© Wuhan University 2025

0 Introduction

The Navier-Stokes equations (NSE) are important in fluid mechanics and turbulence. In the last decades, the research on the asymptotic properties of the solution for NSE has attracted the attention of many scholars^[1-7]. Especially in the past years, the NSE with nonlinear damping and delay has been studied^[8-16], where the damping comes from the resistance of the flow or the friction effect. The delays denote some type of external forces that can be applied to control one system, and these forces can be regarded as the present state of the system or its history.

In recent years, the research on 2D g-Navier-Stokes equations (gNSE) has been paid more attention by scholars. It is derived from 3D NSE by the vertical mean operator in Ref. [17], and its form is as follows:

$\{\begin{array}{l}\frac{\partial u}{\partial t}-\mu \mathrm{\Delta }u+(u\cdot \nabla u)+\nabla p=f\begin{array}{ccc}& \mathrm{i}\mathrm{n}& \mathrm{\Omega }\end{array},\\ \nabla \cdot gu=\mathrm{0}\begin{array}{ccc}& \mathrm{i}\mathrm{n}& \mathrm{\Omega }\end{array},\end{array}$(1)

where $g=g(x_{\mathrm{1}},x_{\mathrm{2}})$ is a suitable smooth real-valued function defined on $(x_{\mathrm{1}},x_{\mathrm{2}})\in \mathrm{\Omega }$ and $\mathrm{\Omega }$ is the bounded domain in ${\mathbb{R}}^{\mathrm{2}}$. We study the 2D gNSE as a small perturbation of the usual NSE, so we want to understand the NSE completely by studying the 2D gNSE systematically. Therefore, the research on the gNSE has a theoretical basis and practical significance.

There are many researches on g-Navier-Stokes equations recently^[18-26]. In Ref. [18], the well-posedness of solutions for gNSE was proved on ${\mathbb{R}}^{\mathrm{2}}$ for $n=\mathrm{2,3}$. In Ref. [19], Roh obtained the existence of the global attractors of gNSE and proved that the semiflows were robust to g. Moreover, the existence of global solutions and the global attractor of gNSE were proved, and the dimension of the global attractor was estimated in Ref. [20]. Also, the global attractor of gNSE with linear dampness on ${\mathbb{R}}^{\mathrm{2}}$ were proved, and the estimation of dimensions was also obtained in Ref. [21]. We investigated the existence of a pullback attractor for the 2D non-autonomous gNSE on the bounded domains in Ref. [22]. Quyet proved the existence of a pullback attractor in V_g for the process in Ref. [23]. Recently, we have discussed the uniform attractor and pullback attractor of gNSE with damping and time delay in Refs. [24-26], and obtained the uniform attractor and pullback attractor using the method of the energy equation and the pullback condition, respectively.

It is well-known that the research of global attractors is an important problem in infinite-dimensional dynamical systems (IDDS). In addition, the dimension is a very important property of the global attractor. However, as far as we know, the global attractor of 2D autonomous gNSE with damping and delay has been studied rarely. Therefore, in this article, we research global attractor of the 2D autonomous gNSE, which has nonlinear damping and time delay on some bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^{\mathrm{2}}$, and its usual form is as follows:

$\{\begin{array}{l}\frac{\partial u}{\partial t}-\nu \mathrm{\Delta }u+(u\cdot \nabla u)+c|u{|}^{\beta -\mathrm{1}}u+\nabla p=f(x,t)+h(t,u_t)\\ \begin{array}{ccc}& \mathrm{i}\mathrm{n}& \mathrm{\Omega }\end{array}\times (\mathrm{0},\mathrm{\infty }),\\ \nabla \cdot gu=\mathrm{0}\begin{array}{ccc}& \mathrm{i}\mathrm{n}& \mathrm{\Omega }\end{array}\times (\mathrm{0},\mathrm{\infty }),\\ u(x,t)=\mathrm{0}\begin{array}{ccc}& \mathrm{o}\mathrm{n}& \partial \mathrm{\Omega }\end{array},\\ u(x,\mathrm{0})=u_{\mathrm{0}}(x)\begin{array}{ccc}& \mathrm{i}\mathrm{n}& \mathrm{\Omega }\end{array},\end{array}$(2)

where $p(x,t)\in \mathbb{R}$ denotes the pressure and $u(x,t)\in {\mathbb{R}}^{\mathrm{2}}$ as the velocity, $\nu >\mathrm{0}$ and $c|u{|}^{\beta -\mathrm{1}}u$ are nonlinear dampings, $\beta \ge \mathrm{1}$ and $c>\mathrm{0}$ are constants, $\mathrm{0}<m_{\mathrm{0}}\le g=g(x_{\mathrm{1}},x_{\mathrm{2}})\le M_{\mathrm{0}}$, where $g=g(x_{\mathrm{1}},x_{\mathrm{2}})$ is a real-valued smooth function. $f=f(x,t)$ is the external force, $h=h(t,u_t)$ is another external force term with time delay, $u_t$ is the function which can be defined by $u_t(\theta )=u(t+\theta )$. $\forall \theta \in (-r,\mathrm{0}),r>\mathrm{0}$ is a constant.

Definition 1^[26] Let $C_{H_g}=C^{\mathrm{0}}([-h,\mathrm{0}];H_g)$, $C_{V_g}=C^{\mathrm{0}}([-h,\mathrm{0}];V_g)$, $h:\mathbb{R}\times C_{H_g}\to (L^{\mathrm{2}}{(\mathrm{\Omega }))}^{\mathrm{2}}$ satisfies the following assumptions:

(I) $h(\mathrm{0})=\mathrm{0}$;

(II) there exists $L_g>\mathrm{0}$, such that $\forall \xi ,\eta \in C_{H_g},|h(\xi )-h(\eta )|\le L_g||\xi -\eta |{|}_{C_{H_g}}$;

(III) there exists $C_g>\mathrm{0}$, such that $\forall t\in [\mathrm{0},T],u,v\in C$

$(-r,T;H_g)$

${\int }_{\mathrm{0}}^t|h(u_s)-h(v_s)|\mathrm{d}s\le C_g{\int }_{-r}^t|u(s)-v(s){|}^{\mathrm{2}}\mathrm{d}s.$

Definition 2^[26] Let $u_{\mathrm{0}}\in H_g,f\in L^{\mathrm{2}}(\tau ,T;V_g^{\text{'}})$, $h:\mathbb{R}\times C_{H_g}\to (L^{\mathrm{2}}{(\mathrm{\Omega }))}^{\mathrm{2}}$ satisfies (I)-(III).

For any $\tau \in \mathbb{R},u\in L^{\mathrm{\infty }}(\tau ,T;V_g)\bigcap L^{\mathrm{2}}(\tau ,T;V_g)\bigcap L^{\beta +\mathrm{1}}$

$(\tau ,T;L^{\beta +\mathrm{1}}(\tau ,T;L^{\beta +\mathrm{1}}(\mathrm{\Omega }))$, $\forall T>\tau$ is called a weak solution of (2) if it fulfills

$\begin{array}{l}\frac{\mathrm{d}}{\mathrm{d}t}u(t)+\nu A_{g}u(t)+B(u(t))+c|u{|}^{\beta -\mathrm{1}}u+\nu R(u(t))\\ =f(x,t)+h(t,u_t)\begin{array}{cc}\mathrm{o}\mathrm{n}& D\text{'}(\tau ,+\mathrm{\infty };V_g^{\text{'}}),[u(\tau )=u\mathrm{0}].\end{array}\end{array}$

Theorem 1^[26] Let $\beta \ge \mathrm{1},f\in L^{\mathrm{2}}(\tau ,T;V_g^{\text{'}})$, $h:\mathbb{R}\times C_{H_g}\to (L^{\mathrm{2}}{(\mathrm{\Omega }))}^{\mathrm{2}}$ satisfies (I)-(III), then for every $u_{\tau }\in V_g$, equation (2) has the only weak solution $u(t)=u(t;\tau ,u_{\tau })\in$

$L^{\mathrm{\infty }}(\tau ,T;V_g)\bigcap L^{\mathrm{2}}(\tau ,T;V_g)\bigcap L^{\beta +\mathrm{1}}(\tau ,T;L^{\beta +\mathrm{1}}(\mathrm{\Omega }))$, and $u(t)$ continuously depends on the initial value in $V_g$.

The main results of this article are as follows:

Lemma 1   For any $\varphi \in H_g,f\in (L^{\mathrm{2}}{(\mathrm{\Omega }))}^{\mathrm{2}}$, let $h:C_{H_g}\to (L^{\mathrm{2}}{(\mathrm{\Omega }))}^{\mathrm{2}}$ and satisfy (I)-(III), then $S_{\tau }=\{\varphi \in C_{H_g}:||\varphi |{|}_{C_{H_g}}\le \tau k=\frac{\tau |f|}{\lambda -L_g}\}$ is positive invariant set of $S(t)$, where $\tau$ is a constant and $\tau >\mathrm{1}$.

Lemma 2   For any $\varphi \in H_g,f\in (L^{\mathrm{2}}{(\mathrm{\Omega }))}^{\mathrm{2}}$, let $h:C_{H_g}\to (L^{\mathrm{2}}{(\mathrm{\Omega }))}^{\mathrm{2}}$ and satisfy (I)-(III), then $S=\{\varphi \in C_{H_g}:||\varphi |{|}_{C_{H_g}}\le k=\frac{|f|}{\lambda -L_g}\}$ is a global absorbed set of $S(t)$ in $C_{H_g}$.

Lemma 3   For any $\varphi \in H_g,f\in (L^{\mathrm{2}}{(\mathrm{\Omega }))}^{\mathrm{2}}$, let $h:C_{H_g}\to (L^{\mathrm{2}}{(\mathrm{\Omega }))}^{\mathrm{2}}$ and satisfy (I)-(III), and $|\nabla g{|}_{\mathrm{\infty }}^{\mathrm{2}}<\frac{{\lambda }_{\mathrm{1}}{m}_{\mathrm{0}}^{\mathrm{2}}}{\mathrm{4}}$, then $\mathrm{\Gamma }=\{\varphi \in C_{V_g}:||\varphi |{|}_{C_{V_g}}\le \rho \}$ is a global absorbed set of $S(t)$ in $C_{V_g}$, where $\rho$ is a given constant.

Theorem 2   For any $\varphi \in H_g,f\in (L^{\mathrm{2}}{(\mathrm{\Omega }))}^{\mathrm{2}}$, let $h:C_{H_g}\to (L^{\mathrm{2}}{(\mathrm{\Omega }))}^{\mathrm{2}}$ and satisfy (I)-(III), then equation (2) possesses a global attractor $A_{\mathrm{0}}$ in $C_{H_g}$, which is compact and connected in $C_{H_g}$, and it attracts any bounded set in $C_{H_g}$.

Theorem 3   Let $f\in H_g$, then the semigroup $S(t)$ corresponding to equation (2) has a global attractor $A_{\mathrm{1}}$ in $C_{V_g}$ satisfying

1) $A_{\mathrm{1}}$ is compact in $C_{V_g}$;

2) $S(t)A_{\mathrm{1}}=A_{\mathrm{1}}$;

3) For any bounded subset $B_{\mathrm{1}}$ of $C_{V_g}$,

$\underset{t\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\mathrm{d}\mathrm{i}\mathrm{s}{t}_{V_g}(s(t)B_{\mathrm{1}},A_{\mathrm{1}})=\underset{t\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\underset{\xi \to B_{\mathrm{1}}}{\mathrm{s}\mathrm{u}\mathrm{p}}\mathrm{d}\mathrm{i}\mathrm{s}{t}_{V_g}(S(t)\xi ,A_{\mathrm{1}})=\mathrm{0}$

Lemma 4   Let $f\in H_g$ and $f$ be sufficiently small, B is bounded subset in $H_g$. Let $u(t)=S(t)u_{\mathrm{0}}$ be a corresponding solution to equation (2), $u_{\mathrm{0}}\in B$, then there exist $t_{\mathrm{0}}$ and constant $M>\mathrm{0}$, such that $||u(t)||=||S(t)u_{\mathrm{0}}||\le M,\forall u_{\mathrm{0}}\in B\subset H_g,\forall t\ge t_{\mathrm{0}}$.

Theorem 4   Let $f\in H_g$ and $f$ be sufficiently small, then $A_{\mathrm{0}}=A_{\mathrm{1}}$, where $A_{\mathrm{0}}$ and $A_{\mathrm{1}}$ denote global attractors in $C_{H_g}$ and $C_{V_g}$, respectively.

This article is organized as follows. In Section 1, we give some results on the classical theory of the global attractor. In Section 2, we obtain the global attractor of 2D gNSE with damping and time delay. In Section 3, we give some relevant conclusions.

1 Preliminaries

We define $L^{\mathrm{2}}(g)=(L^{\mathrm{2}}{(\mathrm{\Omega }))}^{\mathrm{2}}$ and $H_{\mathrm{0}}^{\mathrm{1}}(g)=(H_{\mathrm{0}}^{\mathrm{1}}{(\mathrm{\Omega }))}^{\mathrm{2}}$, the inner product of $L^{\mathrm{2}}(g)$ is $(u,v)=\underset{\mathrm{\Omega }}{\int }u\cdot vg\mathrm{d}x$ and inner product of $H_{\mathrm{0}}^{\mathrm{1}}(g)$ is $((u,v))=\underset{\mathrm{\Omega }}{\int }{\displaystyle \sum _{j=\mathrm{1}}^{\mathrm{2}}}\nabla u_j\cdot \nabla v_{j}g\mathrm{d}x$, corresponding norm is $|\cdot |={(\cdot ,\cdot )}^{\mathrm{1}/\mathrm{2}}$ and $||\cdot ||=({(\cdot ,\cdot ))}^{\mathrm{1}/\mathrm{2}}$ respectively.

Let $M=\{v\in (D{(\mathrm{\Omega }))}^{\mathrm{2}}:\nabla \cdot gv=\mathrm{0}\begin{array}{ccc}& \mathrm{i}\mathrm{n}& \mathrm{\Omega }\}\end{array}$ and $D(\mathrm{\Omega })$ is the space of $C^{\mathrm{\infty }}$ functions that have compact support contained in $\mathrm{\Omega }\subset {\mathbb{R}}^{\mathrm{2}}$. $H_g$ is the closure of M in $L{}_{}{}^{\mathrm{2}}(g)$ which is endowed with the inner product and norm of $L{}_{}{}^{\mathrm{2}}(g)$, $V_g$ is the closure of M in $H_{\mathrm{0}}^{\mathrm{1}}(g)$ endowed with the inner product and norm of $H_{\mathrm{0}}^{\mathrm{1}}(g)$.

Let ${\lambda }_{\mathrm{1}}>\mathrm{0}$, we have

$\underset{\mathrm{\Omega }}{\int }{\varphi }^{\mathrm{2}}g\mathrm{d}x\le \frac{\mathrm{1}}{{\lambda }_{\mathrm{1}}}\underset{\mathrm{\Omega }}{\int }|\nabla \varphi {|}^{\mathrm{2}}g\mathrm{d}x,\begin{array}{cc}& \forall \varphi \in H_{\mathrm{0}}^{\mathrm{1}}\end{array}(\mathrm{\Omega }),$(3)

then

$|u{|}^{\mathrm{2}}\le \frac{\mathrm{1}}{{\lambda }_{\mathrm{1}}}||u|{|}^{\mathrm{2}},\begin{array}{cc}& \forall u\in V_g\end{array}.$(4)

The g-Laplacian operator is defined as follows:

$-{\mathrm{\Delta }}_{g}u=-\frac{(\nabla \cdot g\nabla )u}{g}=-\mathrm{\Delta }u-\frac{\mathrm{1}}{g}\nabla g\cdot \nabla u.$

The first formula in (2) can be expressed as follows:

$\frac{\partial u}{\partial t}-\nu {\mathrm{\Delta }}_{g}u+\nu \frac{\nabla g}{g}\cdot \nabla u+(u,\nabla )u+c|u{|}^{\beta -\mathrm{1}}u+\nabla p=f+h(t,u_t).$(5)

The g-orthogonal projection is defined by $P_g:L^{\mathrm{2}}(g)\to H_g$ and g-Stokes operator is defined by $A_{g}u=-P_g(\frac{\mathrm{1}}{g}(\nabla \cdot (g\nabla u)))$. Applying the projection $P_g$ to (2), for $\forall v\in V_g,\forall t>\mathrm{0}$, we obtain

$\begin{array}{l}\frac{\mathrm{d}}{\mathrm{d}t}(u,v)+\nu ((u,v))+b_g(u,u,v)+c(|u{|}^{\beta -\mathrm{1}}u,v)+\nu (Ru,v)\\ =〈f,v〉+〈h(t,u_t),v〉,\end{array}$(6)

$u(\mathrm{0})=u_{\mathrm{0}},$(7)

where $b_g:V_g\times V_g\times V_g\to \mathbb{R}$ and $b_g(u,v,w)={\displaystyle \sum _{i,j=\mathrm{1}}^{\mathrm{2}}}\int u_i\frac{\partial v_j}{\partial x}{w}_{j}g\mathrm{d}x$, $Ru=P_g[\frac{\mathrm{1}}{g}(\nabla g\cdot \nabla )u]$, $\forall u\in V_g$. We define $G(u)=P_{g}F(u)$ and $F(u)=c|u{|}^{\beta -\mathrm{1}}u$, then the formulations (6) and (7) are equivalent to the following equations

$\frac{\mathrm{d}u}{\mathrm{d}t}+\nu A_{g}u+Bu+G(u)+\nu Ru=f+h,$(8)

$u(\mathrm{0})=u_{\mathrm{0}},$(9)

where $A_g:V_g\to V_g^{\text{'}}$, for $\forall u,v\in V_g$, we have $〈A_{g}u,v〉=((u,v))$. $B(u)=B(u,u)=P_g(u\cdot \nabla )u$ is a bilinear operator, and $B:V_g\times V_g\to V_g^{\text{'}}$ with $〈B(u,v),w〉=b_g(u,v,w),$$\forall u,v,w$

$\in V_g$

For any $u,v\in D(A_g)$, we have $|B(u,v)|\le C|u{|}^{\mathrm{1}/\mathrm{2}}|A_{g}u{|}^{\mathrm{1}/\mathrm{2}}||v||$, where C denotes positive constant. From Refs. [10,17,19, 27], we have the following inequality:

$|\phi {|}_{L^{\mathrm{\infty }}{(\mathrm{\Omega })}^{\mathrm{2}}}\le C||\phi ||{(\mathrm{1}+\mathrm{l}\mathrm{n}\frac{|A_g\phi {|}^{\mathrm{2}}}{{\lambda }_{\mathrm{1}}||\phi |{|}^{\mathrm{2}}})}^{\mathrm{1}/\mathrm{2}},\begin{array}{cc}& \forall \phi \in D(A_g),\end{array}$(10)

$|B(u,v)|\le |(u\cdot \nabla )v|\le |u{|}_{L^{\mathrm{\infty }}(\mathrm{\Omega })}|\nabla v|,$(11)

$|B(u,v)|\le C||u||||v||{(\mathrm{1}+\mathrm{l}\mathrm{n}\frac{|A_{g}u{|}^{\mathrm{2}}}{{\lambda }_{\mathrm{1}}||u|{|}^{\mathrm{2}}})}^{\mathrm{1}/\mathrm{2}},$(12)

$||B(u)|{|}_{V_g^{\text{'}}}\le c|u|||u||,\begin{array}{c}\left|\right|Ru|{|}_{V_g^{\text{'}}}\end{array}\le \frac{|\nabla g{|}_{\mathrm{\infty }}}{m_{\mathrm{0}}{\lambda }_{\mathrm{1}}^{\mathrm{1}/\mathrm{2}}}||u||,\begin{array}{cc}& \forall u\in V_g\end{array}.$(13)

Definition 3^[28] Let M be a complete metric space, a parameter family $S(t),t\ge \mathrm{0}$ of maps $S(t):M\to M,t\ge \mathrm{0}$ is called $C^{\mathrm{0}}$ semigroup if

1) $S(\mathrm{0})$ is the identity map on M,

2) $S(t+s)=S(t)S(s)$ for all $t,s\ge \mathrm{0}$,

3) the function $S(t)x$ is continuous at each point $(t,x)\in [\mathrm{0},\mathrm{\infty })\times M$.

Definition 4^[28] Let $S(t),t\ge \mathrm{0}$ be a $C^{\mathrm{0}}$ semigroup in a complete metric space M.

A subset $B_{\mathrm{0}}$ of M is called an absorbing set in M, if for any bounded subset B of M, there exists some $t_{\mathrm{1}}\ge \mathrm{0}$, such that $S(t)B\subset B_{\mathrm{0}}$, for all $t\ge t_{\mathrm{1}}$.

Definition 5^[28] Let $S(t),t\ge \mathrm{0}$ be a $C^{\mathrm{0}}$ semigroup in a complete metric space M.

A subset $A$ of M is called a global attractor for the semigroup if $A$ is compact and enjoys the following properties:

1) $A$ is an invariant set, i.e., $S(t)A=A$ for any $t\ge \mathrm{0}$;

2) $A$ attracts all bounded sets of M. That is, for any bounded subset B of M, $t\to \mathrm{\infty },d(S(t)B,A)\to \mathrm{0},$where $d(B,A)$ is the semidistance of two sets $B$ and $A$ : $d(B,A)=\underset{x\in B}{\mathrm{s}\mathrm{u}\mathrm{p}}\underset{y\in A}{\mathrm{i}\mathrm{n}\mathrm{f}}d(x,y)$.

Theorem 5^[28] If $S(t)$ is dissipative and $B$ is a compact absorbing set, then there exists a global attractor $A=\omega (B)=\underset{t\ge \mathrm{0}}{\cup }S(t)B$.

2 Proofs of the Main Results

In this section, we will obtain the global attractors of 2D gNSE in $C_{H_g}$ and $C_{V_g}$ respectively.

Proof of Lemma 1   Taking the inner product of

$\frac{\mathrm{d}u}{\mathrm{d}t}+\nu A_{g}u+B(u)+G(u)+\nu Ru=f+h(t,u_t)$

with u, we have

$\begin{array}{l}\frac{\mathrm{1}}{\mathrm{2}}\frac{\mathrm{d}|u{|}^{\mathrm{2}}}{\mathrm{d}t}+\nu ||u|{|}^{\mathrm{2}}+(Bu,u)+(G(u),u)+\nu (Ru,u)\\ =(f,u)+(h(t,u_t),u)\end{array}$

that is

$\frac{\mathrm{d}|u{|}^{\mathrm{2}}}{\mathrm{d}t}+\mathrm{2}(G(u),u)=\mathrm{2}(f,u)-\mathrm{2}\nu (Ru,u)-\mathrm{2}\nu ||u|{|}^{\mathrm{2}}+\mathrm{2}(h(u,u_t{),u)}_{}.$

$\mathrm{S}\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{2}(G(u),u)=\mathrm{2}c|u{|}^{\beta -\mathrm{1}}|u{|}^{\mathrm{2}}=\mathrm{2}c|u{|}_{\beta +\mathrm{1}}^{\beta +\mathrm{1}}>\mathrm{0},(\beta \ge \mathrm{1},c>\mathrm{0}),$

$\frac{\mathrm{d}|u{|}^{\mathrm{2}}}{\mathrm{d}t}\le \mathrm{2}|f|\cdot |u|+\mathrm{2}\nu |Ru|\cdot |u|-\mathrm{2}\nu ||u|{|}^{\mathrm{2}}+\mathrm{2}|h(t,u_t)|\cdot |u|$

$\le \mathrm{2}|u|\cdot (|f|+|h(t,u_t)|)-\mathrm{2}\nu ||u|{|}^{\mathrm{2}}+\mathrm{2}\nu \frac{|\nabla g{|}_{\mathrm{\infty }}}{m_{\mathrm{0}}{\lambda }_{\mathrm{1}}^{\mathrm{1}/\mathrm{2}}}||u|{|}^{\mathrm{2}}$

$\le \mathrm{2}|u|\cdot (|f|+|h(t,u_t)|)-\mathrm{2}\nu {\lambda }_{\mathrm{1}}(\mathrm{1}-|\frac{|\nabla g{|}_{\mathrm{\infty }}}{m_{\mathrm{0}}{\lambda }_{\mathrm{1}}^{\mathrm{1}/\mathrm{2}}})|u{|}^{\mathrm{2}}.$

$\mathrm{L}\mathrm{e}\mathrm{t}\lambda =\nu {\lambda }_{\mathrm{1}}(\mathrm{1}-|\frac{|\nabla g{|}_{\mathrm{\infty }}}{m_{\mathrm{0}}{\lambda }_{\mathrm{1}}^{\mathrm{1}/\mathrm{2}}}),\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}$

$\frac{\mathrm{d}|u{|}^{\mathrm{2}}}{\mathrm{d}t}+\mathrm{2}\lambda |u{|}^{\mathrm{2}}\le \mathrm{2}|u|(|f|+|h(t,u_t)|).$

For any $t\ge \mathrm{0}$, by the Growall inequality, we obtain $|u{|}^{\mathrm{2}}\le ||\varphi |{|}_{C_{H_g}}^{\mathrm{2}}{\mathrm{e}}^{-\mathrm{2}\lambda t}+\mathrm{2}{\int }_{\mathrm{0}}^t{\mathrm{e}}^{-\mathrm{2}\lambda (t-s)}[|u(s)|(|f|+|h(s,u_s)|]\mathrm{d}s$

$\le ||\varphi |{|}_{C_{H_g}}^{\mathrm{2}}{\mathrm{e}}^{-\mathrm{2}\lambda t}+\mathrm{2}\lambda {\int }_{\mathrm{0}}^t{\mathrm{e}}^{-\mathrm{2}\lambda (t-s)}[\frac{L_g||u_s|{|}_{C_{H_g}}}{\lambda }+\frac{|f|}{\lambda }]|u(s)|\mathrm{d}s.$

For $\forall t\ge \mathrm{0}$, in the following we will prove $||u|{|}_{C_{H_g}}<\tau k$ when $||\varphi |{|}_{C_{H_g}}<\tau k$.

We do this by contradiction. Suppose there exists $t_{\mathrm{1}}>\mathrm{0}$, such that $|u(t_{\mathrm{1}},x)|=\tau k$. When $t<t_{\mathrm{1}}$, we have $|u(t,x)|<\tau k$, that is, for any $\mathrm{0}\le t\le t_{\mathrm{1}}$, we have $|u(t,x)|\le \tau k$.

It can be obtained from the above formula that for any $t\ge \mathrm{0}$, we deduce

$|u(t_{\mathrm{1}},x){|}^{\mathrm{2}}\le {\tau }^{\mathrm{2}}{k}^{\mathrm{2}}{\mathrm{e}}^{-\mathrm{2}\lambda t_{\mathrm{1}}}+\mathrm{2}\lambda {\int }_{\mathrm{0}}^{t_{\mathrm{1}}}{\mathrm{e}}^{-\mathrm{2}\lambda (t_{\mathrm{1}}-s)}\tau k(\frac{L_g\tau k}{\lambda }+\frac{|f|}{\lambda })\mathrm{d}s.$

$\mathrm{L}\mathrm{e}\mathrm{t}k=\frac{|f|}{\lambda -L_g},\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{i}\mathrm{s}k=\frac{L_{g}k}{\lambda }+\frac{|f|}{\lambda }.\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{n}$

$|u(t,x){|}^{\mathrm{2}}\le {\tau }^{\mathrm{2}}k(\frac{L_{g}k}{\lambda }+\frac{|f|}{\lambda }){\mathrm{e}}^{-\mathrm{2}\lambda t_{\mathrm{1}}}+\tau k(\frac{\tau k{L}_g}{\lambda }+\frac{|f|}{\lambda })\mathrm{2}\lambda {\mathrm{e}}^{-\mathrm{2}\lambda t_{\mathrm{1}}}{\int }_{\mathrm{0}}^{t_{\mathrm{1}}}{\mathrm{e}}^{\mathrm{2}\lambda s}\mathrm{d}s$

$\le {\tau }^{\mathrm{2}}k(\frac{L_{g}k}{\lambda }+\frac{|f|}{\lambda }){\mathrm{e}}^{-\mathrm{2}\lambda t_{\mathrm{1}}}+\tau k(\frac{\tau k{L}_g}{\lambda }+\frac{|f|}{\lambda })\mathrm{2}\lambda {\mathrm{e}}^{-\mathrm{2}\lambda t_{\mathrm{1}}}\frac{{\mathrm{e}}^{\mathrm{2}\lambda t_{\mathrm{1}}}-\mathrm{1}}{\mathrm{2}\lambda }$

$={\tau }^{\mathrm{2}}k(\frac{L_{g}k}{\lambda }+\frac{|f|}{\lambda }){\mathrm{e}}^{-\mathrm{2}\lambda t_{\mathrm{1}}}+\tau k(\frac{\tau k{L}_g}{\lambda }+\frac{|f|}{\lambda })(\mathrm{1}-{\mathrm{e}}^{-\mathrm{2}\lambda t_{\mathrm{1}}})$

$\le \tau (\tau -\mathrm{1})k\frac{|f|}{\lambda }+\tau k(\frac{\tau k{L}_g}{\lambda }+\frac{|f|}{\lambda })={\tau }^{\mathrm{2}}{k}^{\mathrm{2}}.$

This is a contradiction, so $S_{\tau }$ is a positive invariant set.

Proof of Lemma 2   For any $\varphi \in C_{H_g}$, there exists $\tau \ge \mathrm{1}$, such that $\varphi \in S_{\tau }$. From Lemma 1, we have $|u(t,x)|\le \tau k$, so there exists $\sigma >\mathrm{0}$, such that $\underset{t\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\mathrm{s}\mathrm{u}\mathrm{p}|u|=\sigma$.

For arbitrarily small $\epsilon >\mathrm{0}$, there exists $t_{\mathrm{2}}\ge \mathrm{0}$, such that $|u|\le (\sigma +\epsilon ),\begin{array}{c}t\ge t_{\mathrm{2}}\end{array}$.

For $\lambda >\mathrm{0}$, for the above $\epsilon$ and $k$, we have $T>\mathrm{0}$ and ${\tau }^{\mathrm{2}}{k}^{\mathrm{2}}{\mathrm{e}}^{-\mathrm{2}\lambda T}+\mathrm{2}\lambda {\int }_T^{\mathrm{\infty }}{\mathrm{e}}^{-\mathrm{2}\lambda s}\tau k(\frac{L_g\tau k}{\lambda }+\frac{|f|}{\lambda })\mathrm{d}s\le \epsilon$.

Hence, as $t\ge T+t_{\mathrm{2}}$,

$|u{|}^{\mathrm{2}}\le ||\varphi |{|}_{C_{H_g}}^{\mathrm{2}}{\mathrm{e}}^{-\mathrm{2}\lambda t}+\mathrm{2}\lambda \{{\int }_{\mathrm{0}}^{t-T}+{\int }_{t-T}^t\}{\mathrm{e}}^{-\mathrm{2}\lambda (t-s)}|u(s)|(\frac{L_g}{\lambda }||u|{|}_{C_{H_g}}+\frac{|f|}{\lambda })\mathrm{d}s\le \epsilon +\mathrm{2}\lambda (\sigma +\epsilon ){\int }_{t-T}^t{\mathrm{e}}^{-\mathrm{2}\lambda (t-s)}(\frac{L_g}{\lambda }||u|{|}_{C_{H_g}}+\frac{|f|}{\lambda })\mathrm{d}s$

$\le \epsilon +(\sigma +\epsilon )[\frac{L_g}{\lambda }(\sigma +\epsilon )+\frac{|f|}{\lambda }].$

Let $\epsilon \to \mathrm{0}$, then ${\sigma }^{\mathrm{2}}=\sigma (\frac{\sigma L_g}{\lambda }+\frac{|f|}{\lambda })$, therefore $\sigma \le k=\frac{|f|}{\lambda -L_g}$, that is, $\underset{t\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\mathrm{s}\mathrm{u}\mathrm{p}|u|\le k=\frac{|f|}{\lambda -L_g}$, so $S$ is the global absorbed set of the semigroup $S(t)$ in $C_{H_g}$.

Proof of Lemma 3   Taking the inner product of (8) with $A_{g}u$,

$\frac{\mathrm{1}}{\mathrm{2}}\frac{\mathrm{d}}{\mathrm{d}t}||u|{|}^{\mathrm{2}}+\nu |A_{g}u{|}^{\mathrm{2}}+b(u,u,A_{g}u)+(G(u),A_{g}u)+\nu (Ru,A_{g}u)=(f,A_{g}u)+(h(t,u_t),A_{g}u).$

Since $|b(u,u,A_{g}u)|\le c_{\mathrm{1}}|u{|}^{\frac{\mathrm{1}}{\mathrm{2}}}||u|||A_{g}u{|}^{\frac{\mathrm{2}}{\mathrm{3}}}$$\le \frac{\nu }{\mathrm{4}}|A_{g}u{|}^{\mathrm{2}}+\frac{c_{\mathrm{1}}^{\text{'}}}{{\nu }^{\mathrm{3}}}|u{|}^{\mathrm{2}}||u|{|}^{\mathrm{4}},$ where $c_{\mathrm{1}}^{\text{'}}=\frac{\mathrm{27}{c}_{\mathrm{1}}^{\mathrm{4}}}{\mathrm{4}}$,

$|(G(u),A_{g}u)|\le c|u{|}^{\beta -\mathrm{1}}|u||A_{g}u|\le c(\frac{|u{|}^{\mathrm{2}\beta }}{\mathrm{2}\nu }+\frac{\nu }{\mathrm{2}}|A_{g}u{|}^{\mathrm{2}}).$

$|(f,A_{g}u)+(h(t,u_t),A_{g}u)|\le |A_{g}u|(|f|+|h(t,u_t)|)$

$\le \frac{\nu }{\mathrm{8}}|A_{g}u{|}^{\mathrm{2}}+\frac{\mathrm{4}}{\nu }(|f{|}^{\mathrm{2}}+L_g^{\mathrm{2}}||u|{|}_{C_{H_g}}^{\mathrm{2}}).$

$\nu |(Ru,A_{g}u)|\le \nu |Ru|\cdot |A_{g}u|\le \frac{\nu }{\mathrm{8}}|A_{g}u{|}^{\mathrm{2}}+\mathrm{2}\nu |Ru{|}^{\mathrm{2}}$

$\le \frac{\nu }{\mathrm{8}}|A_{g}u{|}^{\mathrm{2}}+\mathrm{2}\nu \frac{|\nabla g{|}_{\mathrm{\infty }}^{\mathrm{2}}}{m_{\mathrm{0}}^{\mathrm{2}}{\lambda }_{\mathrm{1}}}||u|{|}^{\mathrm{2}}$

Then

$\begin{array}{l}\frac{\mathrm{d}}{\mathrm{d}t}||u|{|}^{\mathrm{2}}+\mathrm{2}\nu |A_{g}u{|}^{\mathrm{2}}\\ \le -\mathrm{2}b(u,u,A_{g}u)-\mathrm{2}(G(u),A_{g}u)-\mathrm{2}\nu (Ru,A_{g}u)+\mathrm{2}(h(t,u_t),A_{g}u)\end{array}$

$\le \frac{\nu }{\mathrm{2}}|A_{g}u{|}^{\mathrm{2}}+\frac{\mathrm{2}{c}_{\mathrm{1}}^{\text{'}}}{{\nu }^{\mathrm{3}}}|u{|}^{\mathrm{2}}||u|{|}^{\mathrm{4}}+\frac{c|u{|}^{\mathrm{2}\beta }}{\mathrm{2}\nu }+\frac{c\nu }{\mathrm{2}}|A_{g}u{|}^{\mathrm{2}}+\frac{\nu }{\mathrm{4}}|A_{g}u{|}^{\mathrm{2}}+\frac{\mathrm{4}\nu |\nabla g{|}_{\mathrm{\infty }}^{\mathrm{2}}}{m_{\mathrm{0}}^{\mathrm{2}}{\lambda }_{\mathrm{1}}}||u|{|}^{\mathrm{2}}+\frac{\nu }{\mathrm{4}}|A_{g}u{|}^{\mathrm{2}}+\frac{\mathrm{8}}{\nu }(|f{|}^{\mathrm{2}}+L_g^{\mathrm{2}}||u|{|}_{L_{H_g}}^{\mathrm{2}})$

$\begin{array}{l}\le (\nu +\frac{c\nu }{\mathrm{2}})|A_{g}u{|}^{\mathrm{2}}+\frac{c|u{|}^{\mathrm{2}\beta }}{\mathrm{2}\nu }+\frac{\mathrm{2}{c}_{\mathrm{1}}^{\text{'}}}{{\nu }^{\mathrm{3}}}|u{|}^{\mathrm{2}}||u|{|}^{\mathrm{4}}\\ +\frac{\mathrm{4}\nu |\nabla g{|}_{\mathrm{\infty }}^{\mathrm{2}}}{m_{\mathrm{0}}^{\mathrm{2}}{\lambda }_{\mathrm{1}}}||u|{|}^{\mathrm{2}}+\frac{\mathrm{8}}{\nu }(|f{|}^{\mathrm{2}}+L_g^{\mathrm{2}}||u|{|}_{L_{H_g}}^{\mathrm{2}})\end{array}$