hfill \\ \hfill & {\epsilon }_k^{n/(n-2)}\sum _{j\in {\mathrm{Y}}_{{\epsilon }_k}}{\int }_0^M{\int }_{\partial G_{{\epsilon }_k}^j}t{B}_{{\epsilon }_k}^j\left(x\right)\overline{u}(x,t)·u_{{\epsilon }_k}(x,t)d\omega dt|+\hfill \\ \hfill & |{\epsilon }_k^{n/(n-2)}\sum _{j\in {\mathrm{Y}}_{{\epsilon }_k}}{\int }_0^M{\int }_{\partial G_{{\epsilon }_k}^j}t{B}_{{\epsilon }_k}^j\left(x\right)\overline{u}(x,t)·u_{{\epsilon }_k}(x,t)d\omega dt-\hfill \\ \hfill & {\int }_0^M{\int }_{\mathsf{\Omega }}tV\left(x\right)\overline{u}(x,t)·u_{{\epsilon }_k}(x,t)dxdt|+\hfill \\ \hfill & |{\int }_0^M{\int }_{\mathsf{\Omega }}tV\left(x\right)\overline{u}(x,t)·u_{{\epsilon }_k}(x,t)dxdt-\hfill \\ \hfill & {\int }_0^M{\int }_{\mathsf{\Omega }}tV\left(x\right)\overline{u}(x,t)·\overline{u}(x,t)dxdt|=I_1+I_2+I_3.\hfill \end{array}$$

(64)

Applying the Cauchy–Bunyakovsky inequality to the terms $I_1$ and $I_3$, and having the strong convergence $u_{{\epsilon }_k}(x,s)\to \overline{u}(x,s)$ in ${\mathbf{L}}_2(\mathsf{\Omega }\times [0,M])$, we show that $I_1\to 0$ and $I_3\to 0$ as ${\epsilon }_k\to 0+$. The rest term $I_2$ approaches zero by Lemma 4.4 proven in [13]. The Inequality (62) is proven.

Consider again the function $F\left(v\right)$ (see Function (31)). Since the function $F\left(v\right)$ is continuous, we conclude from Convergence (55) that

$$tF\left(u_{{\epsilon }_k}(x,t)\right)\to tF\left(\overline{u}(x,t)\right)\mathrm{for}\mathrm{almost}\mathrm{all}(x,t)\in \mathsf{\Omega }\times [0,M].$$

(65)

Consider the sequence of measures $\left\{{\mu }_k\right\}$ and the measure $\mu$ on $\mathsf{\Omega }\times [0,M]$ defined by the formulas

$${\mu }_k\left(E\right)={\int }_E{a}_{{\epsilon }_k}\left(x\right)dxdt,\mu \left(E\right)={\int }_E\overline{a}\left(x\right)dxdt,\forall E\in \mathsf{\Sigma },$$

(66)

where $\mathsf{\Sigma }$ is the standard Lebesque sigma-algebra on $\mathsf{\Omega }\times [0,M].$ It follows that from Assumption (9) the sequence measure ${\mu }_k$ converges to the measure $\mu ,$ that is,

$${\mu }_k\left(E\right)\to \mu \left(E\right)(k\to \infty ),\forall E\in \mathsf{\Sigma }.$$

(67)

It is sufficient to take in Assumption (9) the function $\phi \left(x\right)={\chi }_E\left(x\right).$

Applying Lemma 2 with $f_k=t(F\left(u_{{\epsilon }_k}(x,t)\right)+C)\ge 0$ and $f=t(F\left(\overline{u}(x,t)\right)+C)$ (see Inequality (5)) and having the Formulas (66) and (67), we obtain the inequality

$${\int }_0^M{\int }_{\mathsf{\Omega }}t\overline{a}\left(x\right)F\left(\overline{u}(x,t)\right)dxdt\le \underset{k\to \infty }{lim\; inf}{\int }_0^M{\int }_{\mathsf{\Omega }}t{a}_{\epsilon }\left(x\right)F\left(u_{{\epsilon }_k}(x,t)\right)dxdt$$

(68)

We now apply the energy equalities for the weak solutions $u_{{\epsilon }_k}(·)$ and $\overline{u}(·)$, and obtain, similar to Equalities (33) and (34), the next formulas

$$\begin{array}{cc}\hfill & \frac{1}{2}\parallel u_{{\epsilon }_k}{\left(M\right)\parallel }^2+{\int }_0^M{\int }_{\mathsf{\Omega }}t\lambda \nabla u_{{\epsilon }_k}·\nabla u_{{\epsilon }_k}dxdt+\sum _{i=1}^N{\gamma }_i{\int }_0^M{\int }_{\mathsf{\Omega }}t{a}_{{\epsilon }_k}\left(x\right){\left|u_{{\epsilon }_k}^i\right|}^{p_i}dxdt+\hfill \\ \hfill & {\int }_0^M{\int }_{\mathsf{\Omega }}t{a}_{{\epsilon }_k}\left(x\right)F\left(u_{{\epsilon }_k}\right)dxds+{\epsilon }^{n/(n-2)}\sum _{j\in {\mathrm{Y}}_{\epsilon }}{\int }_0^M{\int }_{\partial G_{{\epsilon }_k}^j}t{B}_{{\epsilon }_k}^j\left(x\right)u_{{\epsilon }_k}·u_{{\epsilon }_k}d\omega dt=\hfill \\ \hfill & \frac{1}{2}{\int }_0^M{\int }_{\mathsf{\Omega }}{\left|u_{{\epsilon }_k}\right|}^{2}dxdt+{\int }_0^M{\int }_{\mathsf{\Omega }}{g}_{{\epsilon }_k}\left(x\right)·u_{{\epsilon }_k}dxdt,\hfill \end{array}$$

(69)

$$\begin{array}{cc}\hfill & \frac{1}{2}\parallel \overline{u}{\left(M\right)\parallel }^2+{\int }_0^M{\int }_{\mathsf{\Omega }}t\lambda \nabla \overline{u}·\nabla \overline{u}dxdt+\sum _{i=1}^N{\gamma }_i{\int }_0^M{\int }_{\mathsf{\Omega }}t\overline{a}\left(x\right){\left|{\overline{u}}^i\right|}^{p_i}dxdt+\hfill \\ \hfill & {\int }_0^M{\int }_{\mathsf{\Omega }}t\overline{a}\left(x\right)F\left(\overline{u}\right)dxdt+{\int }_0^M{\int }_{\mathsf{\Omega }}tV\left(x\right)\overline{u}(x,t)·\overline{u}(x,t)dxdt=\hfill \\ \hfill & \frac{1}{2}{\int }_0^M{\int }_{\mathsf{\Omega }}{\left|\overline{u}\right|}^{2}dxdt+{\int }_0^M{\int }_{\mathsf{\Omega }}\overline{g}\left(x\right)·\overline{u}dxdt.\hfill \end{array}$$

(70)

Consider the difference

$$\begin{array}{cc}\hfill & \left|{\int }_0^M{\int }_{\mathsf{\Omega }}{g}_{{\epsilon }_k}\left(x\right)·u_{{\epsilon }_k}dxdt-{\int }_0^M{\int }_{\mathsf{\Omega }}\overline{g}\left(x\right)·\overline{u}dxdt\right|=\hfill \\ \hfill & \left|{\int }_0^M{\int }_{\mathsf{\Omega }}{g}_{{\epsilon }_k}\left(x\right)·(u_{{\epsilon }_k}-\overline{u})dxdt+{\int }_0^M{\int }_{\mathsf{\Omega }}\left(g_{{\epsilon }_k}\left(x\right)-\overline{g}\left(x\right)\right)·\overline{u}dxdt\right|=\hfill \\ \hfill & \parallel g_{{\epsilon }_k}{(·)\parallel }_{{\mathbf{L}}_2}{\parallel u_{{\epsilon }_k}-\overline{u}\parallel }_{{\mathbf{L}}_2}+\left|{\int }_0^M{\int }_{\mathsf{\Omega }}\left(g_{{\epsilon }_k}\left(x\right)-\overline{g}\left(x\right)\right)·\overline{u}dxdt\right|.\hfill \end{array}$$

(71)

Recall that $u_{{\epsilon }_k}(·)\to \overline{u}(·)$ strongly in the space ${\mathbf{L}}_2(\mathsf{\Omega }\times [0,M])$ and $g_{{\epsilon }_k}(·)\rightharpoondown \overline{g}(·)$ weakly in ${\mathbf{L}}_2(\mathsf{\Omega }\times [0,M])$ (Assumption (9)) and, hence, the functions $g_{{\epsilon }_k}(·)$ are uniformly bounded in ${\mathbf{L}}_2(\mathsf{\Omega }\times [0,M])$. Consequently, both summand in Formula (71) approaches zero and, hence,

$${\int }_0^M{\int }_{\mathsf{\Omega }}{g}_{{\epsilon }_k}\left(x\right)·u_{{\epsilon }_k}dxdt\to {\int }_0^M{\int }_{\mathsf{\Omega }}\overline{g}\left(x\right)·\overline{u}dxdtas{\epsilon }_k\to 0+.$$

(72)

Thus, the right-hand side of Equation (69) tends to the right-hand side of Equation (70). Then the left-hand side of Equation (69) also tends to the left-hand side of Equation (70). Combining this observations with the Inequalities (59)–(62) and (68), we conclude that

$$\begin{array}{c}\underset{k\to \infty }{lim}\parallel u_{{\epsilon }_k}{\left(M\right)\parallel }^2={\parallel \overline{u}\left(M\right)\parallel }^2,\\ \underset{k\to \infty }{lim}{\int }_0^M{\int }_{\mathsf{\Omega }}t\lambda \nabla u_{{\epsilon }_k}·\nabla u_{{\epsilon }_k}dxdt={\int }_0^M{\int }_{\mathsf{\Omega }}t\lambda \nabla \overline{u}·\nabla \overline{u}dxdt,\\ \underset{k\to \infty }{lim}{\int }_0^M{\int }_{\mathsf{\Omega }}ta\left(x\right)|u_{{\epsilon }_k}^i{|}^{p_i}dxdt={\int }_0^M{\int }_{\mathsf{\Omega }}ta\left(x\right){\left|{\overline{u}}^i\right|}^{p_i}dxdt,i=1,\cdots ,N,\\ \underset{k\to \infty }{lim}{\int }_0^M{\int }_{\mathsf{\Omega }}ta\left(x\right)F\left(u_{{\epsilon }_k}\right)dxdt={\int }_0^M{\int }_{\mathsf{\Omega }}ta\left(x\right)F\left(\overline{u}\right)dxdt,\\ \underset{k\to \infty }{lim}{\epsilon }^{n/(n-2)}\sum _{j\in {\mathrm{Y}}_{\epsilon }}{\int }_0^M{\int }_{\partial G_{{\epsilon }_k}^j}t{B}_{{\epsilon }_k}^j\left(x\right)u_{{\epsilon }_k}·u_{{\epsilon }_k}d\omega dt=\\ {\int }_0^M{\int }_{\mathsf{\Omega }}tV\left(x\right)\overline{u}(x,t)·\overline{u}(x,t)dxdt.\end{array}$$

Using the reasoning as in the end of the proof of Theorem 1, we deduce that $u_{{\epsilon }_k}(·)\to \overline{u}(·)$ strongly in the space ${\mathbf{L}}_{\mathbf{p}}(\mathsf{\Omega }\times [0,M])\cap {\mathbf{L}}_2(0,M;\mathbf{V})\cap C([0,M];\mathbf{H})$ and ${\partial }_t{u}_{{\epsilon }_k}(·)\to {\partial }_t\overline{u}(·)$ strongly in the space $\mathcal{L}(0,M)$ as ${\epsilon }_k\to 0+$.

In fact, we showed that

$$u_{{\epsilon }_k}(·)\to \overline{u}(·)ask\to \infty \mathrm{in}\mathrm{the}\mathrm{strong}\mathrm{topology}{\mathsf{\Theta }}^{\mathrm{loc}}.$$

Recall that $\overline{u}(·)\in \overline{\mathcal{K}}.$ The assumption $u_{{\epsilon }_k}\left(s\right)\notin {\mathcal{O}}^{\prime }\left(\overline{\mathcal{K}}\right)$ yields $\overline{u}\notin {\mathcal{O}}^{\prime }\left(\overline{\mathcal{K}}\right)$ and, moreover, $\overline{u}\notin \overline{\mathcal{K}}$. We arrive to a contradiction.

We have proven Convergence (50) and, consequently Convergence (49). Theorem 3 is proven. □

In the final part of this paper we consider the reaction–diffusion equations for which the uniqueness theorem of the Cauchy problem takes place. It is sufficient to consider the case, when the non-linear term $f\left(v\right)$ in Equation (1) is bounded as follows

$$(f\left(v_1\right)-f\left(v_2\right),v_1-v_2)\ge -C{|v_1-v_2|}^2\forall v_1,v_2\in {\mathbb{R}}^N$$

(73)

([20,40]). It was proven in [40] that Equation (73) implies that the Problem (1)–(4) and (35)–(37) generate dynamic semigroups in $\mathbf{H}$, which have the global attractors ${\mathcal{A}}_{\epsilon }$ and $\overline{\mathcal{A}}$, that are bounded in the space $\mathbf{V}={\mathbf{H}}_0^1\left(\mathsf{\Omega }\right)$ (see also [19,21]). Moreover, the following identities holds:

$${\mathcal{A}}_{\epsilon }=\{u\left(0\right)\mid u\in {\mathfrak{A}}_{\epsilon }\},\overline{\mathcal{A}}=\{u\left(0\right)\mid u\in \overline{\mathfrak{A}}\}.$$

In this case, the Convergence (49) implies the following corollary.

Corollary 1.

Let the assumptions of Theorem 2 be true. Then,

$${\mathrm{dist}}_{\mathbf{H}}({\mathcal{A}}_{\epsilon },\overline{\mathcal{A}})\to 0,\epsilon \to 0+.$$

5. Example

In conclusion, we consider the following complex Ginzburg–Landau equation in ${\mathsf{\Omega }}_{\epsilon }⋐{\mathbb{R}}^3(n=3)$, which is an important particular case of the general reaction–diffusions system:

$${\partial }_{t}u=(1+i\alpha )\mathrm{\Delta }u+R_{\epsilon }\left(x\right)u-(1+i{\beta }_{\epsilon }\left(x\right)){\left|u\right|}^{2}u+g_{\epsilon }\left(x\right),x\in {\mathsf{\Omega }}_{\epsilon },$$

(74)

where $u=u_{\epsilon }(x,t)=u_1(x,t)+i{u}_2(x,t)$ is an unknown complex function, $R_{\epsilon }\left(x\right)=R\left(x,\frac{x}{\epsilon }\right)$ and ${\beta }_{\epsilon }\left(x\right)=\beta \left(x,\frac{x}{\epsilon }\right)$ are known real-bounded coefficients, $\alpha$ is a real constant. The real functions $R(x,y),\beta \left(x,y\right)\in C(\mathsf{\Omega }\times {\mathbb{R}}^3)$ and satisfy Condition (8). The known complex function $g_{\epsilon }\left(x\right)=g_{\epsilon }\left(x,\frac{x}{\epsilon }\right)=g_1\left(x,\frac{x}{\epsilon }\right)+i{g}_2\left(x,\frac{x}{\epsilon }\right)\in L_2({\mathsf{\Omega }}_{\epsilon };\mathbb{C})$ and satisfies Condition (9). The Equation (74) can be written in the vector form

$${\partial }_t\left(\begin{array}{c}{u}_1\hfill \\ u_2\hfill \end{array}\right)=\left(\begin{array}{cc}1& -\alpha \\ \alpha & 1\end{array}\right)\left(\begin{array}{c}\mathrm{\Delta }{u}_1\hfill \\ \mathrm{\Delta }{u}_2\hfill \end{array}\right)+R_{\epsilon }\left(\begin{array}{c}{u}_1\hfill \\ u_2\hfill \end{array}\right)-{\left|u\right|}^2\left(\begin{array}{cc}1& -{\beta }_{\epsilon }\\ {\beta }_{\epsilon }& 1\end{array}\right)\left(\begin{array}{c}{u}_1\hfill \\ u_2\hfill \end{array}\right)+\left(\begin{array}{c}{g}_{\epsilon 1}\hfill \\ g_{\epsilon 2}\hfill \end{array}\right),$$

(75)

that is, a particular case of the System (1) for $N=2$ with $\lambda =\left(\begin{array}{cc}1& -\alpha \\ \alpha & 1\end{array}\right),f_{\epsilon }(u_1,u_2)={\left|u\right|}^2\left(\begin{array}{cc}1& -{\beta }_{\epsilon }\\ {\beta }_{\epsilon }& 1\end{array}\right)\left(\begin{array}{c}{u}_1\hfill \\ u_2\hfill \end{array}\right)-R_{\epsilon }\left(\begin{array}{c}{u}_1\hfill \\ u_2\hfill \end{array}\right),g_{\epsilon }\left(x\right)=\left(\begin{array}{c}{g}_{\epsilon 1}\hfill \\ g_{\epsilon 2}\hfill \end{array}\right).$ Clearly, the matrix $\lambda$ has a positive symmetric part $\frac{1}{2}(\lambda +{\lambda }^{*})=I.$ For the vector function $f_{\epsilon }(u_1,u_2),$ we have

$$\begin{array}{ccc}\hfill f_{\epsilon 1}\left(u_1\right)u_1+f_{\epsilon 2}\left(u_2\right)u_2& =& \left(\right|u_1{|}^2+|u_2{|}^2{)}^2-R_{\epsilon }{\left|u\right|}^2\ge \frac{1}{2}\left(\right|u_1{|}^4+|u_2{|}^4-R_0^2),\hfill \\ \hfill |f_{\epsilon }|& \le & C\left(\left(\right|u_1{|}^2+|u_2{{|}^2)}^{3/2}+1\right),\hfill \end{array}$$

Therefore, Condition (7) holds for $p=4.$ We supplement the System (75) with boundary Conditions (2) and (3) and the initial Condition (4) kipping all the notations of Section 2.

Theorems 1–3 are applicable to the considered initial-boundary value problem for the Ginzburg–Landau equation in the corresponding trajectory space. The corresponding homogenized Ginzburg–Landau equations reads

$${\partial }_{t}u=(1+i\alpha )\mathrm{\Delta }u+\overline{R}\left(x\right)u+V\left(x\right)u-(1+i\overline{\beta }\left(x\right)){\left|u\right|}^{2}u+\overline{g}\left(x\right),x\in {\mathsf{\Omega }}_{\epsilon },$$

(76)

where $\overline{R}\left(x\right),\overline{\beta }\left(x\right),$ and $\overline{g}\left(x\right)$ are the means of the functions $R_{\epsilon }\left(x\right),{\beta }_{\epsilon }\left(x\right),$ and $g_{\epsilon }\left(x\right)$, while $V\left(x\right)$ is the limit potential (the diagonal $2\times 2$-matrix) is defined by the Formula (38) ($N=2$), where $(v^1(x,y),v^2(x,y))$ is the solution of the corresponding inner Problem (35)–(37) in ${\mathbb{R}}_y^3\backslash G_0.$

Finally, we note that under the condition

$$|{\beta }_{\epsilon }|\le \sqrt{3},$$

the considered initial-boundary value problem for the Ginzburg–Landau Equations (74) and (76) are uniquely solvable (see [20]) and, therefore, Corollary 1 holds for this equations.

6. Conclusions

In this paper, we studied 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 dynamic system. By means of generalized Fatou’s lemma we prove the strong convergence of the trajectory attractors of the considered system to the trajectory attractors of the corresponding homogenized reaction–diffusion system.