© Wuhan University 2023

0 Introduction

In this paper, we study a class of three-dimensional Brinkman-Forchheimer equation as follows:

$\{\begin{array}{cc}{u}_t-\gamma \mathrm{\Delta }u+au+b\left|u\right|u+c{\left|u\right|}^{\mathrm{2}}u+\nabla p=f(x),& (x,t)\in \mathrm{\Omega }\times {\mathbb{R}}^{+}\\ \mathrm{d}\mathrm{i}\mathrm{v}u=\mathrm{0},& (x,t)\in \mathrm{\Omega }\times {\mathbb{R}}^{+}\\ {u(x,t)|}_{\partial \mathrm{\Omega }}=\mathrm{0},& t\in {\mathbb{R}}^{+}\\ u(x,\mathrm{0})=u_{\mathrm{0}}(x),& x\in \mathrm{\Omega }\end{array}$(1)

where $\mathrm{\Omega }\subseteq {\mathbb{R}}^{\mathrm{3}}$ is an open set, not necessarily bounded, which is sufficiently regular and satisfies Poincaré's inequality. Here $u=u(x,t)=(u_{\mathrm{1}}(x,t),u_{\mathrm{2}}(x,t),u_{\mathrm{3}}(x,t))$ is the fluid velocity vector, $\gamma$ is the Brinkman coefficient, $a>\mathrm{0}$ is the Darcy coefficient, $b>\mathrm{0},c>\mathrm{0}$ are the Forchheimer coefficients, $p$ is the pressure and $u_{\mathrm{0}}=u_{\mathrm{0}}(x)$ is the initial data.

Brinkman-Forchheimer equation describes the motion of fluid flow in a saturated porous medium^[1,2], and has been studied by many researchers^[3-6]. From the physical viewpoint, Gilver and Altobelli^[7] obtained a determination of effective viscosity for the Brinkman-Forchheimer flows model. Nield^[8] dealt with the momentum equation in a porous medium, involving the fluid mechanics of the interface region between a porous medium and a fluid layer. Vafai and Kim^[9,10] obtained an exact solution to Brinkman-Forchheimer equation by using a generalized momentum equation. From the mathematical viewpoint, the discussion of three-dimensional Brinkman-Forchheimer equations is mainly concerned with the well-posedness, regularity and long-time behavior of solutions. The global attractor is the core concept of infinite dimensional dynamical systems. If a system has a global attractor, the attractor will contain all possible limit states of the solution of the system. For the autonomous Brinkman-Forchheimer equation, Uğurlu^[11] showed the existence of global attractor of the system (1) in $(H_{\mathrm{0}}^{\mathrm{1}}{(\mathrm{\Omega }))}^{\mathrm{3}}$. If the term "$c{\left|u\right|}^{\mathrm{2}}u$" in system (1) is replaced by "$c{\left|u\right|}^{\beta }u$", Ouyang and Yang^[12] proved the existence of global attractor in $(H_{\mathrm{0}}^{\mathrm{1}}{(\mathrm{\Omega }))}^{\mathrm{3}}$ when $\mathrm{1}<\beta <\frac{\mathrm{4}}{\mathrm{3}}$ by condition-(C) method. In Ref.[13], the existence of $D$-pullback attractors for the three-dimensional non-autonomous Brinkman-Forchheimer equation is deduced by establishing the $D$-pullback asymptotical compactness of $\theta$-cocycle. In Ref.[14], Song et al discussed the $L^{\mathrm{2}}$-decay of the weak solution of the Brinkman-Forchheimer equation in three-dimensional full space. In Ref.[15], Qiao et al proved the existence of the global attractor for the strong solution of the Brinkman-Forchheimer equation in a three-dimensional bounded domain. In Ref.[16], Song and Wu discussed a non-autonomous Brinkman-Forchheimer equation with singularly oscillating external force in 3D bounded domains. To the best of our knowledge, there is no discussion of the existence of global attractors in three-dimensional unbounded domains for Brinkman-Forchheimer equation. In this paper, we will discuss the existence of global attractor of system (1) in three-dimensional unbounded domains that satisfy the Poincaré's inequality. We will use the method of uniformly estimating the tail of the solution to obtain the asymptotic compactness of the corresponding solution operator of the equation. This method was first proposed by Wang in Ref.[17].

The structure of this paper is arranged as follows: In Section 1, we give some function space symbols and some inequalities that will be used later. Meanwhile, we provide some uniform estimates of the solution of equation (1), which will be used in the following two sections. In Section 2, we estimate the boundedness of the tail of the solution in $(H_{\mathrm{0}}^{\mathrm{1}}{(\mathrm{\Omega }))}^{\mathrm{3}}$. In Section 3 we prove the asymptotic compactness of the solution in $(H_{\mathrm{0}}^{\mathrm{1}}{(\mathrm{\Omega }))}^{\mathrm{3}}$ and then obtain the existence of global attractor.

1 Preliminaries

Let ${\mathbf{L}}^p(\mathrm{\Omega })=(L^p{(\mathrm{\Omega }))}^{\mathrm{3}},{\mathbf{H}}^{\mathrm{2}}(\mathrm{\Omega })=(H^{\mathrm{2}}{(\mathrm{\Omega }))}^{\mathrm{3}}.$ Throughout this paper, we use ${\Vert \cdot \Vert }_p$ to denote the norm in ${\mathbf{L}}^p(\mathrm{\Omega }).$$C$ stands for a generic positive constant, depending on $\mathrm{\Omega }$ and some constants, but independent of time $t$.

The Hausdorff semidistance in $X$ from set $B_{\mathrm{1}}$ to set $B_{\mathrm{2}}$ is defined as

$\mathrm{d}\mathrm{i}\mathrm{s}{\mathrm{t}}_X(B_{\mathrm{1}},B_{\mathrm{2}})=\underset{b_{\mathrm{1}}\in B_{\mathrm{1}}}{\mathrm{s}\mathrm{u}\mathrm{p}}\underset{b_{\mathrm{2}}\in B_{\mathrm{2}}}{\mathrm{i}\mathrm{n}\mathrm{f}}{\Vert b_{\mathrm{1}}-b_{\mathrm{2}}\Vert }_X.$

We set $E=\{u|u\in (C_{\mathrm{0}}^{\mathrm{\infty }}{(\mathrm{\Omega }))}^{\mathrm{3}},\mathrm{d}\mathrm{i}\mathrm{v}u=\mathrm{0}\},$$H$ is the closure of the set $E$ in $(L^{\mathrm{2}}{(\mathrm{\Omega }))}^{\mathrm{3}}$ topology, and $V$ is the closure of the set $E$ in $(H_{\mathrm{0}}^{\mathrm{1}}{(\mathrm{\Omega }))}^{\mathrm{3}}$ topology. $H^{\text{'}}$ and $V^{\text{'}}$ are the dual spaces of $H$ and $V$, clearly, $V$↪$H\equiv H^{\text{'}}$↪$V^{\text{'}}$, where the injection is dense and continuous. We denote by $(\cdot ,\cdot )$ and $\Vert \cdot \Vert$ the inner product and norm in $H$. That is,

$(u,v)={\displaystyle \sum _{j=\mathrm{1}}^{\mathrm{3}}}{\int }_{\mathrm{\Omega }}{u}_j(x)v_j(x)\mathrm{d}x,\Vert u{\Vert }^{\mathrm{2}}=(u,u),u,v\in H,$

$((\cdot ,\cdot ))$ and ${\Vert \cdot \Vert }_V$ denote the inner product and norm in $V,$ that is,

$((u,v))={\displaystyle \sum _{i,j=\mathrm{1}}^{\mathrm{3}}}{\int }_{\mathrm{\Omega }}\frac{\partial u_j}{\partial x_i}\frac{\partial v_j}{\partial x_i}\mathrm{d}x,u,v\in V,$

and

${\Vert u\Vert }_V^{\mathrm{2}}={\displaystyle \sum _{i=\mathrm{1},j=\mathrm{1}}^{\mathrm{3}}}{\int }_{\mathrm{\Omega }}{\Vert {\partial }_i{u}_j\Vert }^{\mathrm{2}},u\in V.$

We call $u\in L^{\mathrm{\infty }}(\mathrm{0},T;H)\bigcap L^{\mathrm{2}}(\mathrm{0},T;V)\bigcap L^{\mathrm{4}}(\mathrm{0},T;{\mathbf{L}}^{\mathrm{4}}(\mathrm{\Omega }))$ a weak solution of problem (1) on $[\mathrm{0},T]$, if

$\{\begin{array}{c}\frac{\mathrm{d}}{\mathrm{d}t}(u,v)+\gamma ((u,v))+a(u,v)+(b\left|u\right|u,v)+(c{\left|u\right|}^{\mathrm{2}}u,v)=(f,v),\forall v\in V,\forall t>\mathrm{0}\\ u(\mathrm{0})=u_{\mathrm{0}}\end{array}$(2)

The weak form (2) is equivalent to the following functional equation:

$\{\begin{array}{c}\frac{\mathrm{d}u}{\mathrm{d}t}+\gamma Au+au+B(u)=f,\forall t>\mathrm{0}\\ u(\mathrm{0})=u_{\mathrm{0}}\end{array}$(3)

Here $Au=-\tilde{P}\mathrm{\Delta }u$ is the Stokes operator defined as $<Au,v>=((u,v)).$$\tilde{P}$ is the orthogonal projection from ${\mathbf{L}}^{\mathrm{2}}(\mathrm{\Omega })$ to $H$. $F(u)=b\left|u\right|u+c{\left|u\right|}^{\mathrm{2}}u,$$B(u)=\tilde{P}F(u),f\in H.$

Then we introduce some useful inequalities and lemmas.

Ladyzhenskaya's inequality^[18]:

${\Vert u\Vert }_{\mathrm{3}}\le C{\Vert u\Vert }^{\frac{\mathrm{1}}{\mathrm{2}}}{\Vert u\Vert }_{V^{}}^{\frac{\mathrm{1}}{\mathrm{2}}},\forall u\in V$(4)

${\Vert u\Vert }_{\mathrm{4}}\le C{\Vert u\Vert }^{\frac{\mathrm{1}}{\mathrm{4}}}{\Vert u\Vert }_{V^{}}^{\frac{\mathrm{3}}{\mathrm{4}}},\forall u\in V$(5)

Sobolev's inequality^[19]:

${\Vert u\Vert }_{\mathrm{6}}\le C{\Vert u\Vert }_{V^{}},\forall u\in V$(6)

Lemma 1   (Grownwall's inequality) If $y(t),h(t)\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^{\mathrm{1}}([\mathrm{0},T];\mathbb{R})$, and $y(t)$ is an absolutely continuous function on $[\mathrm{0},T]$, and the following inequality holds:

$y^{\text{'}}(t)+ky(t)\le h(t),$

where $k\ge \mathrm{0}$, then

$y^{\text{'}}(t)\le y(t_{\mathrm{0}}){\mathrm{e}}^{-k(t-t_{\mathrm{0}})}+{\int }_{t_{\mathrm{0}}}^t{\mathrm{e}}^{-k(t-s)}h(s)\mathrm{d}s.$

Especially if $h(t)=C$, then

$y(t)\le y(t_{\mathrm{0}}){\mathrm{e}}^{-k(t-t_{\mathrm{0}})}+C{k}^{-\mathrm{1}}.$

Now we give the existence and uniqueness theorem of the strong solution of equation (1).

Theorem 1   Suppose $u_{\mathrm{0}}\in V\bigcap {\mathbf{L}}^{\mathrm{4}}(\mathrm{\Omega })$ and $f\in H$. Then there exists a strong solution of equation (1) satisfying

$u\in L^{\mathrm{\infty }}(\mathrm{0},T;V)\bigcap L^{\mathrm{\infty }}(\mathrm{0},T;{\mathbf{L}}^{\mathrm{4}}(\mathrm{\Omega }))\bigcap L^{\mathrm{2}}(\mathrm{0},T;{\mathbf{H}}^{\mathrm{2}}(\mathrm{\Omega }))$

$\nabla u\left|u\right|\in L^{\mathrm{2}}(\mathrm{0},T;H),u_t\in L^{\mathrm{2}}(\mathrm{0},T;H).$

Proof   The proof of this theorem is similar to the proof of Theorem 3.2 in Ref.[15], so here we omit the proof process.

Then we give some uniform estimates of the solution below.

Proposition 1   Suppose $u_{\mathrm{0}}\in V\bigcap {\mathbf{L}}^{\mathrm{4}}(\mathrm{\Omega })$ and $f\in H$. Then there exists a time $t_{\mathrm{0}}$, a positive constant ${\rho }_{\mathrm{1}}$ such that

$\Vert u(t)\Vert \le {\rho }_{\mathrm{1}},\forall t>t_{\mathrm{0}}.$

Proposition 2   Suppose $u_{\mathrm{0}}\in V\bigcap {\mathbf{L}}^{\mathrm{4}}(\mathrm{\Omega })$ and $f\in H$. Then there exists a time $t_{\mathrm{1}}$, a positive constant ${\rho }_{\mathrm{2}}$ such that

${\Vert \nabla u(t)\Vert }^{\mathrm{2}}+{\Vert u\Vert }_{\mathrm{4}}^{\mathrm{4}}\le {\rho }_{\mathrm{2}},\forall t>t_{\mathrm{1}}.$

Proposition 3   Suppose $u_{\mathrm{0}}\in V\bigcap {\mathbf{L}}^{\mathrm{4}}(\mathrm{\Omega })$ and $f\in H$. Then there exists a time $t_{\mathrm{2}}$, a positive constant ${\rho }_{\mathrm{3}}$ such that

$\Vert u_t(s)\Vert \le {\rho }_{\mathrm{3}},\forall s>t_{\mathrm{2}}.$

The propositions 1, 2, and 3 are proved in Ref.[15], and it is easy to verify that they are still valid in the three-dimensional unbounded domain that satisfies the Poincaré's inequality.

Proposition 4   Suppose $u_{\mathrm{0}}\in V\bigcap {\mathbf{L}}^{\mathrm{4}}(\mathrm{\Omega })$ and $f\in H$. Then there exists a positive constant ${\rho }_{\mathrm{4}}$ such that

$\Vert \mathrm{\Delta }u(t)\Vert \le {\rho }_{\mathrm{4}},\forall t>t_{\mathrm{2}}.$

Proof   Applying Minkowski's inequality, from (1) we have

$\gamma \Vert \mathrm{\Delta }u\Vert \le \Vert u_t\Vert +a\Vert u\Vert +\Vert f\Vert +b\Vert \left|u\right|u\Vert +c\Vert {\left|u\right|}^{\mathrm{2}}u\Vert$(7)

According to Ladyzhenskaya's inequality we have

$b\Vert \left|u\right|u\Vert =b{\Vert u\Vert }_{\mathrm{4}}^{\mathrm{2}}\le C{\Vert u\Vert }^{\frac{\mathrm{1}}{\mathrm{2}}}{\Vert u\Vert }_V^{\frac{\mathrm{3}}{\mathrm{2}}},\forall u\in V$(8)

And by Sobolev's inequality, we get

$c\Vert {\left|u\right|}^{\mathrm{2}}u\Vert =c{\Vert u\Vert }_{\mathrm{6}}^{\mathrm{3}}\le C{\Vert u\Vert }_V^{\mathrm{3}},\forall u\in V$(9)

Substituting (8) and (9) into (7), we get

$\gamma \Vert \mathrm{\Delta }u\Vert \le \Vert u_t\Vert +a\Vert u\Vert +\Vert f\Vert +C{\Vert u\Vert }^{\frac{\mathrm{1}}{\mathrm{2}}}{\Vert u\Vert }_V^{\frac{\mathrm{3}}{\mathrm{2}}}+C{\Vert u\Vert }_V^{\mathrm{3}}.$

So there must exist a positive constant ${\rho }_{\mathrm{4}}$ such that $\Vert \mathrm{\Delta }u(t)\Vert \le {\rho }_{\mathrm{4}},\forall t>t_{\mathrm{2}}.$

Proposition 5   When $t\ge \mathrm{0}$, the map $S(t):V\to V$ is a Lipschitz continuous map on $V.$

Proof   Assuming that $u,v$ are two solutions of equation (1), with initial values $u_{\mathrm{0}}$ and $v_{\mathrm{0}}$, respectively. Let $w=u-v,w_{\mathrm{0}}=u_{\mathrm{0}}-v_{\mathrm{0}}$, then we have

$\frac{\mathrm{d}w}{\mathrm{d}t}+\gamma Aw+aw=-B(u)+B(v)$(10)

Taking the inner product of (10) with $Aw$ in $H$, we get

$\begin{array}{l}\frac{\mathrm{1}}{\mathrm{2}}\frac{\mathrm{d}}{\mathrm{d}t}{\Vert w\Vert }_V^{\mathrm{2}}+\gamma {\Vert Aw\Vert }^{\mathrm{2}}+a{\Vert w\Vert }_V^{\mathrm{2}}=-(F(u)-F(v),Aw)\\ \le \Vert F(u)-F(v)\Vert \Vert Aw\Vert \le \frac{\gamma }{\mathrm{2}}{\Vert Aw\Vert }^{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}\gamma }{\Vert F(u)-F(v)\Vert }^{\mathrm{2}}\end{array}$(11)

Since

$\begin{array}{l}{\Vert F(u)-F(v)\Vert }^{\mathrm{2}}={{\int }_{\mathrm{\Omega }}\left|b\left|u\right|u-b\left|v\right|v+c{\left|u\right|}^{\mathrm{2}}u-c{\left|v\right|}^{\mathrm{2}}v\right|}^{\mathrm{2}}\mathrm{d}x\\ \le \mathrm{2}\left({\int }_{\mathrm{\Omega }}{\left|c{\left|u\right|}^{\mathrm{2}}u-c{\left|v\right|}^{\mathrm{2}}v\right|}^{\mathrm{2}}\mathrm{d}x+{\int }_{\mathrm{\Omega }}{\left|b\left|u\right|u-b\left|v\right|v\right|}^{\mathrm{2}}\mathrm{d}x\right)\\ \le C\left({\int }_{\mathrm{\Omega }}[{\left|u\right|}^{\mathrm{2}}{\left|w\right|+\left|{\left|u\right|}^{\mathrm{2}}-{\left|v\right|}^{\mathrm{2}}\right|\left|v\right|]}^{\mathrm{2}}\mathrm{d}x+{\int }_{\mathrm{\Omega }}{[\left|u\right|\left|w\right|+\left|\left|u\right|-\left|v\right|\right|\left|v\right|]}^{\mathrm{2}}\mathrm{d}x\right)\\ \le C\left({\int }_{\mathrm{\Omega }}{\left|u\right|}^{\mathrm{4}}{\left|w\right|}^{\mathrm{2}}\mathrm{d}x+{\int }_{\mathrm{\Omega }}{(\left|u\right|+\left|v\right|)}^{\mathrm{2}}{\left|v\right|}^{\mathrm{2}}{\left|w\right|}^{\mathrm{2}}\mathrm{d}x+{\int }_{\mathrm{\Omega }}{\left|u\right|}^{\mathrm{2}}{\left|w\right|}^{\mathrm{2}}\mathrm{d}x+{\int }_{\mathrm{\Omega }}{\left|v\right|}^{\mathrm{2}}{\left|w\right|}^{\mathrm{2}}\mathrm{d}x\right)\left(\mathrm{12}\right)\end{array}$

and in the last step of the above inequality, we used the following simple inequality

$\left|x^p-y^p\right|\le Cp({\left|x\right|}^{p-\mathrm{1}}+{\left|y\right|}^{p-\mathrm{1}})\left|x-y\right|,\forall x,y\ge \mathrm{0}$

So we have

$\begin{array}{l}{\Vert F(u)-F(v)\Vert }^{\mathrm{2}}\le C{\Vert u\Vert }_{\mathrm{6}}^{\mathrm{4}}{\Vert w\Vert }_{\mathrm{6}}^{\mathrm{2}}+C{\Vert \left|u\right|+\left|v\right|\Vert }_{\mathrm{6}}^{\mathrm{2}}{\Vert v\Vert }_{\mathrm{6}}^{\mathrm{2}}{\Vert w\Vert }_{\mathrm{6}}^{\mathrm{2}}+C{\Vert u\Vert }_{\mathrm{4}}^{\mathrm{2}}{\Vert w\Vert }_{\mathrm{4}}^{\mathrm{2}}+C{\Vert v\Vert }_{\mathrm{4}}^{\mathrm{2}}{\Vert w\Vert }_{\mathrm{4}}^{\mathrm{2}}\\ \le C{\Vert u\Vert }_{\mathrm{6}}^{\mathrm{4}}{\Vert w\Vert }_V^{\mathrm{2}}+C{\Vert \left|u\right|+\left|v\right|\Vert }_{\mathrm{6}}^{\mathrm{2}}{\Vert v\Vert }_V^{\mathrm{2}}{\Vert w\Vert }_V^{\mathrm{2}}+C{\Vert u\Vert }_V^{\mathrm{2}}{\Vert w\Vert }_V^{\mathrm{2}}+C{\Vert v\Vert }_V^{\mathrm{2}}{\Vert w\Vert }_V^{\mathrm{2}}\end{array}$(13)

Substituting (13) into (11), we get

$\frac{\mathrm{d}}{\mathrm{d}t}{\Vert w\Vert }_V^{\mathrm{2}}+\gamma {\Vert Aw\Vert }^{\mathrm{2}}+\mathrm{2}a{\Vert w\Vert }_V^{\mathrm{2}}\le C\left({\Vert u\Vert }_{\mathrm{6}}^{\mathrm{4}}+({\Vert u\Vert }_{\mathrm{6}}^{\mathrm{2}}+{\Vert v\Vert }_{\mathrm{6}}^{\mathrm{2}}){\Vert v\Vert }_V^{\mathrm{2}}+{\Vert u\Vert }_V^{\mathrm{2}}+{\Vert v\Vert }_V^{\mathrm{2}}\right){\Vert w\Vert }_V^{\mathrm{2}}$(14)

Applying Gronwall's inequality to (14), we have

${\Vert w(t)\Vert }_V^{\mathrm{2}}\le {\Vert w_{\mathrm{0}}\Vert }_V^{\mathrm{2}}\mathrm{e}\mathrm{x}\mathrm{p}\left\{C{\int }_{\mathrm{0}}^t[{\Vert u\Vert }_{\mathrm{6}}^{\mathrm{4}}+({\Vert u\Vert }_{\mathrm{6}}^{\mathrm{2}}+{\Vert v\Vert }_{\mathrm{6}}^{\mathrm{2}}){\Vert v\Vert }_V^{\mathrm{2}}+{\Vert u\Vert }_V^{\mathrm{2}}+{\Vert v\Vert }_V^{\mathrm{2}}]\mathrm{d}s\right\}$(15)

According to Sobolev's inequality: ${\Vert u\Vert }_{\mathrm{6}}\le C{\Vert u\Vert }_V,{\Vert v\Vert }_{\mathrm{6}}\le C{\Vert v\Vert }_V,$and because of $u,v\in L^{\mathrm{\infty }}(\mathrm{0},T;V)$, it yields

${\int }_{\mathrm{0}}^t[{\Vert u\Vert }_{\mathrm{6}}^{\mathrm{4}}+({\Vert u\Vert }_{\mathrm{6}}^{\mathrm{2}}+{\Vert v\Vert }_{\mathrm{6}}^{\mathrm{2}}){\Vert v\Vert }_V^{\mathrm{2}}+{\Vert u\Vert }_V^{\mathrm{2}}+{\Vert v\Vert }_V^{\mathrm{2}}]\mathrm{d}s<+\mathrm{\infty }.$

The proof is completed.

2 Uniform Estimate on the Tail of the Solution

In this section, we will employ the technique of uniform estimate on the tail of solution to establish the $(H_{\mathrm{0}}^{\mathrm{1}}{(\mathrm{\Omega }))}^{\mathrm{3}}$-asymptotic compactness of the Brinkman-Forchheimer equation in a three-dimensional unbounded domain satisfying Poincaré's inequality.

Given $k>\mathrm{0}$, we denote by ${\mathrm{\Omega }}_k$ the set ${\mathrm{\Omega }}_k=\{x\in \mathrm{\Omega },|x|\le k\}$ and $\mathrm{\Omega }/{\mathrm{\Omega }}_k$ the complement of ${\mathrm{\Omega }}_k$. For our purpose, we choose a cut-off function $\theta$ with two order continuous derivative such that $\mathrm{0}\le \theta (s)\le \mathrm{1}$ and

$\{\begin{array}{cc}\theta \left(s\right)=\mathrm{0},& \mathrm{i}\mathrm{f}\left|s\right|<\mathrm{1}\\ \theta (s)=\mathrm{1},& \mathrm{i}\mathrm{f}\left|s\right|>\mathrm{2}\end{array}$(16)

We have the following lemma.

Lemma 2   Suppose $f\in H$, and $u_{\mathrm{0}}\in B$, which is a bounded set in $V\bigcap {\mathbf{L}}^{\mathrm{4}}(\mathrm{\Omega })$. Then for every $\epsilon \ge \mathrm{0}$, there exist $T^{\text{'}}(\epsilon )>\mathrm{0}$ and $k^{\text{'}}(\epsilon )>\mathrm{0}$ such that ${\int }_{\Omega /{\Omega }_k}|\nabla u(t){|}^{\mathrm{2}}\mathrm{d}x\le \epsilon ,\forall t\ge T^{\text{'}},k\ge k^{\text{'}},$ where $T^{\text{'}}(\epsilon )$ and $k^{\text{'}}(\epsilon )$ depend on $\epsilon$.

Proof   We will use the tail estimation method, which has been used by Wang in Ref.[17] to establish the existence of global attractor for reaction-diffusion equation in unbounded domain. The method has also been used in Ref.[20] to discuss the existence of global attractor for Newton Boussinesq equation in two-dimensional channel.

Multiplying the first equation of (1) by $-{\theta }^{\mathrm{2}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})\mathrm{\Delta }u$, we have

$\begin{array}{ll}& (u_t,-{\theta }^{\mathrm{2}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})\mathrm{\Delta }u)+(-\gamma \mathrm{\Delta }u,-{\theta }^{\mathrm{2}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})\mathrm{\Delta }u)+(au,-{\theta }^{\mathrm{2}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})\mathrm{\Delta }u)+(b|u|u,-{\theta }^{\mathrm{2}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})\mathrm{\Delta }u)\\ & +(c|u{|}^{\mathrm{2}}u,-{\theta }^{\mathrm{2}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})\mathrm{\Delta }u)+(\nabla p,-{\theta }^{\mathrm{2}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})\mathrm{\Delta }u)=(f,-{\theta }^{\mathrm{2}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})\mathrm{\Delta }u)(\mathrm{17})\end{array}$

For the first term on the left-hand side of (17), applying Green's formula, we have

$\begin{array}{l}(u_t,-{\theta }^{\mathrm{2}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})\mathrm{\Delta }u)={\int }_{\Omega }-{\theta }^{\mathrm{2}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})\mathrm{\Delta }u\cdot \frac{\partial u}{\partial t}\mathrm{d}x={\displaystyle \sum _{i=\mathrm{1}}^{\mathrm{3}}}{\int }_{\mathrm{\Omega }}\frac{\partial u}{\partial x_i}\frac{\partial ({\theta }^{\mathrm{2}}(\frac{|x^{\mathrm{2}}|}{k^{\mathrm{2}}})u_t)}{\partial x_i}\mathrm{d}x\\ ={\displaystyle \sum _{i=\mathrm{1}}^{\mathrm{3}}}{\int }_{\mathrm{\Omega }}\frac{\partial u}{\partial x_i}[\mathrm{4}\theta (\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}}){\theta }^{\text{'}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})\frac{x_i}{k^{\mathrm{2}}}{u}_t+{\theta }^{\mathrm{2}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})\frac{{\partial }^{\mathrm{2}}u}{\partial t\partial x_i}]\mathrm{d}x(\mathrm{18})\\ =\frac{\mathrm{1}}{\mathrm{2}}\frac{\mathrm{d}}{\mathrm{d}t}{\int }_{\mathrm{\Omega }}{\theta }^{\mathrm{2}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})|\nabla u{|}^{\mathrm{2}}\mathrm{d}x+\mathrm{4}{\displaystyle \sum _{i=\mathrm{1}}^{\mathrm{3}}}{\int }_{\mathrm{\Omega }}\theta (\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}}){\theta }^{\text{'}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})\frac{\partial u}{\partial x_i}{u}_t\frac{x_i}{k^{\mathrm{2}}}\mathrm{d}x\end{array}$

For the second term on the left-hand side of (17), we obtain

$(-\gamma \mathrm{\Delta }u,-{\theta }^{\mathrm{2}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})\mathrm{\Delta }u)=\gamma {\int }_{\mathrm{\Omega }}{\theta }^{\mathrm{2}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})|\mathrm{\Delta }u{|}^{\mathrm{2}}\mathrm{d}x$(19)

For the third term on the left-hand side of (17), we get

$\begin{array}{l}(au,-{\theta }^{\mathrm{2}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})\mathrm{\Delta }u)=-a{\int }_{\Omega }\Delta u{\theta }^{\mathrm{2}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})u\mathrm{d}x=a{\displaystyle \sum _{i=\mathrm{1}}^{\mathrm{3}}}{\int }_{\mathrm{\Omega }}\frac{\partial u}{\partial x_i}[\frac{\partial u}{\partial x_i}{\theta }^{\mathrm{2}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})+\mathrm{2}\theta (\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}}){\theta }^{\text{'}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})\frac{\mathrm{2}{x}_i}{k^{\mathrm{2}}}u]\mathrm{d}x\\ =a{\displaystyle \sum _{i=\mathrm{1}}^{\mathrm{3}}}{\int }_{\mathrm{\Omega }}{\theta }^{\mathrm{2}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})|\frac{\partial u}{\partial x_i}{|}^{\mathrm{2}}\mathrm{d}x+\mathrm{4}a{\displaystyle \sum _{i=\mathrm{1}}^{\mathrm{3}}}{\int }_{\mathrm{\Omega }}\frac{\partial u}{\partial x_i}\theta (\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}}){\theta }^{\text{'}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})\frac{x_i}{k^{\mathrm{2}}}u\mathrm{d}x\\ =a{\int }_{\mathrm{\Omega }}{\theta }^{\mathrm{2}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})|\nabla u{|}^{\mathrm{2}}\mathrm{d}x+\mathrm{4}a{\displaystyle \sum _{i=\mathrm{1}}^{\mathrm{3}}}{\int }_{\mathrm{\Omega }}\frac{\partial u}{\partial x_i}\theta (\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}}){\theta }^{\text{'}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})\frac{x_i}{k^{\mathrm{2}}}u\mathrm{d}x\end{array}$(20)

Taking (17)-(20) into account, we get

$\begin{array}{l}\frac{\mathrm{1}}{\mathrm{2}}\frac{\mathrm{d}}{\mathrm{d}t}{\int }_{\mathrm{\Omega }}{\theta }^{\mathrm{2}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})|\nabla u{|}^{\mathrm{2}}\mathrm{d}x+a{\int }_{\mathrm{\Omega }}{\theta }^{\mathrm{2}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})|\nabla u{|}^{\mathrm{2}}\mathrm{d}x+\gamma {\int }_{\mathrm{\Omega }}{\theta }^{\mathrm{2}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})|\mathrm{\Delta }u{|}^{\mathrm{2}}\mathrm{d}x\\ =-\mathrm{4}{\displaystyle \sum _{i=\mathrm{1}}^{\mathrm{3}}}{\int }_{\mathrm{\Omega }}\theta (\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}}){\theta }^{\text{'}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})\frac{\partial u}{\partial x_i}{u}_t\frac{x_i}{k^{\mathrm{2}}}\mathrm{d}x-\mathrm{4}a{\displaystyle \sum _{i=\mathrm{1}}^{\mathrm{3}}}{\int }_{\mathrm{\Omega }}\frac{\partial u}{\partial x_i}\theta (\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}}){\theta }^{\text{'}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})\frac{x_i}{k^{\mathrm{2}}}u\mathrm{d}x\\ +b{\int }_{\mathrm{\Omega }}{\theta }^{\mathrm{2}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})\mathrm{\Delta }u|u|u\mathrm{d}x+c{\int }_{\mathrm{\Omega }}{\theta }^{\mathrm{2}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})\mathrm{\Delta }u|u{|}^{\mathrm{2}}u\mathrm{d}x+{\int }_{\mathrm{\Omega }}{\theta }^{\mathrm{2}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})\mathrm{\Delta }u\nabla p\mathrm{d}x-{\int }_{\mathrm{\Omega }}f{\theta }^{\mathrm{2}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})\mathrm{\Delta }u\mathrm{d}x\end{array}$(21)

We now estimate the right-hand side of (21) term by term. Applying Young's inequality and Holder's inequality, we find

$\begin{array}{l}|{\int }_{\mathrm{\Omega }}\theta (\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})\cdot {\theta }^{\text{'}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})\frac{\partial u}{\partial x_i}{u}_t\frac{x_i}{k^{\mathrm{2}}}\mathrm{d}x|=|{\int }_{\mathrm{\Omega }(k\le |x|\le \sqrt[]{\mathrm{2}}k)}\theta (\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})\cdot {\theta }^{\text{'}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})\frac{\partial u}{\partial x_i}{u}_t\frac{x_i}{k^{\mathrm{2}}}\mathrm{d}x|\\ \le \frac{C}{k}{\int }_{\Omega (k\le |x|\le \sqrt[]{\mathrm{2}}k)}|\frac{\partial u}{\partial x_i}||u_t|\mathrm{d}x\le \frac{C}{k}\Vert \nabla u\Vert \Vert u_t\Vert \end{array}$(22)

$\begin{array}{l}|{\int }_{\mathrm{\Omega }}\frac{\partial u}{\partial x_i}\theta (\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}}){\theta }^{\text{'}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})\frac{x_i}{k^{\mathrm{2}}}u\mathrm{d}x|=|{\int }_{\mathrm{\Omega }(k\le |x|\le \sqrt[]{\mathrm{2}}k)}\frac{\partial u}{\partial x_i}\theta (\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}}){\theta }^{\text{'}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})\frac{x_i}{k^{\mathrm{2}}}u\mathrm{d}x|\\ \le \frac{C}{k}{\int }_{\Omega (k\le |x|\le \sqrt[]{\mathrm{2}}k)}|\frac{\partial u}{\partial x_i}||u|\mathrm{d}x\le \frac{C}{k}\Vert \nabla u\Vert \Vert u\Vert \end{array}$(23)

$\begin{array}{l}|b{\int }_{\mathrm{\Omega }}{\theta }^{\mathrm{2}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})\mathrm{\Delta }u|u|u\mathrm{d}x|\le b({\int }_{\mathrm{\Omega }}{\theta }^{\mathrm{2}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})|\mathrm{\Delta }u{|}^{\mathrm{2}}{\mathrm{d}x)}^{\frac{\mathrm{1}}{\mathrm{2}}}({\int }_{\mathrm{\Omega }}{\theta }^{\mathrm{2}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}}){(|u|u)}^{\mathrm{2}}{\mathrm{d}x)}^{\frac{\mathrm{1}}{\mathrm{2}}}\\ \le \frac{\gamma }{\mathrm{4}}{\int }_{\mathrm{\Omega }}{\theta }^{\mathrm{2}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})|\mathrm{\Delta }u{|}^{\mathrm{2}}\mathrm{d}x+C{\int }_{\mathrm{\Omega }}{\theta }^{\mathrm{2}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}}){(|u|u)}^{\mathrm{2}}\mathrm{d}x.\end{array}$

Since ${\int }_{\mathrm{\Omega }}{\theta }^{\mathrm{2}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}}){(|u|u)}^{\mathrm{2}}\mathrm{d}x\le {\int }_{\mathrm{\Omega }(|x|\ge k)}{(|u|u)}^{\mathrm{2}}\mathrm{d}x,$ so we have

$|b{\int }_{\mathrm{\Omega }}{\theta }^{\mathrm{2}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})\mathrm{\Delta }u|u|u\mathrm{d}x|\le \frac{\gamma }{\mathrm{4}}{\int }_{\mathrm{\Omega }}{\theta }^{\mathrm{2}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})|\mathrm{\Delta }u{|}^{\mathrm{2}}\mathrm{d}x+C{\int }_{\mathrm{\Omega }(|x|\ge k)}{(|u|u)}^{\mathrm{2}}\mathrm{d}x$(24)

Similar with (24), we have

$|c{\int }_{\mathrm{\Omega }}{\theta }^{\mathrm{2}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})\mathrm{\Delta }u|u{|}^{\mathrm{2}}u\mathrm{d}x|\le \frac{\gamma }{\mathrm{4}}{\int }_{\mathrm{\Omega }}{\theta }^{\mathrm{2}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})|\mathrm{\Delta }u{|}^{\mathrm{2}}\mathrm{d}x+C{\int }_{\mathrm{\Omega }(|x|\ge k)}(|u{|}^{\mathrm{2}}{u)}^{\mathrm{2}}\mathrm{d}x$(25)

For the sixth term and the fifth term on the right-hand side of (21) we have

$\begin{array}{l}|{\int }_{\mathrm{\Omega }}f{\theta }^{\mathrm{2}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})\mathrm{\Delta }u\mathrm{d}x|=|{\int }_{\mathrm{\Omega }(|x|\ge k)}f{\theta }^{\mathrm{2}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})\mathrm{\Delta }u\mathrm{d}x|\\ \le {({\int }_{\mathrm{\Omega }(|x|\ge k)}{f}^{\mathrm{2}}\mathrm{d}x)}^{\frac{\mathrm{1}}{\mathrm{2}}}({\int }_{\mathrm{\Omega }(|x|\ge k)}{\theta }^{\mathrm{4}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})|\mathrm{\Delta }u{|}^{\mathrm{2}}{\mathrm{d}x)}^{\frac{\mathrm{1}}{\mathrm{2}}}\le C{\int }_{\Omega (|x|\ge k)}|f{|}^{\mathrm{2}}\mathrm{d}x+\frac{\gamma }{\mathrm{4}}{\int }_{\mathrm{\Omega }}{\theta }^{\mathrm{2}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})|\mathrm{\Delta }u{|}^{\mathrm{2}}\mathrm{d}x\end{array}$(26)

$|{\int }_{\mathrm{\Omega }}{\theta }^{\mathrm{2}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})\mathrm{\Delta }u\nabla p\mathrm{d}x|\le \frac{\gamma }{\mathrm{4}}{\int }_{\mathrm{\Omega }}{\theta }^{\mathrm{2}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})|\mathrm{\Delta }u{|}^{\mathrm{2}}\mathrm{d}x+C{\int }_{\mathrm{\Omega }}{\theta }^{\mathrm{2}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})|\nabla p{|}^{\mathrm{2}}\mathrm{d}x\le \frac{\gamma }{\mathrm{4}}{\int }_{\mathrm{\Omega }}{\theta }^{\mathrm{2}}(\frac{|x{|}^{\mathrm{2}}}{k^{\mathrm{2}}})|\mathrm{\Delta }u{|}^{\mathrm{2}}\mathrm{d}x+C{\int }_{\Omega (|x|\ge k)}|\nabla p{|}^{\mathrm{2}}\mathrm{d}x$

And because

$\begin{array}{l}{\int }_{\Omega (|x|\ge k)}|\nabla p{|}^{\mathrm{2}}\mathrm{d}x={{\int }_{\Omega (|x|\ge k)}\left|f-u_t+\gamma \Delta u-au-b|u|u-c|u{|}^{\mathrm{2}}u\right|}^{\mathrm{2}}\mathrm{d}x\\ \le C({\int }_{\Omega (|x|\ge k)}|f{|}^{\mathrm{2}}\mathrm{d}x+{\int }_{\Omega (|x|\ge k)}|u_t{|}^{\mathrm{2}}\mathrm{d}x+{\int }_{\Omega (|x|\ge k)}|\mathrm{\Delta }u{|}^{\mathrm{2}}\mathrm{d}x\\ +{\int }_{\Omega (|x|\ge k)}|u{|}^{\mathrm{2}}\mathrm{d}x+{{\int }_{\mathrm{\Omega }(|x|\ge k)}\left||u|u\right|}^{\mathrm{2}}\mathrm{d}x+{{\int }_{\mathrm{\Omega }(|x|\ge k)}\left||u{|}^{\mathrm{2}}u\right|}^{\mathrm{2}}\mathrm{d}x)\end{array}$