Attractors of the 3D Magnetohydrodynamics Equations with Damping

Abstract

The three-dimensional magnetohydrodynamics equation with damping is considered in this paper. Global attractor of the 3D magnetohydrodynamics equations with damping is proved for $4\le \beta <5$ with any $\alpha >0$.

1 Introduction

In this paper, we consider the following three-dimensional magnetohydrodynamics (MHD) equations with damping:

$$\begin{aligned} \begin{array}{l} \left\{ \begin{array}{l} \partial _{t}u-\nu \Delta u+(u\cdot \nabla )u-(b\cdot \nabla )b+\alpha |u|^{\beta -1}u+\nabla (p+\frac{|b|^{2}}{2})=f_{1}(x),\\ \partial _{t}b-\kappa \Delta b+(u\cdot \nabla )b-(b\cdot \nabla )u=f_{2}(x),\\ \nabla \cdot u=0,~~~~\nabla \cdot b=0,\\ u|_{\partial D}=b|_{\partial D}=0,\\ u|_{t=0}=u_{0},~~~b|_{t=0}=b_{0}, \end{array} \right. \end{array} \end{aligned}$$

(1.1)

where $D\subseteq \mathbb {R}^{3}$ is a bounded domain with the boundary $\partial D$ and $t>0$. u, b are the fluid velocity and magnetic field, respectively. $f_{1}(x),f_{2}(x)$ are the external body force. p is the pressure, $\beta \ge 1$ is constant, and $\alpha$ is the damping coefficient. The constants $\nu ,\kappa \ge 0$ are kinematic viscosity and magnetic resistivity. For simplicity, we set $\nu =\kappa =1$.

The magnetohydrodynamic model has been investigated by many authors. In [12], Sermange and Temam have proved the well-posedness of solutions for MHD system in a bounded and a periodic domain. At the same time, regularity properties and attractors were also obtained. In [1], the pullback attractors and well-posedness of solutions of 2D MHD system were proved by using the Galerkin method. Based on the well-posedness of strong solutions of 3D MHD system that is difficult problem, many authors in [3, 6, 18, 19, 21] have studied the attractors and invariant measures of solutions of 3D-modified MHD system. Our result improves the early results in [21].

The damping term is very important for proving the well-posedness of 3D MHD system. In previous years, well-posedness and regularity of solutions of 3D Navier–Stokes system with damping were proved in [2, 20]. In [4, 5, 7,8,9, 11], the global existence of strong solution for 3D Navier–Stokes system with damping was proved for $\beta >3$ with any $\alpha >0$ and $\alpha \ge \frac{1}{4}$ as $\beta =3$. Moreover, the global well-posedness of the 3D magnetohydrodynamics equations with damping was proved for $\beta \ge 4$ with any $\alpha >0$ in [16]. Based on [10], global well-posedness of the 3D magneto–micropolar equations with damping was proved for $\beta \ge 4$ with any $\alpha >0$. The existence and regularity of the trajectory attractor of 3D modified Navier–Stokes equations were proved in [17].

To obtain the existence of attractors for the three-dimensional magnetohydrodynamics equations with damping, we overcome the main difficulty lies in dealing with the nonlinear term $(u\cdot \nabla )u$, $(u\cdot \nabla )b$, $(b\cdot \nabla )u$, $(b\cdot \nabla )b$ and $F(u)=\alpha |u|^{\beta -1}u$. C is a nonnegative constant which may change from line to line.

This paper is organized as follows. In Sect. 2, we give some preliminaries and main Theorem 2.1. In Sect. 3, the uniform estimate of solutions for system (1.1) is proved. In Sect. 4, the existence of a global attractor for system (1.1) is proved.

2 Preliminaries

In this paper, the inner products and norms are defined by

$$\begin{aligned} (u,v)=\int _{D}u\cdot v\mathrm{d}x,~~\forall u,v\in H,~~((u,v))=\int _{D}\nabla u\cdot \nabla v\mathrm{d}x,~~\forall u,v\in V, \end{aligned}$$

and $||\cdot ||^{2}=(\cdot ,\cdot )$, $||\nabla \cdot ||^{2}=((\cdot ,\cdot ))$, $\mathcal {V}=\{u\in (C_0^\infty (D))^3:\mathrm{div}u=0\}$, $H=$ the closure of $\mathcal {V}$ in $(L^{2}(D))^{3}$ and $V=$ the closure of $\mathcal {V}$ in $(H^{1}_{0}(D))^{3}$. The $L^{p}-$norm is given by $||\cdot ||_{p}$. By using the Poincaré inequality, there exists a positive constant $\lambda _{1}$ such that

$$\begin{aligned} \sqrt{\lambda _{1}}(||u||+||b||)\le ||\nabla u||+||\nabla b||,~~~\forall u,b\in V, \end{aligned}$$

(2.1)

here, $\lambda _{1}$ represents the minimum of between the first eigenvalue of $-\Delta u$ and the first eigenvalue of $-\Delta b$. Let $F(u)=\alpha |u|^{\beta -1}u$ and $D(A)=H^{2}(D)\cap V$. Here, P is the orthogonal projection of $(L^{2}(D))^{3}$ onto H such that $Au=-P\Delta u$ and $Ab=-P\Delta b$. For $u,v\in V$, we define the bilinear form $B(u,v)=P((u\cdot \nabla )v)$. We define $g(u)=PF(u)$. Now, we rewrite system (1.1) as follows in the abstract form:

$$\begin{aligned} \begin{array}{l} \left\{ \begin{array}{l} \partial _{t}u+Au+B(u,u)-B(b,b)+g(u)=f_{1},\\ \partial _{t}b+Ab+B(u,b)-B(b,u)=f_{2},\\ u|_{t=0}=u_{0},~~~b|_{t=0}=b_{0}. \end{array} \right. \end{array}\end{aligned}$$

(2.2)

Now, we introduce the main result as follows.

Theorem 2.1

Assume that $4\le \beta <5$ with any $\alpha >0$, $(u_{0},b_{0})\in V\times V$ and $f_{1},f_{2}\in H$. The operator $\{S(t)\}_{t\ge 0}$ of the three-dimensional MHD equations with damping system (2.2) satisfies

$$\begin{aligned} S(t)(u_{0},b_{0})=(u(t),b(t)). \end{aligned}$$

$\{S(t)\}_{t\ge 0}$ is defined in the space $V\times V$. System (2.2) has a $(V\times V,H^{2}\times H^{2})-$global attractor that satisfies the following.

• (i) The global attractor $\mathcal {A}$ is invariant and compact in $H^{2}\times H^{2}$.

• (ii) The global attractor $\mathcal {A}$ attracts bounded subset of $V\times V$ in relation to the norm topology of $H^{2}\times H^{2}$.

3 Uniform Estimate

Firstly, we will prove the uniform estimates of strong solutions for system (2.2) as $t\rightarrow \infty$. We show the existence of attractors by using the following estimates.

Lemma 3.1

Let $(u_{0},b_{0})\in V\times V$ and $f_{1},f_{2}\in H$ for $4\le \beta <5$ with any $\alpha >0$. There exists a constant $t_{0}$ such that

$$\begin{aligned}&\displaystyle ||u(t)||^{2}+||b(t)||^{2}\le C,\end{aligned}$$

(3.1)

$$\begin{aligned}&\displaystyle \int _{t}^{t+1}(||\nabla u(s)||^{2}+||\nabla b(s)||^{2} +||u(s)||_{\beta +1}^{\beta +1})\mathrm{d}s \le C. \end{aligned}$$

(3.2)

Proof

Multiplying the first equation of (2.2) by u and the second equation of (2.2) by b, integrating over D, then we have

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}(||u(t)||^{2}+||b(t)||^{2})+2\left( ||\nabla u||^{2}+||\nabla b||^{2} +\alpha ||u||_{\beta +1}^{\beta +1}\right) =2(f_{1},u)+2(f_{2},b)\\&\quad \le ||\nabla u||^{2}+||\nabla b||^{2}+\frac{1}{\lambda _{1}}(||f_{1}||^{2}+||f_{2}||^{2}). \end{aligned}$$

Hence,

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}(||u(t)||^{2}+||b(t)||^{2})&+||\nabla u||^{2}+||\nabla b||^{2} +\alpha ||u||_{\beta +1}^{\beta +1} \le \frac{1}{\lambda _{1}}(||f_{1}||^{2}+||f_{2}||^{2}), \end{aligned}$$

(3.3)

and

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}(||u(t)||^{2}+||b(t)||^{2})+\lambda _{1}(||u||^{2}+||b||^{2})\le \frac{1}{\lambda _{1}}(||f_{1}||^{2}+||f_{2}||^{2}). \end{aligned}$$

(3.4)

Applying the Gronwall inequality, it is easy to get

$$\begin{aligned} ||u(t)||^{2}+||b(t)||^{2}\le (||u_{0}||^{2}+||b_{0}||^{2})e^{-\lambda _{1}t}+\frac{1}{\lambda _{1}^{2}}(||f_{1}||^{2}+||f_{2}||^{2}). \end{aligned}$$

(3.5)

Let $t_{0}=\max \{-\frac{1}{\lambda _{1}}\ln \frac{||f_{1}||^{2}+||f_{2}||^{2}}{\lambda _{1}^{2} (||u_{0}||^{2}+||b_{0}||^{2})},0\}$. For any $t\ge t_{0}$,

$$\begin{aligned} ||u(t)||^{2}+||b(t)||^{2}\le \frac{2}{\lambda _{1}^{2}}(||f_{1}||^{2}+||f_{2}||^{2})\le C. \end{aligned}$$

(3.6)

Integrating (3.3) on $[t,t+1]$ and applying above inequality (3.6), we get for any $t\ge t_{0}$,

$$\begin{aligned} \int _{t}^{t+1}(||\nabla u(s)||^{2}+||\nabla b(s)||^{2} +\alpha ||u(s)||_{\beta +1}^{\beta +1})\mathrm{d}s&\le ||u(t)||^{2}+||b(t)||^{2} +\frac{1}{\lambda _{1}}(||f_{1}||^{2}+||f_{2}||^{2})\\&\le C. \end{aligned}$$

Lemma 3.2

Let $(u_{0},b_{0})\in V\times V$ and $f_{1},f_{2}\in H$ for $4\le \beta <5$ with any $\alpha >0$. There exists a $t_{1}$ such that for every $t\ge t_{1}$,

$$\begin{aligned} ||\nabla u(t)||^{2}+||\nabla b(t)||^{2}+||u||_{\beta +1}\le C. \end{aligned}$$

(3.7)

Proof

Inspired by [16], it is easy to get for $\beta \ge 4$ with any $\alpha >0$,

$$\begin{aligned}&||\nabla u(t)||^{2}+||\nabla b(t)||^{2}+\int _{0}^{t}(||\Delta u(s)||^{2}+||\Delta b(s)||^{2})\mathrm{d}s\nonumber \\&\quad +\int _{0}^{t}(|||u|^{\frac{\beta -1}{2}}\nabla u||^{2}+ ||\nabla |u|^{\frac{\beta +1}{2}}||^{2})\mathrm{d}s\le C. \end{aligned}$$

(3.8)

Multiplying the $L^{2}-$inner product of the first equation of (1.1) by $u_{t}$, then we get

$$\begin{aligned}&||u_{t}||^{2}+\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}||\nabla u||^{2} +\frac{\alpha }{\beta +1}\frac{\mathrm{d}}{\mathrm{d}t}||u||_{\beta +1}^{\beta +1}=-\int _{D}(u\cdot \nabla )u u_{t}\mathrm{d}x+\int _{D}(b\cdot \nabla )b u_{t}\mathrm{d}x+(f_{1},u_{t})\nonumber \\&\quad \le \frac{1}{2}||u_{t}||^{2}+||f_{1}||^{2}+C||u\cdot \nabla u||^{2}+C||b\cdot \nabla b||^{2}. \end{aligned}$$

(3.9)

It is easy to get that

$$\begin{aligned} ||u_{t}||^{2}+\frac{\mathrm{d}}{\mathrm{d}t}||\nabla u||^{2} +\frac{\mathrm{d}}{\mathrm{d}t}||u||_{\beta +1}^{\beta +1} \le C||f_{1}||^{2}+C||u\cdot \nabla u||^{2}+C||b\cdot \nabla b||^{2}. \end{aligned}$$

(3.10)

For the second term on the right-hand side of inequality (3.10), inspired by Theorem 1.1 in [10], we deduce

$$\begin{aligned} C||u\cdot \nabla u||^{2}&\le C\int _{D}|u|^{2}|\nabla u|^{\frac{4}{\beta -1}}|\nabla u|^{2-\frac{4}{\beta -1}}\mathrm{d}x\nonumber \\&\le C(|||u|^{\frac{\beta -1}{2}}\nabla u||^{2}+||\nabla u||^{2}). \end{aligned}$$

(3.11)

Inspired by Theorem 1.2 in [16] and Theorem 1.1 in [7], we get for $\beta \ge 4$

$$\begin{aligned} ||b(t)||_{3\frac{\beta +1}{\beta -1}}\le C. \end{aligned}$$

(3.12)

For the third term on the right-hand side of inequality (3.10), using (3.12), then we get

$$\begin{aligned} C||b\cdot \nabla b||^{2}&\le C||b||_{\frac{3(\beta +1)}{\beta -1}}^{2}||\nabla b||^{2}_{\frac{6(\beta +1)}{\beta +5}}\nonumber \\&\le C||b||_{\frac{3(\beta +1)}{\beta -1}}^{2}||\nabla b||^{\frac{4}{\beta +1}} ||\Delta b||^{\frac{2(\beta -1)}{\beta +1}}\nonumber \\&\le C(||\nabla b||^{2}+||\Delta b||^{2}). \end{aligned}$$

(3.13)

Integrating (3.10) on [0, t] and applying inequalities (3.11) and (3.13) to get

$$\begin{aligned} ||u||_{\beta +1}\le C,~~\forall t\ge t_{0}+1\equiv t_{1}. \end{aligned}$$

(3.14)

Lemma 3.3

Let $(u_{0},b_{0})\in V\times V$ and $f_{1},f_{2}\in H$ for $4\le \beta <5$ with any $\alpha >0$. There exists a $t_{2}$ such that for every $t\ge t_{2}$,

$$\begin{aligned} \int _{t}^{t+1}(||\Delta u||^{2}+||\Delta b||^{2}+|||u|^{\frac{\beta -1}{2}}\nabla u||^{2})\mathrm{d}s\le C. \end{aligned}$$

(3.15)

Proof

By (3.8), there exists a $t_{2}$ such that for any $t\ge t_{2}$

$$\begin{aligned} \int _{t}^{t+1}(||\Delta u||^{2}+||\Delta b||^{2}+|||u|^{\frac{\beta -1}{2}}\nabla u||^{2})\mathrm{d}s\le C. \end{aligned}$$

(3.16)

Lemma 3.4

Let $(u_{0},b_{0})\in V\times V$ and $f_{1},f_{2}\in H$ for $4\le \beta <5$ with any $\alpha >0$. There exists a $t_{3}$ such that for every $t\ge t_{3}$,

$$\begin{aligned} ||u_{t}||^{2}+||b_{t}||^{2}\le C. \end{aligned}$$

(3.17)

Proof

Multiplying the first equation of (1.1) by $u_{t}$ and the second equation of (1.1) by $b_{t}$, integrating the result on D, then we have

$$\begin{aligned}&||u_{t}||^{2}+||b_{t}||^{2}+\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}(||\nabla u||^{2}+||\nabla b||^{2}) +\frac{\alpha }{\beta +1}\frac{\mathrm{d}}{\mathrm{d}t}||u||^{\beta +1}_{\beta +1}\nonumber \\&\le \frac{||u_{t}||^{2}+||b_{t}||^{2}}{2}+||f_{1}||^{2}+||f_{2}||^{2}\nonumber \\&\quad +C||u\cdot \nabla u||^{2}+C||b\cdot \nabla b||^{2}+C||b\cdot \nabla u||^{2}+C||u\cdot \nabla b||^{2}\nonumber \\&=\frac{||u_{t}||^{2}+||b_{t}||^{2}}{2}+||f_{1}||^{2}+||f_{2}||^{2}+\sum \limits _{i=1}^{4}I_{i}. \end{aligned}$$

(3.18)

For $I_{1}$, inspired by Theorem 1.1 in [10], we get

$$\begin{aligned} I_{1}&\le C\int _{D}|u|^{2}|\nabla u|^{\frac{4}{\beta -1}}|\nabla u|^{2-\frac{4}{\beta -1}}\mathrm{d}x\nonumber \\&\le C(|||u|^{\frac{\beta -1}{2}}\nabla u||^{2}+||\nabla u||^{2}). \end{aligned}$$

(3.19)

For $I_{2}$, by Sobolev inequality and (3.12), we get

$$\begin{aligned} I_{2}&\le C||b||_{\frac{3(\beta +1)}{\beta -1}}^{2}||\nabla b||^{2}_{\frac{6(\beta +1)}{\beta +5}}\nonumber \\&\le C||b||_{\frac{3(\beta +1)}{\beta -1}}^{2}||\nabla b||^{\frac{4}{\beta +1}} ||\Delta b||^{\frac{2(\beta -1)}{\beta +1}}\nonumber \\&\le C(||\nabla b||^{2}+||\Delta b||^{2}). \end{aligned}$$

(3.20)

Similarly, we also have

$$\begin{aligned} I_{3}&\le C||b||_{\frac{3(\beta +1)}{\beta -1}}^{2}||\nabla u||^{\frac{4}{\beta +1}} ||\Delta u||^{\frac{2(\beta -1)}{\beta +1}}\nonumber \\&\le C(||\nabla u||^{2}+||\Delta u||^{2}). \end{aligned}$$

(3.21)

For $I_{4}$, applying the Sobolev inequality, we get

$$\begin{aligned} I_{4}&\le C||u||_{\beta +1}^{2}||\nabla b||^{2}_{\frac{2(\beta +1)}{\beta -1}}\nonumber \\&\le C||u||_{\beta +1}^{2}||\nabla b||^{\frac{2(\beta -2)}{\beta +1}} ||\Delta b||^{\frac{6}{\beta +1}}\nonumber \\&\le C(||\nabla b||^{2}+||\Delta b||^{2}). \end{aligned}$$

(3.22)

Adding up (3.18)–(3.22), we get

$$\begin{aligned}&||u_{t}||^{2}+||b_{t}||^{2}+\frac{\mathrm{d}}{\mathrm{d}t}(||\nabla u||^{2}+||\nabla b||^{2}) +\frac{\mathrm{d}}{\mathrm{d}t}||u||^{\beta +1}_{\beta +1}\nonumber \\&\le C(||f_{1}||^{2}+||f_{2}||^{2})+C(||\nabla b||^{2}+||\Delta b||^{2}+||\nabla u||^{2}+||\Delta u||^{2}+|||u|^{\frac{\beta -1}{2}}\nabla u||^{2}). \end{aligned}$$

(3.23)

By Lemma 3.2–Lemma 3.3, integrating (3.23) in time from t to $t+1$, it is easy to get

$$\begin{aligned} \int _{t}^{t+1}(||u_{t}(s)||^{2}+||b_{t}(s)||^{2})\mathrm{d}s\le C. \end{aligned}$$

(3.24)

We apply $\partial _{t}$ to the first equation of (2.2) and multiply the $L^{2}-$inner product by $u_{t}$. Similarly, we apply $\partial _{t}$ to the second equation of (2.2) and multiply the $L^{2}-$inner product by $b_{t}$. Then we get

$$\begin{aligned}&\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}(||u_{t}||^{2}+||b_{t}||^{2}) +||\nabla u_{t}||^{2}+||\nabla b_{t}||^{2}\le |\int _{D}u_{t}\nabla u u_{t}\mathrm{d}x| +|\int _{D}b_{t}\nabla b u_{t}\mathrm{d}x|\nonumber \\&+|\int _{D}u_{t}\nabla b b_{t}\mathrm{d}x|+|\int _{D}b_{t}\nabla u b_{t}\mathrm{d}x|-\int _{D}F'(u)u_{t}u_{t}\mathrm{d}x=\sum \limits _{i=5}^{9}I_{i}. \end{aligned}$$

(3.25)

For $I_{9}$, by Lemma 2.4 in [14], we have $I_{9}\le 0$.

For $I_{5}$, by using Sobolev inequality and Lemma 3.2, we get

$$\begin{aligned} I_{5}&\le C||u_{t}||^{\frac{1}{2}}||\nabla u_{t}||^{\frac{3}{2}}||\nabla u||\nonumber \\&\le \frac{1}{4}||\nabla u_{t}||^{2}+C||u_{t}||^{2}||\nabla u||^{4}\nonumber \\&\le \frac{1}{4}||\nabla u_{t}||^{2}+C||u_{t}||^{2}. \end{aligned}$$

(3.26)

For $I_{8}$, similarly, then we get

$$\begin{aligned} I_{8}\le \frac{1}{4}||\nabla b_{t}||^{2}+C||b_{t}||^{2}. \end{aligned}$$

(3.27)

For $I_{6}$ and $I_{7}$, by Gagliardo–Nirenberg inequality and Lemma 3.2, we get

$$\begin{aligned} I_{6}+I_{7}&\le C||u_{t}||_{4}||b_{t}||_{4}||\nabla b||\nonumber \\&\le C||u_{t}||^{\frac{1}{4}}||\nabla u_{t}||^{\frac{3}{4}}||b_{t}||^{\frac{1}{4}}||\nabla b_{t}||^{\frac{3}{4}}||\nabla b||\nonumber \\&\le \frac{1}{4}||\nabla u_{t}||^{2}+\frac{1}{4}||\nabla b_{t}||^{2}+C(||u_{t}||^{2}+||b_{t}||^{2})||\nabla b||^{4}\nonumber \\&\le \frac{1}{4}||\nabla u_{t}||^{2}+\frac{1}{4}||\nabla b_{t}||^{2}+C(||u_{t}||^{2}+||b_{t}||^{2}). \end{aligned}$$

(3.28)

Adding up (3.25)–(3.28), we get

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}(||u_{t}||^{2}+||b_{t}||^{2}) +||\nabla u_{t}||^{2}+||\nabla b_{t}||^{2}\le C(||u_{t}||^{2}+||b_{t}||^{2}). \end{aligned}$$

(3.29)

Applying the uniform Gronwall’s lemma, there exists a $t_{3}$ such that for any $s\ge t_{3}$

$$\begin{aligned} ||u_{t}(s)||^{2}+||b_{t}(s)||^{2}\le C. \end{aligned}$$

(3.30)

Lemma 3.5

Let $(u_{0},b_{0})\in V\times V$ and $f_{1},f_{2}\in H$ for $4\le \beta <5$ with any $\alpha >0$. There exists a $t_{4}$ such that for every $t\ge t_{4}$,

$$\begin{aligned} ||Au(t)||+||Ab(t)||\le C. \end{aligned}$$

(3.31)

Proof

By using the Minkowski inequality, we deduce

$$\begin{aligned} ||Au||+||Ab||&\le ||u_{t}||+||b_{t}||+||f_{1}||+||f_{2}||+||B(u,u)|| \nonumber \\&\quad +||B(b,b)||+||B(u,b)||+||B(b,u)||+\alpha ||u|u|^{\beta -1}||\nonumber \\&=||u_{t}||+||b_{t}||+||f_{1}||+||f_{2}||+\sum \limits _{i=1}^{5}J_{i}. \end{aligned}$$

(3.32)

By the Sobolev inequality, we have

$$\begin{aligned} J_{1}\le C||u||_{\infty }||\nabla u||\le C||\nabla u||^{\frac{3}{2}}||Au||^{\frac{1}{2}} \le \frac{1}{4}||Au||+C||\nabla u||^{3}. \end{aligned}$$

(3.33)

For $J_{2}$, similarly, then we also have

$$\begin{aligned} J_{2}\le \frac{1}{4}||Ab||+C||\nabla b||^{3}. \end{aligned}$$

(3.34)

For $J_{3}$, applying the Sobolev inequality, we deduce

$$\begin{aligned} J_{3}\le C||u||_{\infty }||\nabla b||\le C||\nabla u||^{\frac{1}{2}}||Au||^{\frac{1}{2}}||\nabla b|| \le \frac{1}{8}||Au||+C||\nabla u||^{2}+C||\nabla b||^{4}. \end{aligned}$$

(3.35)

For $J_{4}$, similarly, then we also have

$$\begin{aligned} J_{4} \le \frac{1}{4}||Ab||+C||\nabla b||^{2}+C||\nabla u||^{4}. \end{aligned}$$

(3.36)

For $J_{5}$, since $\frac{\beta -3}{2}<1$ for $4\le \beta <5$, applying the Young’s inequality, we get

$$\begin{aligned} J_{5}=\alpha ||u||^{\beta }_{2\beta }\le C||\Delta u||^{\frac{\beta -3}{2}}||\nabla u||^{\frac{\beta +3}{2}} \le \frac{1}{8}||Au||+C||\nabla u||^{\frac{\beta +3}{5-\beta }}. \end{aligned}$$

(3.37)

Substituting (3.33)–(3.37) into (3.32), it is easy to get that for any $t\ge t_{4}$,

$$\begin{aligned} ||Au||+||Ab||\le C. \end{aligned}$$

(3.38)

Lemma 3.6

Let $(u_{0},b_{0})\in V\times V$ and $f_{1},f_{2}\in H$ for $4\le \beta <5$ with any $\alpha >0$. There exists a $t_{5}$ such that for every $t\ge t_{5}$,

$$\begin{aligned} ||\nabla u_{t}||^{2}+||\nabla b_{t}||^{2}&\le C,\end{aligned}$$

(3.39)

$$\begin{aligned} \int _{t}^{t+1}(||\nabla u_{t}(s)||^{2}+||\nabla b_{t}(s)||^{2})\mathrm{d}s&\le C. \end{aligned}$$

(3.40)

Proof

We integrate inequality (3.29) from t to $t+1$ and use the Lemma 3.4 to get

$$\begin{aligned} \int _{t}^{t+1}(||\nabla u_{t}(s)||^{2}+||\nabla b_{t}(s)||^{2})\mathrm{d}s&\le ||u_{t}(t)||^{2}+||b_{t}(t)||^{2} +C\int _{t}^{t+1}(||u_{t}||^{2}+||b_{t}||^{2})\mathrm{d}s\nonumber \\&\le C. \end{aligned}$$

(3.41)

By virtue of Lemma 3.5, then we deduce

$$\begin{aligned} ||u(t)||_{D(A)}+||b(t)||_{D(A)}\le C. \end{aligned}$$

Applying the Agmon inequality, it is easy to get

$$\begin{aligned} ||u(t)||_{\infty }+||b(t)||_{\infty }\le C. \end{aligned}$$

(3.42)

We apply $\partial _{t}$ to the first equation of (2.2) and multiply the $L^{2}-$inner product by $Au_{t}$. Similarly, we apply $\partial _{t}$ to the second equation of (2.2) and multiply the $L^{2}-$inner product by $Ab_{t}$. Then we also have

$$\begin{aligned}&\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}(||\nabla u_{t}||^{2}+||\nabla b_{t}||^{2}) +||Au_{t}||^{2}+||Ab_{t}||^{2}\nonumber \\&\le |\int _{D}u_{t}\nabla uAu_{t}\mathrm{d}x|+|\int _{D}u\nabla u_{t}Au_{t}\mathrm{d}x|+|\int _{D}b_{t}\nabla bAu_{t}\mathrm{d}x|+|\int _{D}b\nabla b_{t}Au_{t}\mathrm{d}x|\nonumber \\&+ |\int _{D}u_{t}\nabla bAb_{t}\mathrm{d}x|+|\int _{D}u\nabla b_{t}Ab_{t}\mathrm{d}x|+|\int _{D}b_{t}\nabla uA b_{t}\mathrm{d}x|+|\int _{D}b\nabla u_{t}Ab_{t}\mathrm{d}x|\nonumber \\&+|\int _{D}F'(u)u_{t}Au_{t}\mathrm{d}x|=\sum \limits _{i=1}^{9}K_{i}. \end{aligned}$$

(3.43)

For $K_{1}$ and $K_{2}$, applying the Sobolev inequality and Lemma 3.5, we get

$$\begin{aligned} K_{1}&\le C||\nabla u_{t}||||\nabla u||^{\frac{1}{2}}||Au||^{\frac{1}{2}}||Au_{t}||\nonumber \\&\le \frac{1}{16}||Au_{t}||^{2}+C||\nabla u_{t}||^{2}, \end{aligned}$$

(3.44)

and

$$\begin{aligned} K_{2}&\le C||\nabla u||||\nabla u_{t}||^{\frac{1}{2}}||Au_{t}||^{\frac{3}{2}}\nonumber \\&\le \frac{1}{16}||Au_{t}||^{2}+C||\nabla u_{t}||^{2}. \end{aligned}$$

(3.45)

For $K_{3}$ and $K_{4}$, we get by using the similar method

$$\begin{aligned} K_{3}&\le C||\nabla b_{t}||||\nabla b||^{\frac{1}{2}}||Ab||^{\frac{1}{2}}||Au_{t}||\nonumber \\&\le \frac{1}{16}||Au_{t}||^{2}+C||\nabla b_{t}||^{2}, \end{aligned}$$

(3.46)

and

$$\begin{aligned} K_{4}&\le C||\nabla b||||\nabla b_{t}||^{\frac{1}{2}}||Ab_{t}||^{\frac{1}{2}}||Au_{t}||\nonumber \\&\le \frac{1}{16}||Au_{t}||^{2}+\frac{1}{4}||Ab_{t}||^{2}+C||\nabla b_{t}||^{2}. \end{aligned}$$

(3.47)

Similarly, we deduce

$$\begin{aligned} \sum \limits _{i=5}^{8}K_{i}&\le C||\nabla u_{t}||||\nabla b||^{\frac{1}{2}}||Ab||^{\frac{1}{2}}||A b_{t}||+C||\nabla u||||\nabla b_{t}||^{\frac{1}{2}}||Ab_{t}||^{\frac{3}{2}}\nonumber \\&+C||\nabla b_{t}||||\nabla u||^{\frac{1}{2}}||Au||^{\frac{1}{2}}||Ab_{t}|| +C||\nabla b||||\nabla u_{t}||^{\frac{1}{2}}||Au_{t}||^{\frac{1}{2}}||Ab_{t}||\nonumber \\&\le \frac{1}{8}||Au_{t}||^{2}+\frac{1}{4}||Ab_{t}||^{2}+C(||\nabla u_{t}||^{2}+||\nabla b_{t}||^{2}). \end{aligned}$$

(3.48)

For $K_{9}$, by (3.42), we have

$$\begin{aligned} K_{9}&\le C||u||^{\beta -1}_{\infty }||u_{t}||||Au_{t}||\nonumber \\&\le \frac{1}{8}||Au_{t}||^{2}+C||u_{t}||^{2}~~\mathrm{for}~t\ge t_{4}. \end{aligned}$$

(3.49)

Summing up (3.43)–(3.49), we get

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}(||\nabla u_{t}||^{2}+||\nabla b_{t}||^{2}) \le C(||\nabla u_{t}||^{2}+||\nabla b_{t}||^{2}). \end{aligned}$$

(3.50)

By the uniform Gronwall’s lemma, there exists a $t_{5}$ such that for every $t\ge t_{5}$,

$$\begin{aligned} ||\nabla u_{t}||^{2}+||\nabla b_{t}||^{2}\le C. \end{aligned}$$

(3.51)

4 Global Attractors

In this section, we will show the existence of a global attractor for system (2.2) in $H^{2}\times H^{2}$. Inspired by [13, 14], we introduce the following the main lemmas.

Lemma 4.1

$\{S(t)\}_{t\ge 0}$ is Lipschitz continuous in $V\times V$.

Proof

Let $(u_{1},b_{1})$ and $(u_{2},b_{2})$ be two solutions of system (2.2) with initial values $(u_{01},b_{01})$ and $(u_{02},b_{02})$. We set $\bar{u}=u_{1}-u_{2}$ and $\bar{b}=b_{1}-b_{2}$. We multiply the inner product with $A\bar{u}$ and $A\bar{b}$, respectively. Then we get

$$\begin{aligned}&\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}(||\nabla \bar{u}||^{2}+||\nabla \bar{b}||^{2})+||A\bar{u}||^{2}+||A\bar{b}||^{2} \nonumber \\&\le \alpha \int _{D}||u_{1}|^{\beta -1}u_{1}-|u_{2}|^{\beta -1}u_{2}||A\bar{u}|\mathrm{d}x +\int _{D}|\bar{u}\nabla u_{1}A\bar{u}|\mathrm{d}x+\int _{D}|u_{2}\nabla \bar{u}A\bar{u}|\mathrm{d}x\nonumber \\&\quad +\int _{D}|\bar{b}\nabla b_{1}A\bar{u}|\mathrm{d}x+\int _{D}|b_{2}\nabla \bar{b}A\bar{u}|\mathrm{d}x +\int _{D}|u_{1}\nabla \bar{b}A\bar{b}|\mathrm{d}x+\int _{D}|\bar{u}\nabla b_{2}A\bar{b}|\mathrm{d}x\nonumber \\&\quad +\int _{D}|\bar{b}\nabla u_{1}A\bar{b}|\mathrm{d}x+\int _{D}|b_{2}\nabla \bar{u} A\bar{b}|\mathrm{d}x=\sum \limits _{i=1}^{9}L_{i}. \end{aligned}$$

(4.1)

Inspired by [13, 14], since $\int _{0}^{t}(||u_{1}||^{2(\beta -1)}_{3(\beta -1)}+||\nabla u_{2}||^{2}(| |u_{1}||^{2(\beta -2)}_{6(\beta -2)} +||u_{2}||^{2(\beta -2)}_{6(\beta -2)}))\mathrm{d}s<C$ for $4\le \beta <5$, then we get

$$\begin{aligned} L_{1}\le \frac{1}{8}||A\bar{u}||^{2}+C(||u_{1}||^{2(\beta -1)}_{3(\beta -1)}+||\nabla u_{2}||^{2}(||u_{1}||^{2(\beta -2)}_{6(\beta -2)}+||u_{2}||^{2(\beta -2)}_{6(\beta -2)}))||\nabla \bar{u}||^{2}. \end{aligned}$$

(4.2)

For $L_{2}$ and $L_{3}$, applying the Sobolev inequality, we have

$$\begin{aligned} L_{2}&\le C||\nabla \bar{u}||||\nabla u_{1}||^{\frac{1}{2}}||Au_{1}||^{\frac{1}{2}}||A\bar{u}||\nonumber \\&\le \frac{1}{16}||A\bar{u}||^{2}+C||\nabla u_{1}||||Au_{1}||||\nabla \bar{u}||^{2}, \end{aligned}$$

(4.3)

and

$$\begin{aligned} L_{3}&\le C||\nabla u_{2}||||\nabla \bar{u}||^{\frac{1}{2}}||A\bar{u}||^{\frac{3}{2}}\nonumber \\&\le \frac{1}{16}||A\bar{u}||^{2}+C||\nabla u_{2}||^{4}||\nabla \bar{u}||^{2}. \end{aligned}$$

(4.4)

Similarly, for the rest of terms $L_{4}-L_{9}$, we get

$$\begin{aligned} L_{4}+L_{5}&\le C||\nabla \bar{b}||||\nabla b_{1}||^{\frac{1}{2}}||Ab_{1}||^{\frac{1}{2}}||A\bar{u}|| +C||\nabla b_{2}||||\nabla \bar{b}||^{\frac{1}{2}}||A\bar{b}||^{\frac{1}{2}}||A\bar{u}||\nonumber \\&\le \frac{1}{8}||A\bar{u}||^{2}+\frac{1}{8}||A\bar{b}||^{2}+C(||\nabla b_{1}||||Ab_{1}||+||\nabla b_{2}||^{4})||\nabla \bar{b}||^{2},\end{aligned}$$

(4.5)

$$\begin{aligned} L_{6}+L_{7}&\le C||\nabla u_{1}||||\nabla \bar{b}||^{\frac{1}{2}}||A\bar{b}||^{\frac{3}{2}} +C||\nabla \bar{u}||||\nabla b_{2}||^{\frac{1}{2}}||Ab_{2}||^{\frac{1}{2}}||A\bar{b}||\nonumber \\&\le \frac{1}{8}||A\bar{b}||^{2}+C(||\nabla u_{1}||^{4}+||\nabla b_{2}||||A b_{2}||)(||\nabla \bar{u}||^{2}+||\nabla \bar{b}||^{2}), \end{aligned}$$

(4.6)

and

$$\begin{aligned} L_{8}+L_{9}&\le C||\nabla \bar{b}||||\nabla u_{1}||^{\frac{1}{2}}||Au_{1}||^{\frac{1}{2}}||A\bar{b}|| +C||\nabla b_{2}||||\nabla \bar{u}||^{\frac{1}{2}}||A\bar{u}||^{\frac{1}{2}}||A\bar{b}||\nonumber \\&\le \frac{1}{8}||A\bar{u}||^{2}+\frac{1}{8}||A\bar{b}||^{2}+C(||\nabla u_{1}||||Au_{1}||+||\nabla b_{2}||^{4})(||\nabla \bar{u}||^{2}+||\nabla \bar{b}||^{2}). \end{aligned}$$

(4.7)

Adding up (4.1)–(4.7), it is easy to get

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}(||\nabla \bar{u}||^{2}+||\nabla \bar{b}||^{2})&\le C[||\nabla u_{1}||^{2}+||\nabla u_{1}||^{4} +||u_{1}||^{2(\beta -1)}_{3(\beta -1)}+||\nabla b_{1}||||Ab_{1}||\nonumber \\&+||\nabla u_{2}||^{2}(||u_{1}||^{2(\beta -2)}_{6(\beta -2)}+||u_{2}||^{2(\beta -2)}_{6(\beta -2)})+||\nabla u_{2}||^{4}\nonumber \\&+||Au_{1}||^{2}+||\nabla b_{2}||^{4}+||\nabla b_{2}||^{2}+||Ab_{2}||^{2}](||\nabla \bar{u}||^{2}+||\nabla \bar{b}||^{2}). \end{aligned}$$

(4.8)

Applying the Gronwall inequality and Lemma 3.1–Lemma 3.6, this completes the proof of Lemma 4.1.

Lemma 4.2

Assume that $\mathcal {A}$ is a $(V\times V,V\times V)-$global attractor for $\{S(t)\}_{t\ge 0}$. $\mathcal {A}$ is a $(V\times V,H^{2}\times H^{2})-$global attractor if and only if

• (i) $\{S(t)\}_{t\ge 0}$ is a bounded $(V\times V,H^{2}\times H^{2})-$absorbing set.

• (ii) $\{S(t)\}_{t\ge 0}$ is $(V\times V,H^{2}\times H^{2})-$asymptotically compact.

Firstly, we will prove the operator $\{S(t)\}_{t\ge 0}$ has a $(V\times V,V\times V)-$global attractor, then by using above Lemma 4.2, we get the attractor is a $(V\times V,H^{2}\times H^{2})-$global attractor. Let

$$\begin{aligned} B_{1}=\{u,b\in V:||\nabla u||^{2}+||\nabla b||^{2}\le C\} \end{aligned}$$

and

$$\begin{aligned} B_{2}=\{u,b\in D(A):||Au||^{2}+||Ab||^{2}\le C\}. \end{aligned}$$

By above Lemma 3.2, we deduce that $B_{1}$ is bounded absorbing set of $\{S(t)\}_{t\ge 0}$ in the space $(V\times V,V\times V)$. By above Lemma 3.5, we get that $B_{2}$ is bounded absorbing set of $\{S(t)\}_{t\ge 0}$ in the space $(V\times V,H^{2}\times H^{2})$. By Lemma 3.5, the $\{S(t)\}_{t\ge 0}$ is $(V\times V,V\times V)$-asymptotically compact. Inspired by [13,14,15], we get a $(V\times V,V\times V)$-global attractor $\mathcal {A}$. Finally, we will show $\{S(t)\}_{t\ge 0}$ is $(V\times V,H^{2}\times H^{2})$-asymptotically compact. We need the following lemma.

Lemma 4.3

Let $(u_{0},b_{0})\in V\times V$ and $f_{1},f_{2}\in H$ for $4\le \beta <5$ with any $\alpha >0$. The dynamical system $\{S(t)\}_{t\ge 0}$ is $(V\times V,H^{2}\times H^{2})$-asymptotically compact.

Proof

Assume that $(u_{0n},b_{0n})$ is a bounded in $V\times V$ and $t_{n}\rightarrow \infty$. We will show $\{S(t_{n})(u_{0n},b_{0n})\}$ has a convergent subsequence in $H^{2}\times H^{2}$. Let

$$\begin{aligned} (u_{n}(t),b_{n}(t))=S(t)(u_{0n},b_{0n}),~~~~(\bar{u}_{n}(t_{n}),\bar{b}_{n}(t_{n})) =(\frac{\partial u_{n}}{\partial t}|_{t=t_{n}},\frac{\partial b_{n}}{\partial t}|_{t=t_{n}}). \end{aligned}$$

For the first equation and the second equation of (2.2), we get

$$\begin{aligned} Au_{n}(t_{n})&=f_{1}-\bar{u}_{n}(t_{n})-B(u_{n}(t_{n}),u_{n}(t_{n})) +B(b_{n}(t_{n}),b_{n}(t_{n}))-g(u_{n}(t_{n})),\nonumber \\ Ab_{n}(t_{n})&=f_{2}-\bar{b}_{n}(t_{n})-B(u_{n}(t_{n}),b_{n}(t_{n})) +B(b_{n}(t_{n}),u_{n}(t_{n})). \end{aligned}$$

By Lemma 3.5 and Lemma 3.6, then there exists a positive constant $T>0$ such that for every $t\ge T$,

$$\begin{aligned} ||\nabla \frac{\partial u_{n}}{\partial t}(t)||+||\nabla \frac{\partial b_{n}}{\partial t}(t)||\le C,~~ ||Au_{n}(t)||+||Ab_{n}(t)||\le C. \end{aligned}$$

(4.9)

When $t_{n}\rightarrow \infty$, there exists a $N>0$ such that $t_{n}\ge T$ for every $n\ge N$. Applying (4.9), we deduce for $n\ge N$,

$$\begin{aligned} ||\nabla \bar{u}_{n}(t_{n})||+||\nabla \bar{b}_{n}(t_{n})||\le C,~~ ||Au_{n}(t_{n})||+||Ab_{n}(t_{n})||\le C. \end{aligned}$$

(4.10)

Applying the compactness of embedding $V\hookrightarrow H$ and $D(A)\hookrightarrow V$ and (4.10), then there exist $(\bar{u},\bar{b})\in V\times V$ and $(\hat{u},\hat{b})\in D(A)\times D(A)$ such that

$$\begin{aligned} u_{n}(t_{n})\rightarrow \hat{u}~~\mathrm{strongly~~in}~~V, \end{aligned}$$

(4.11)

$$\begin{aligned} b_{n}(t_{n})\rightarrow \hat{b}~~\mathrm{strongly~~in}~~V, \end{aligned}$$

(4.12)

$$\begin{aligned} \bar{u}_{n}(t_{n})\rightarrow \bar{u}~~\mathrm{strongly~~in}~~H, \end{aligned}$$

(4.13)

$$\begin{aligned} \bar{b}_{n}(t_{n})\rightarrow \bar{b}~~\mathrm{strongly~~in}~~H. \end{aligned}$$

(4.14)

By (4.10) and $H^{2}\hookrightarrow L^{\infty }$, we get

$$\begin{aligned} ||u_{n}(t_{n})||_{\infty }+||b_{n}(t_{n})||_{\infty }\le C,~~~\forall n\ge N. \end{aligned}$$

(4.15)

Inspired by [13, 14], applying (4.11), we get

$$\begin{aligned} ||F(u_{n}(t_{n}))-F(\hat{u})||^{2}\le C||u_{n}(t_{n})-\hat{u}||^{2}\rightarrow 0, \mathrm{as} n\rightarrow \infty . \end{aligned}$$

Hence,

$$\begin{aligned} g(u_{n}(t_{n}))\rightarrow g(\hat{u})~~\mathrm{strongly~~in}~~H. \end{aligned}$$

(4.16)

Then, by Sobolev inequality, we have

$$\begin{aligned}&||B(u_{n}(t_{n}),u_{n}(t_{n}))-B(\hat{u},\hat{u})||^{2}\nonumber \\&\quad \le C(||(u_{n}(t_{n})\cdot \nabla )(u_{n}(t_{n})-\hat{u})||^{2} +||(u_{n}(t_{n})-\hat{u})\cdot \nabla \hat{u}||^{2})\nonumber \\&\quad \le C(||\nabla u_{n}(t_{n})||^{2}||\nabla (u_{n}(t_{n})-\hat{u})||||A(u_{n}(t_{n})-\hat{u})|| +||\nabla (u_{n}(t_{n})-\hat{u})||^{2}||\nabla \hat{u}||||A\hat{u}||)\nonumber \\&\quad \rightarrow 0,~~~~\mathrm{as}~~n\rightarrow \infty . \end{aligned}$$

(4.17)

Similarly, we have

$$\begin{aligned}&||B(b_{n}(t_{n}),b_{n}(t_{n}))-B(\hat{b},\hat{b})||^{2}\nonumber \\&\le C(||(b_{n}(t_{n})\cdot \nabla )(b_{n}(t_{n})-\hat{b})||^{2} +||(b_{n}(t_{n})-\hat{b})\cdot \nabla \hat{b}||^{2})\nonumber \\&\le C(||\nabla b_{n}(t_{n})||^{2}||\nabla (b_{n}(t_{n})-\hat{b})||||A(b_{n}(t_{n})-\hat{b})|| +||\nabla (b_{n}(t_{n})-\hat{b})||^{2}||\nabla \hat{b}||||A\hat{b}||)\nonumber \\&\rightarrow 0,~~~~\mathrm{as}~~n\rightarrow \infty ,\nonumber \\&||B(u_{n}(t_{n}),b_{n}(t_{n}))-B(\hat{u},\hat{b})||^{2}\end{aligned}$$

(4.18)

$$\begin{aligned}&\le C(||(u_{n}(t_{n})\cdot \nabla )(b_{n}(t_{n})-\hat{b})||^{2} +||(u_{n}(t_{n})-\hat{u})\cdot \nabla \hat{b}||^{2})\nonumber \\&\le C(||\nabla u_{n}(t_{n})||^{2}||\nabla (b_{n}(t_{n})-\hat{b})||||A(b_{n}(t_{n})-\hat{b})|| +||\nabla (u_{n}(t_{n})-\hat{u})||^{2}||\nabla \hat{b}||||A\hat{b}||)\nonumber \\&\rightarrow 0,~~~~\mathrm{as}~~n\rightarrow \infty , \end{aligned}$$

(4.19)

and

$$\begin{aligned}&||B(b_{n}(t_{n}),u_{n}(t_{n}))-B(\hat{b},\hat{u})||^{2}\nonumber \\&\quad \le C(||(b_{n}(t_{n})\cdot \nabla )(u_{n}(t_{n})-\hat{u})||^{2} +||(b_{n}(t_{n})-\hat{b})\cdot \nabla \hat{u}||^{2})\nonumber \\&\quad \le C(||\nabla b_{n}(t_{n})||^{2}||\nabla (u_{n}(t_{n})-\hat{u})||||A(u_{n}(t_{n})-\hat{u})|| +||\nabla (b_{n}(t_{n})-\hat{b})||^{2}||\nabla \hat{u}||||A\hat{u}||)\nonumber \\&\quad \rightarrow 0,~~~~\mathrm{as}~~n\rightarrow \infty . \end{aligned}$$

(4.20)

(4.17)–(4.20) imply that

$$\begin{aligned} -B(u_{n}(t_{n}),u_{n}(t_{n})) +B(b_{n}(t_{n}),b_{n}(t_{n}))\rightarrow -B(\hat{u},\hat{u}) +B(\hat{b},\hat{b})~\mathrm{strongly~~in}~H,\end{aligned}$$

(4.21)

$$\begin{aligned} -B(u_{n}(t_{n}),b_{n}(t_{n})) +B(b_{n}(t_{n}),u_{n}(t_{n}))\rightarrow -B(\hat{u},\hat{b}) +B(\hat{b},\hat{u})~\mathrm{strongly~~in}~H. \end{aligned}$$

(4.22)

Applying (4.13), (4.14), (4.16), (4.21) and (4.22), then we get

$$\begin{aligned} Au_{n}(t_{n})&\rightarrow f_{1}-\bar{u}-B(\hat{u},\hat{u}) +B(\hat{b},\hat{b})-g(\hat{u})~~\mathrm{strongly~~in}~~H,\end{aligned}$$

(4.23)

$$\begin{aligned} Ab_{n}(t_{n})&\rightarrow f_{2}-\bar{b}-B(\hat{u},\hat{b}) +B(\hat{b},\hat{u})~~\mathrm{strongly~~in}~~H, \end{aligned}$$

(4.24)

as $n\rightarrow \infty$. We get $\{S(t)\}_{t\ge 0}$ is $(V\times V,H^{2}\times H^{2})$-asymptotically compact.

Proof of Theorem 2.1

Applying Lemma 3.5, we get $B_{2}=\{u,b\in D(A):||Au||^{2}+||Ab||^{2}\le C\}$ denotes a bounded $(V\times V,H^{2}\times H^{2})-$absorbing set. Next, applying Lemma 4.3, we obtain the $\{S(t)\}_{t\ge 0}$ is $(V\times V,H^{2}\times H^{2})$-asymptotically compact. Finally, by Lemma 4.2, $\mathcal {A}$ is a $(V\times V,H^{2}\times H^{2})-$global attractor.