Non-autonomous 3D Brinkman-Forchheimer equation with singularly oscillating external force and its uniform attractor

1.   Introduction

For $\rho\in [0, 1)$ and $\varepsilon > 0$, we consider the following non-autonomous Brinkman-Forchheimer equation with a singularly oscillating external force on $\Omega\subset\mathbb{R}^3$:

where $\Omega$ is an open, bounded domain of $\mathbb{R}^3$ with sufficiently smooth boundary $\partial\Omega$. $u = (u_1, u_2, u_3)$ is the fluid velocity vector, $\nu$ is the Brinkman coefficient, $a > 0$ is the Darcy coefficient, $b > 0, c > 0$ are the Forchheimer coefficients, $p$ is the pressure, $\beta\in (1, \frac{4}{3}]$ is a constant.

Along with (1.1), we consider the averaged Brinkman-Forchheimer equation

without by rapid and singular oscillations, which formally corresponds to $\varepsilon = 0$.

The Brinkman-Forchheimer equation describes the motion of fluid flow in a saturated porous medium and has been studied by many researchers. We should note that most of the previous studies have been focused on physical viewpoint or numerical simulation viewpoint(see ^{[1,2,3,4,5,6,7,8]}), or continuous dependence of solutions on the coefficients $\nu, b$ and $c$ (see ^{[9,10,11,12,13,14,15]}). The asymptotic behavior of solutions was examined in ^{[16,17,18,19,20,21,22]}, where ^{[16,17,18,19,20,21]} were mainly for the case of the parameter $\beta = 2$. In ^[16], using condition (C) method, U$\breve{g}$urlu showed the existence of global attractor in $H_0^1(\Omega)$. In ^[17], Wang and Lin showed that Brinkman-Forchheimer equation has a global attractor in $H^2(\Omega)$ by a very clever way. In ^[18], using the method of regularization of solutions and the compact embedding to deduce the uniformly $\omega$-limit compactness of the associated evolutionary process, You, Zhao and Zhou proved the existence of uniform attractor in $H_0^1(\Omega)$. In ^[19], the existence of $\mathcal{D}$-pullback attractors was deduced by establishing the $\mathcal{D}$-pullback asymptotical compactness of $\theta$-cocycle. In ^[20], Song and Qiao proved the existence and structure of the uniform attractor in $H_0^1(\Omega)$ for the processes associated to the fluid when the external force $f_0(x, t)$ is translation compact in $L_\mathrm{loc}^2(\mathbb{R}, (L^2(\Omega))^3)$ and investigated the averaging problems of the equations with oscillating external forces. In ^[21], Zhang, Su and Wen investigated the existence of global attractor and uniform attractor for the 3D autonomous and nonautonomous Brinkman-Forchheimer equations. For $\beta\neq 2$, in ^[22], using condition (C) method, Ouyang and Yang proved the existence of global attractor in $H_0^1(\Omega)$ when $1 < \beta\leq\frac{4}{3}$.

Stability of attractors for a dynamical system with some oscillating (or perturbed) external forces is very important in natural phenomenon. Indeed, this issue has been considered by some mathematicians and engineers. Chepyzhov et al. ^[23] studied the non-autonomous sine-Gordon type equations with rapidly oscillating external force. Efendiev and Zelik ^[24,25] considered the reaction-diffusion systems with rapidly oscillating coefficients and nonlinear rapidly oscillating in time. Chepyzhov and Vishik ^[26,27,28] investigated the Navier-Stokes equations with terms that rapidly oscillate with respect to spatial and time variables. Qin et al. ^[29] investigated the uniform attractors for a 3D non-autonomous Navier-Stokes-Voight equation with singularly oscillating forces. Anh and Toan ^[30] considered the nonclassical diffusion equation on $\mathbb{R}^N(N\geq 3)$ with a singularly oscillating external force. Medjo ^[31] and ^[32] used different methods to discuss non-autonomous planetary 3D geostrophic equation with oscillating external force and its global attractor. Medjo ^[33] investigated a non-autonomous two-phase flow model with oscillating external force and its global attractor. As far as we know, there is almost no paper dealing with 3D Brinkman-Forchheimer equations with rapidly oscillating terms have been published.

Motivated by ^[22] and ^{[23,24,25,26,27,28,29,30,31,32,33]}, we consider the properties of (1.1), depending on the small parameter $\varepsilon$, which reflects the rate of fast time oscillation in the term $\varepsilon^{-\rho}f_1(\frac{t}{\varepsilon}, x)$, having the growing amplitude of order $\varepsilon^{-\rho}$. By using the method in ^{[27,29,30,32,33]}, under suitable assumptions on the external force, we prove the stability of the uniform attractor $\mathcal{A}^\varepsilon(0 < \varepsilon\leq 1)$ associated to problem (1.1)-(1.2) as $\varepsilon\rightarrow 0^+$ in space $H$. The uncertainty of parameter $\beta$ brings a lot of trouble to our proof, because when $\beta$ is too large, some Sobolev embedding inequlaities can not be used. In the proof process of this paper, we finally determine that $1 < \beta\leq\frac{4}{3}$, just like in ^[22].

The main purpose of this paper is to show:

(1) the uniform (w.r.t.$\varepsilon$) boundedness of the family $\mathcal{A}^\varepsilon$ in $H$ which is defined in Section 3:

(2) the convergence of $\mathcal{A}^\varepsilon$ to $\mathcal{A}^0$ as $\varepsilon\rightarrow 0^+$ in the standard Hausdorff semidistance in $H$, i.e., :

This paper is organized as follows. In Section 2, we present the notations and preliminaries that are required for this study. In Section 3, we show the existence of uniform attractor $\mathcal{A}^\varepsilon$ and we demonstrate the structure of the uniform attractor. In Section 4, we verify the uniform boundedness of the uniform attractor $\mathcal{A}^\varepsilon$. In Section 5, we prove the convergence $\mathcal{A}^\varepsilon\rightarrow \mathcal{A}^0$ as $\varepsilon\rightarrow 0^+$.

2.   Notations and preliminaries

Given a space $X$, we usually denote the norm in $X$ by $\parallel\cdot\parallel_X$, and we indicate by

the Hausdorff semidistance in $X$ from a set $B_1$ to a set $B_2$. Throughout this paper, we set $\mathbb{R}_\tau = [\tau, +\infty), \tau\in\mathbb{R}$. $C$ will stand for a generic positive constant, which is different from line to line or even in the same line.

The mathematical setting of our problem is similar to that of the Navier-Stokes equations. Let us introduce the following spaces

where $\mathrm{cl}_X$ denotes the closure in the space $X$. Operator $P$ is the Helmholtz-Leray orthogonal projection from $(L^2(\Omega))^3$ onto $H$. $A: = -P\Delta$ is the Stokes operator subject to the nonslip homogeneous Dirichlet boundary with the domain $(H^2(\Omega))^3\cap V$, and $A$ is a self-adjoint positively defined operator on $H$. We define, for $\sigma\in\mathbb{R}$, the scale of Hilbert spaces

with inner products and norms

In particular,

and we have the generalized Poincar$\acute{e}$ inequality

where $\lambda_1$ is the first eigenvalue of the Stokes operator $A$.

In this paper, we use $(\cdot, \cdot)$ and $\parallel\cdot\parallel$ denote the product and the norm of $H$, i.e.,

$((\cdot, \cdot))$ and $\parallel\cdot\parallel_V$ denote the product and norm of $V$, i.e.,

In this paper, $\mathbf{L}^p(\Omega) = (L^p(\Omega))^3$, and we use $\parallel\cdot\parallel_p$ to denote the norm in $\mathbf{L}^p(\Omega)$ (for $p\neq 2$). Assumptions on the external forces The function $f_0(x, t)$ and $f_1(x, t)$ are taken from the space $L_b^2(\mathbb{R}; H)$ of translation bounded functions in $L_{\mathrm{loc}}^2(\mathbb{R}; H)$, with

for some constants $M_0, M_1\geq 0$.

Putting

It is easy to check that $f^\varepsilon\in L_b^2(\mathbb{R}; H)$, and

Now we recall some inequalities and a Gronwall-type lemma that will be needed in the sequel.

Lemma 2.1. (^[34]) Let $p\in[2, \infty)$. Then for every $a, b\in\mathbb{R}$,

Lemma 2.2. (^[27]) For every $\tau\in\mathbb{R}$, every nonnegative locally summable function $\varphi$ on $\mathbb{R}_\tau$ and every $\beta > 0$, we have

for all $t\geq\tau$.

Lemma 2.3. (^[23]) Let a real function $z(t), t\geq 0$ be uniformly continuous and satisfy the inequality

where $\gamma > 0, f(t)\geq 0$ for all $t\geq 0$ and $f\in L_{\mathrm{loc}}^1(\mathbb{R}^+)$. Suppose also that

Then

3.   On the existence and structure of the uniform attractor $\mathcal{A}^\varepsilon$

We rewrite (1.1) and (1.2) in the abstract form

where the pressure $p$ has disappeared by force of the application of the Helmholtz-Leray projection $P$, and $G(u) = P(b|u|u+c|u|^\beta u)$, $f^\varepsilon(x, t) = f_0(x, t)+\varepsilon^{-\rho}f_1(x, \frac{t}{\varepsilon})$ and $\varepsilon > 0$ is fixed.

Proposition 3.1. Given $f^\varepsilon\in L_b^2(\mathbb{R}; H)$ and $u_\tau\in H$. Then the system (3.1) has a unique weak solution $u(t)$ satisfying

Proof. We can prove the global existence and uniqueness result by using the Faedo-Galerkin method (see ^[35,36]).

If the functions $f_0(t)$ and $f_1(t)$ are translation bounded, i.e. conditions (2.2) and (2.3) hold, equation (3.1) generates the dynamical process

acting on $H$ by the formula

where $u(t)$ is the solution to (3.1).

Proposition 3.2. For any $\varepsilon > 0$, the process $\{U_{f^\varepsilon}(t, \tau)\}$ associated to (3.1) is uniformly compact in $H$ and it has a uniform (with respect to $\tau\in \mathbb{\mathbb{R}}$) absorbing set $B_\varepsilon$ in H,

where the constant $C_0$ depends on $\nu$ and $\lambda_1$. Moreover, there exists a uniform attractor $\mathcal{A}^\varepsilon$ in $H$.

Proof. The proof process is similar to the proof in ^[21], so we omit it here.

We consider the hull $\mathcal{H}(f^\varepsilon)$ of $f^\varepsilon(x, t)$ in the space $L_{\mathrm{loc}}^2(\mathbb{R}; H)$:

Recall that $\mathcal{H}(f^\varepsilon)$ is compact in $L_{\mathrm{loc}}^2(\mathbb{R}; H)$ and each element $\hat{f}^\varepsilon\in\mathcal{H}(f^\varepsilon)$ can be written as

with $\hat{f}_0\in\mathcal{H}(f_0)$ and $\hat{f}_1\in\mathcal{H}(f_1)$, where $\mathcal{H}(f_0)$ and $\mathcal{H}(f_1)$ are the hulls of $f_0$ and $f_1$ in $L_{\mathrm{loc}}^2(\mathbb{R}; H)$ respectively.

We also note that (see ^[23])

It follows that

where $C$ is independent of $f_0, f_1, \rho$ and $\varepsilon$.

To describe the structure of the attractor $\mathcal{A}_{\hat{f}^\varepsilon}$, we consider the family of equations

with the external force $\hat{f}^\varepsilon\in\mathcal{H}(f^\varepsilon)$.

For $\hat{f}^\varepsilon\in\mathcal{H}(f^\varepsilon)$, the Eq.(3.8) generates a process $\{U_{\hat{f}^\varepsilon}(t, \tau)\}$ that satisfies the same properties as $\{U_{f^\varepsilon}(t, \tau)\}$. Moreover, the process $\{U_{\hat{f}^\varepsilon}(t, \tau)\}$ has a uniform attractor $\mathcal{A}_{\hat{f}^\varepsilon}$ that satisfies $\mathcal{A}_{\hat{f}^\varepsilon}\subset \mathcal{A}_{f^\varepsilon}$.

Proposition 3.3. Let $f_0(x, t), f_1(x, t)$ be translation compact in the space $L_{\mathrm{loc}}^2(\mathbb{R}; H)$. Then for any fixed $\varepsilon, 0 < \varepsilon\leq 1$, the family of processes $\{U_{\hat{f}^\varepsilon}(t, \tau)\}, \hat{f}^\varepsilon\in\mathcal{H}(f^\varepsilon)$ corresponding to (3.8) has an absorbing set $B_{\varepsilon}$, which is bounded in $H$ and satisfies

The family $\{U_{\hat{f}^\varepsilon}(t, \tau)\}, \hat{f}^\varepsilon\in\mathcal{H}(f^\varepsilon)$ is $(H\times \mathcal{H}(f^\varepsilon); H)$-continuous. That is, if

then

Proof. The first part of the proposition is proved in Proposition 3.2. Now, let us prove the second part. Let $w_n = \hat{u}_n-\hat{u} = U_{\hat{f}^\varepsilon_n}(t, \tau)u_{\tau n}-U_{\hat{f}^\varepsilon}(t, \tau)u_\tau$. Then $w_n$ satisfies

Multiplying (3.12) by $w_n$ we have

Noticing $b|u|u+c|u|^\beta u$ is monotonic, i.e.,

then (3.13) gives

Applying Gronwall Lemma to (3.15) we have

Note that

therefore, it follows from (3.16) that

and (3.11) is proved, i.e., the family of processes $\{U_{\hat{f}^\varepsilon}(t, \tau)\}, \hat{f}^\varepsilon\in\mathcal{H}(f^\varepsilon)$ is $(H\times \mathcal{H}(f^\varepsilon); H)$-continuous.

We denote by $\mathcal{K}_{\hat{f}^\varepsilon}$ the kernel of (3.8) with the external force $\hat{f}^\varepsilon\in\mathcal{H}(f^\varepsilon)$. Let us recall that $\mathcal{K}_{\hat{f}^\varepsilon}$ is the family of all complete solutions $\{\hat{u}(t), t\in \mathbb{R}\}$ of (3.8), which are uniformly bounded in $H$. The set

is called the kernel section of $\mathcal{K}_{\hat{f}^\varepsilon}$ at time $t = s$.

For every $\varepsilon\in [0, 1]$, the following representation of the uniform attractor $\mathcal{A}^\varepsilon$ of equation (3.1) holds:

Actually, $\mathcal{K}_{\hat{f}^\varepsilon}(0)$ can be replaced by $\mathcal{K}_{\hat{f}^\varepsilon}(\tau)$, for an arbitrary $\tau\in\mathbb{R}$.

4.   Uniform boundedness of $\mathcal{A}^\varepsilon$ in $H$

First, we consider the auxiliary linear equation with nonautonomous external force and give some useful estimates and then prove the uniform boundedness of $\mathcal{A}^\varepsilon$ in $H$.

Considering the linear equation

we get the following lemma.

Lemma 4.1. If $K\in L_{\mathrm{loc}}^2(\mathbb{R}; V)$, then the above problem has a unique solution

Moreover, the inequalities

hold for every $t\geq\tau$ and some constant $C > 0$, independent of the initial time $\tau\in\mathbb{R}$.

Proof. Multiplying the equation (4.1) by $\mathcal{V}$ and $A^2\mathcal{V}$ respectively, we have

It follows from (4.6) and (4.7) that

Applying Gronwall lemma it yields

From (4.6) and (4.7) we also can get

Integrating (4.8) and (4.9) on $[t, t+1]$ respectively, we obtain

The proof is finished.

Setting $F(t, \tau) = \int_\tau^t f_1(s)\mathrm{d}s, t\geq\tau$, we assume that

Lemma 4.2. Assume that $f_1\in L_{\mathrm{loc}}^2(\mathbb{R}; H)$ and satisfies (4.11). Then the solution $v(t)$ to the Cauchy problem

with $\varepsilon\in(0, 1]$, satisfies the inequality

where $C > 0$ is a constant independent of $f_1$.

Proof. Without loss of generality, we may assume $\tau = 0$. Denoting $\mathcal{V}(t) = \int_0^t v(s)\mathrm{d}s$, we have, for any $t\geq 0$,

as $v(0) = 0$. Integrating (4.12) in time, we see that the function $\mathcal{V}(t)$ solves the problem

with external force

It follows from (4.11) that

and

By (2.5) we have

So applying Lemma 5.1, we obtain

Hence, on account of (4.14) we have

and

from which we derive the integral estimate

This finishes the proof.

Theorem 4.1. Let (4.11) holds true. Then the uniform attractors $\mathcal{A}^\varepsilon$ are uniformly (w.r.t. $\varepsilon$) bounded in $H$, that is,

Proof. Let $u$ be the solution to (3.1) with initial data $u_\tau\in H$. For $\varepsilon > 0$, we consider the problem

Lemma 4.2 provides the estimate

Then, the function $w(t) = u(t)-v(t)$ clearly satisfies the equation

with initial condition $w|_{t = \tau} = u_\tau$. Taking the inner product of (4.17) with $w$ in $H$, we obtain

By Lemma 2.1, we have

Noticing

and

according to Sobolev inequality

combining (4.20)-(4.23) with (4.19) we have

Case Ⅰ. $1 < \beta < \frac{4}{3}$.

Now we use Gagliardo-Nirenberg inequality to obtain

Considering $1 < \beta < \frac{4}{3}$, so $3\beta-2 < 2$. Combining (4.25), (4.26) with (4.24), and according to (4.16), we obtain

Let $g(s) = \parallel \nabla v(s)\parallel^2+Cl^6\varepsilon^{6(1-\rho)}+C(l^2\varepsilon^{2(1-\rho)})^{\frac{4-\beta}{4-3\beta}}+\frac{1}{a}\parallel f_0\parallel^2$. Noticing (4.16), we have

Applying Lemma 2.3 to (4.27) we have

Case Ⅱ. $\beta = \frac{4}{3}$.

From (4.24)-(4.26) we have

Similar to the derivation of (4.29), we get

Recalling that $u = w+v$, using (4.16), (4.29) and (4.31), we end up with

Thus, for every $\varepsilon\leq \varepsilon_0$, the process $\{U_{f^\varepsilon}(t, \tau)\}$ has the absorbing set

On the other hand, if $\varepsilon_0 < \varepsilon\leq 1$, the process $\{U_{f^\varepsilon}(t, \tau)\}$ possesses also the absorbing set (cf (3.2))

In conclusion, for every $\varepsilon\in [0, 1]$, the bounded set

is an absorbing set for $\{U_{f^\varepsilon}(t, \tau)\}$ which is independent of $\varepsilon$. Since $\mathcal{A}^\varepsilon\subset B^*$, the proof is completed.

5.   Convergence of the uniform attractors

The main result of this section is the following.

Theorem 5.1. Let (4.11) hold. Then, the uniform attractor $\mathcal{A}^\varepsilon$ converges to $\mathcal{A}^0$ as $\varepsilon\rightarrow 0^+$ in the following sense:

The proof of this theorem requires some steps. Now, we shall study the difference of two solutions to (3.1) with $\varepsilon > 0$ and $\varepsilon = 0$, respectively, sharing the same initial data. We denote

with $u_\tau$ belonging to the absorbing set $B^*$ found in the previous section. Owing to (4.32), we have the uniform bound:

for some $R_1 = R_1(l, M_0)$ because the size of $B^*$ depends on $l$ and $M_0$. In particular, for $\varepsilon = 0$, since $u_\tau\in B^*$, we have the bound

for some $R_0 = R_0(l, M_0)$.

Lemma 5.1. For every $\varepsilon\in(0, 1]$, every $\tau\in\mathbb{R}$ and every $u_\tau\in B^*$, the deviation $\tilde{w}(t) = u^\varepsilon(t)-u^0(t)$ with $u^\varepsilon(0) = u^0(0) = u_\tau$, fulfills the estimate

for some positive constant $C$ independent of $\varepsilon$.

Proof. Since the deviation $\tilde{w}(t)$ solves

the difference $q(t) = \tilde{w}(t)-v(t)$, where $v(t)$ is the solution to (4.15), fulfills the Cauchy problem

At this point, we take the scalar product in $H$ of $(5.5)$ with $q$, so getting

Noting the first term on the right-hand side of (5.6) is given by

we now proceed to estimate the first term on the right-hand side of (5.7). Since

and

it follows from (5.8)-(5.10) that

Now, let us estimate the second term on the right-hand side of (5.7). Noting

similar arguments as (5.9) and (5.10), we have

and

Hence, from (5.12)-(5.14) we get

Combining (5.11), (5.15) with (5.7), we have

Noting the second term on the right-hand side of (5.6) is given by

we now proceed to estimate the first term on the right-hand side of (5.17). Since

so we should estimate the right-hand side of the last inequality in (5.18) term by term. Because

and

where the last inequalities in (5.19) and (5.20) are valid only if $\frac{6}{5-3\beta}\leq 6$, i.e. $\beta\leq\frac{4}{3}$, so combining (5.19), (5.20) with (5.18), we have

Now let us estimate the second term on the right-hand side of (5.17). Since

in the second inequality of (5.22) we used the fact that

for any $x, y\geq 0$, where $C$ is an absolute constant. So let us estimate the right-hand side of the last inequality in (5.22) term by term. For the first term, we have

and

similarly,

It follows from (5.23)-(5.25) that

Similar arguments as (5.23)-(5.26), for the second term on the right-hand side of inequality (5.22), we have

Combining (5.26), (5.27) with (5.22), we have

So it follows from (5.17), (5.21) and (5.28) that

Now, considering (5.6), (5.16) and (5.29), and according to (5.1) and (5.2), it yields

Recalling that $\parallel q(\tau)\parallel = 0$ and (4.16), Lemma 2.3 entails

Finally, as $\tilde{w} = q+v$, using (4.16) to control $\parallel v\parallel$, we obtain the desired conclusion (5.3).

In order to study the convergence of the uniform attractors, we actually need a generalization of Lemma 5.1, which applies to the whole family of equations

with the external force $\hat{f} = \hat{f}^\varepsilon\in\mathcal{H}(f^\varepsilon)$. To this end, we observe that every function $\hat{f}_1\in\mathcal{H}(f_1)$ fulfills the inequality (4.11). Defining

we have

For any $\varepsilon\in[0, 1]$, let $\hat{u}^\varepsilon(t) = U_{\hat{f}^\varepsilon}(t, \tau)u_\tau$ be the solution to (5.31) with external force $\hat{f}^\varepsilon = \hat{f}_0+\varepsilon^{-\rho}\hat{f}_1(\cdot/\varepsilon)\in\mathcal{H}(f^\varepsilon)$ and $u_\tau\in B^*$. For $\varepsilon > 0$, we consider the deviation $\hat{w}(t) = \hat{u}^\varepsilon(t)-\hat{u}^0(t)$.

Lemma 5.2. The inequality

holds, where $C$ is independent of $\varepsilon$.

Proof. As the similar argument to the proof of Lemma 5.1, with $\hat{u}^\varepsilon, \hat{f}_0$ and $\hat{f}_1$ in place of $u^\varepsilon, f_0$ and $f_1$, respectively. Noting that (5.2) still holds for $\hat{u}^0$, and the family $\{U_{\hat{f}^\varepsilon}(t, \tau)\}(\hat{f}^\varepsilon\in\mathcal{H}(f^\varepsilon))$ is $(H\times \mathcal{H}(f^\varepsilon), H)$-continuous, and using (5.32) in place of (4.11), finally complete the proof of the lemma.

We can now complete the proof of Theorem 5.1, using the following argument from ^[27], which we report in some detail for the reader's convenience.

Proof of Theorem 5.1 Let $\varepsilon > 0$ and $u^\varepsilon\in \mathcal{A}^\varepsilon$. Thus, in view of (3.18), there exists a complete bounded trajectory $\hat{u}^\varepsilon(t)$ of (5.31), with the external force

such that $\hat{u}^\varepsilon(0) = u^\varepsilon$. For every $L\geq 0$ to be specified later, consider the vector

From the straightforward equality

by applying Lemma 5.2, we have that

On the other hand, the set $\mathcal{A}^0$ attracts $U_{\hat{f}_0}(t, -L)B^*$, uniformly as $\hat{f}_0\in\mathcal{H}(f_0)$. Then, for every $\delta > 0$, there is $T = T(\delta)\geq 0$, independent of $L$, such that

Setting $L = T$, and collecting the two above inequalities, we readily get

Since $u^\varepsilon\in \mathcal{A}^\varepsilon$ and $\delta > 0$ are arbitrary, taking the limit $\varepsilon\rightarrow 0^+$, the conclusion follows.

6.   Conclusion

In this paper, we investigated a class of three-dimensional Brinkman-Forchheimer equation with oscillating external forces $f^\varepsilon(x, t) = f_0(x, t)+\varepsilon^{-\rho}f_1(x, \frac{t}{\varepsilon})$. Based on some translation-compactness assumptions on the external forces, we obtained the uniform boundedness of the uniform attractor $\mathcal{A}^\varepsilon$ of the system (1.1) in $(L^2(\Omega))^3$, and the convergence of $\mathcal{A}^\varepsilon$ to the attractor $\mathcal{A}^0$ of the system (1.2) as $\varepsilon\rightarrow 0^+$. To prove the uniform boundedness and the convergence of the uniform attractors, the Gagliardo-Nirenberg inequality is needed. In the proof process, we concluded that the parameter $\beta\in(1, \frac{4}{3}]$.

Acknowledgments

The authors are thankful to the editors and the anonymous reviewers for their valuable suggestions and comments on the manuscript. The first author was supported in part by the National Natural Science Foundation of China (No. 11601417), Natural Science Basic Research Plan in Shaanxi Province of China (Nos. 2018JM1047, 2019JM-283) and Postdoctoral Fund in Shaanxi Province of China (No. 2016BSHEDZZ112). The second author was supported by the National Natural Science Foundation of China (No. 11771262).

Conflict of interest

The authors declare no conflict of interest in this paper.