In this section, we discuss the existence of pullback attractors for the 3D Navier-Stokes-Voight equations with delays in continuous and sub-linear operators. At first, we propose the assumptions for *g* given in Section 3:

(H1^{′}) For all $u\in V$, $g(\cdot ,u):\mathbb{R}\to H$ is measurable.

(H2^{′}) For all $t\in \mathbb{R}$, $g(t,\cdot ):H\to H$ is continuous.

(H3

^{′}) There are two nonnegative functions

$\gamma (t),\beta (t):\mathbb{R}\to [0,+\mathrm{\infty})$ with

$\gamma (t)\in {L}_{loc}^{p}(\mathbb{R})$ for some

$1\le p\le +\mathrm{\infty}$ and

$\beta (t)\in {L}_{loc}^{1}(\mathbb{R})$ such that for any

$u\in V$,

${\parallel g(t,u)\parallel}_{{V}^{\prime}}^{2}\le \gamma (t){\parallel u\parallel}^{2}+\beta (t)\phantom{\rule{1em}{0ex}}\text{for any}t\in \mathbb{R}.$

To construct a multi-valued process, we introduce symbols ${C}_{V}=C([-h,0];V)$ and ${S}_{V}^{2q}=V\times {L}^{2q}(-h,0;V)$, where $\frac{1}{p}+\frac{1}{q}=1$ as two phase spaces. Let $D(\tau ,{u}_{0},\phi )$ denote the set of global solutions of (1.1) in $[\tau ,+\mathrm{\infty})$ and the initial datum $({u}_{0},\phi )\in {S}_{V}^{2q}$.

By Theorem 3.1, we know there exists a solution to problem (3.2) although we have no discussion on the uniqueness of solutions to problem (3.2). We may define two strict processes,

$({C}_{V},\{U(\cdot ,\cdot )\})$ as

$U(t,\tau )\phi =\{u(t):u\in D(\tau ,\phi (0),\phi )\}\phantom{\rule{1em}{0ex}}\text{for any}\phi \in {C}_{V},$

and

$({S}_{V}^{2q},\{U(\cdot ,\cdot )\})$ as

$U(t,\tau )({u}_{0},\phi )=\{u(t):u\in D(\tau ,{u}^{0},\phi )\}\phantom{\rule{1em}{0ex}}\text{for any}({u}_{0},\phi )\in {S}_{V}^{2q}.$

Considering the regularity of the problem, the asymptotic behavior of the two processes shall be the same, as we shall see in what follows.

In order to simplify the calculation form, we introduce a function

${\kappa}_{\sigma}$. For any

$\sigma >0$, we set

${\kappa}_{\sigma}(t,s)=(d-\frac{\sigma +d}{{\alpha}^{2}})(t-s)-\frac{{e}^{dh}}{{\alpha}^{2}d(1-{\rho}_{0})}{\int}_{s}^{t}\gamma (r)\phantom{\rule{0.2em}{0ex}}dr,\phantom{\rule{1em}{0ex}}\mathrm{\forall}t,s\in \mathbb{R}.$

(4.1)

From (4.1), we can find that

$-{\kappa}_{\sigma}(t,s)={\kappa}_{\sigma}(0,t)-{\kappa}_{\sigma}(0,s),\phantom{\rule{1em}{0ex}}\mathrm{\forall}t,s\in \mathbb{R},$

(4.2)

and for any

*σ*:

$0<\sigma <{\alpha}^{2}d-d$, then

${\kappa}_{\sigma}(0,r)\le {\kappa}_{\sigma}(0,t)+(d-\frac{\sigma +d}{{\alpha}^{2}})h,\phantom{\rule{1em}{0ex}}\mathrm{\forall}r\in [t-h,t],$

(4.3)

where $d=\nu min\{{\lambda}_{1},\frac{1}{{\alpha}^{2}}\}$.

**Lemma 4.1** *Suppose that* (H1′)-(H3′)

*hold*,

*for any initial datum* $({u}_{0},\phi )\in {S}_{V}^{2q}$ *and any* $u\in D(\tau ,{u}_{0},\phi )$,

*it holds* ${\parallel u(t)\parallel}^{2}\le \frac{1}{{\alpha}^{2}}{C}_{\tau}{e}^{-{\kappa}_{\sigma}(t,\tau )}+\frac{1}{{\alpha}^{2}}{\sigma}^{-1}{\int}_{\tau}^{t}{e}^{-{\kappa}_{\sigma}(t,s)}\beta (s)\phantom{\rule{0.2em}{0ex}}ds,$

(4.4)

*where* ${C}_{\tau}=d{\int}_{-h}^{0}{e}^{dr}{\parallel \phi (r)\parallel}^{2}\phantom{\rule{0.2em}{0ex}}dr+{|{u}_{0}|}^{2}+{\alpha}^{2}{\parallel {u}_{0}\parallel}^{2}$.

*Proof* Let

*u* be a solution of (3.2), so

$u\in D(\tau ,{u}_{0},\phi )$. Multiplying (1.1) by

$u(t)$ and using the energy equality and the Poincaré inequality, we have

$\begin{array}{r}\frac{d}{dt}({|u(t)|}^{2}+{\alpha}^{2}{\parallel u(t)\parallel}^{2})+d({|u(t)|}^{2}+{\alpha}^{2}{\parallel u(t)\parallel}^{2})\\ \phantom{\rule{1em}{0ex}}\le 2{\parallel g(t,u(t-\rho (t)))\parallel}_{{V}^{\prime}}{\parallel u(t)\parallel}_{V},\end{array}$

where

$d=\nu min\{{\lambda}_{1},\frac{1}{{\alpha}^{2}}\}$. Thus

$\begin{array}{r}\frac{d}{dt}\left[{e}^{dt}({|u(t)|}^{2}+{\alpha}^{2}{\parallel u(t)\parallel}^{2})\right]\\ \phantom{\rule{1em}{0ex}}\le 2{e}^{dt}{\parallel g(t,u(t-\rho (t)))\parallel}_{{V}^{\prime}}{\parallel u(t)\parallel}_{V}\\ \phantom{\rule{1em}{0ex}}\le 2{e}^{dt}({\gamma}^{\frac{1}{2}}(t)\parallel u(t-\rho (t))\parallel +{\beta}^{\frac{1}{2}}(t))\parallel u(t)\parallel \\ \phantom{\rule{1em}{0ex}}\le 2{e}^{dt}{\gamma}^{\frac{1}{2}}(t)\parallel u(t)\parallel \parallel u(t-\rho (t))\parallel +2{e}^{dt}{\beta}^{\frac{1}{2}}(t)\parallel u(t)\parallel \\ \phantom{\rule{1em}{0ex}}\le {C}_{0}^{-1}{e}^{dt}{\parallel u(t-\rho (t))\parallel}^{2}+({C}_{0}\gamma (t)+\sigma ){e}^{dt}{\parallel u(t)\parallel}^{2}+{\sigma}^{-1}{e}^{dt}\beta (t),\end{array}$

(4.5)

where we have denoted

${C}_{0}=\frac{{e}^{dh}}{d(1-{\rho}_{0})}.$

Considering

$\begin{array}{rl}{\int}_{\tau}^{t}{e}^{ds}{\parallel u(s-\rho (s))\parallel}^{2}\phantom{\rule{0.2em}{0ex}}ds& \le \frac{{e}^{dh}}{1-{\rho}_{0}}\left({\int}_{\tau -h}^{t}{e}^{dr}\parallel u(r)\parallel \phantom{\rule{0.2em}{0ex}}dr\right)\\ =\frac{{e}^{dh}}{1-{\rho}_{0}}({e}^{d\tau}{\int}_{-h}^{0}{e}^{dr}{\parallel \phi (r)\parallel}^{2}\phantom{\rule{0.2em}{0ex}}dr+{\int}_{\tau}^{t}{e}^{dr}{\parallel u(r)\parallel}^{2}\phantom{\rule{0.2em}{0ex}}dr),\end{array}$

(4.6)

and integrating (4.5) from

*τ* to

*t*, we deduce

$\begin{array}{r}{e}^{dt}{|u(t)|}^{2}+{\alpha}^{2}{e}^{dt}{\parallel u(t)\parallel}^{2}\\ \phantom{\rule{1em}{0ex}}\le {C}_{0}^{-1}{\int}_{\tau}^{t}{e}^{ds}{\parallel u(s-\rho (s))\parallel}^{2}\phantom{\rule{0.2em}{0ex}}ds+{\int}_{\tau}^{t}({C}_{0}\gamma (s)+\sigma ){e}^{ds}{\parallel u(s)\parallel}^{2}\phantom{\rule{0.2em}{0ex}}ds+{\sigma}^{-1}{\int}_{\tau}^{t}\beta (s)\phantom{\rule{0.2em}{0ex}}ds\\ \phantom{\rule{2em}{0ex}}+{e}^{d\tau}{|{u}_{0}|}^{2}+{\alpha}^{2}{e}^{d\tau}{\parallel {u}_{0}\parallel}^{2}\\ \phantom{\rule{1em}{0ex}}\le {C}_{0}^{-1}\times \frac{{e}^{dh}}{1-{\rho}_{0}}\times {e}^{d\tau}{\int}_{-h}^{0}{e}^{dr}{\parallel \phi (r)\parallel}^{2}\phantom{\rule{0.2em}{0ex}}dr+{C}_{0}^{-1}\times \frac{{e}^{dh}}{1-{\rho}_{0}}{\int}_{\tau}^{t}{e}^{dr}{\parallel u(r)\parallel}^{2}\phantom{\rule{0.2em}{0ex}}dr\\ \phantom{\rule{2em}{0ex}}+{\int}_{\tau}^{t}({C}_{0}\gamma (s)+\sigma ){e}^{ds}{\parallel u(s)\parallel}^{2}\phantom{\rule{0.2em}{0ex}}ds+{\sigma}^{-1}{\int}_{\tau}^{t}{e}^{ds}\beta (s)\phantom{\rule{0.2em}{0ex}}ds+{e}^{d\tau}{|{u}_{0}|}^{2}+{\alpha}^{2}{e}^{d\tau}{\parallel {u}_{0}\parallel}^{2}\\ \phantom{\rule{1em}{0ex}}=d{e}^{d\tau}{\int}_{-h}^{0}{e}^{dr}{\parallel \phi (r)\parallel}^{2}\phantom{\rule{0.2em}{0ex}}dr+{\int}_{\tau}^{t}{e}^{ds}({C}_{0}\gamma (s)+\sigma +d){\parallel u(s)\parallel}^{2}\phantom{\rule{0.2em}{0ex}}ds\\ \phantom{\rule{2em}{0ex}}+{\sigma}^{-1}{\int}_{\tau}^{t}{e}^{dr}\beta (r)\phantom{\rule{0.2em}{0ex}}dr+{e}^{d\tau}{|{u}_{0}|}^{2}+{\alpha}^{2}{e}^{d\tau}{\parallel {u}_{0}\parallel}^{2},\end{array}$

(4.7)

where we set

${C}_{\tau}=d{\int}_{-h}^{0}{e}^{ds}{\parallel \phi (s)\parallel}^{2}\phantom{\rule{0.2em}{0ex}}ds+{|{u}_{0}|}^{2}+{\alpha}^{2}{\parallel {u}_{0}\parallel}^{2}.$

Observing the above estimates, we easily deduce

$\begin{array}{r}{e}^{dt}{|u(t)|}^{2}+{\alpha}^{2}{e}^{dt}{\parallel u(t)\parallel}^{2}\\ \phantom{\rule{1em}{0ex}}\le {e}^{d\tau}{C}_{\tau}+{\int}_{\tau}^{t}({C}_{0}\gamma (s)+\sigma +d){e}^{dr}{\parallel u(r)\parallel}^{2}\phantom{\rule{0.2em}{0ex}}dr+{\sigma}^{-1}{\int}_{\tau}^{t}{e}^{dr}\beta (r)\phantom{\rule{0.2em}{0ex}}dr.\end{array}$

(4.8)

Applying the Poincaré inequality and the Gronwall inequality to (4.8), we deduce that (4.4) holds. This finishes the proof of this lemma. □

Next, we shall prove that the processes

$({C}_{V},\{U(\cdot ,\cdot )\})$ and

$({S}_{V}^{2q},\{U(\cdot ,\cdot )\})$ defined above are pullback-absorbing. To obtain this, we propose the assumptions (H4′)

$\underset{t\to -\mathrm{\infty}}{lim\hspace{0.17em}sup}\frac{1}{t}{\int}_{0}^{t}\gamma (s)\phantom{\rule{0.2em}{0ex}}ds=\overline{\gamma}\in [0,+\mathrm{\infty}),$

and the relation among constants

$\sigma >0$,

*d* defined above, and

*α* in (1.1) satisfies(H5′)

$d-\frac{\sigma +d}{{\alpha}^{2}}-\frac{{C}_{0}}{{\alpha}^{2}}>0,$

and $\beta (t)$ satisfies

(H6

^{′})

${\int}_{-\mathrm{\infty}}^{0}{e}^{-{\kappa}_{\sigma}(0,r)}\beta (r)\phantom{\rule{0.2em}{0ex}}dr<+\mathrm{\infty},$

where the function ${\kappa}_{\sigma}(t,s)$ is given by (4.1).

Before proving that the two multi-valued processes possess pullback-absorbing sets, we introduce the definition of the two natural tempered universes which shall play the key role for our main purpose.

**Definition 4.1** Suppose that

${\mathcal{R}}_{\sigma}(t)$ is the collection of the sets of all functions

$r:\mathbb{R}\to [0,+\mathrm{\infty})$ such that

$\underset{t\to -\mathrm{\infty}}{lim}{e}^{-{\kappa}_{\sigma}(0,t)}{r}^{2}(t)=0.$

Let ${\mathcal{D}}_{{S}_{V}^{2q}}$ be the class of all families $\stackrel{\u02c6}{D}=\{D(t):t\in \mathbb{R}\}\subset \mathcal{P}({S}_{V}^{2q})$ such that $D(t)\subset {\overline{B}}_{{S}_{V}^{2q}}(0,{r}_{\stackrel{\u02c6}{D}}(t))$ for some ${r}_{\stackrel{\u02c6}{D}}\in {\mathcal{R}}_{\sigma}$. In the same way, let ${\mathcal{D}}_{{C}_{V}}^{\sigma}$ denote the class of all families $\stackrel{\u02c6}{D}=\{D(t):t\in \mathbb{R}\}\subset \mathcal{P}({C}_{V})$ satisfying $D(t)\subset {\overline{B}}_{{C}_{V}}(0,{r}_{\stackrel{\u02c6}{D}}(t))$ for some ${r}_{\stackrel{\u02c6}{D}}\in {\mathcal{R}}_{\sigma}$.

Let ${B}_{0}$ be any fixed bounded subset of ${S}_{V}^{2q}$. Observing that ${\mathcal{D}}_{{C}_{V}}^{\sigma}\subset {\mathcal{D}}_{{S}_{V}^{2q}}$, which is inclusion-closed, by (H4^{′}) and (H5^{′}), we deduce that the family $\stackrel{\u02c6}{B}=\{B(t)\equiv {B}_{0},t\in \mathbb{R}\}$ is contained in ${\mathcal{D}}_{{S}_{V}^{2q}}^{\sigma}$. With regard to ${\mathcal{D}}_{{C}_{V}}$, we use the same method and obtain a similar conclusion if ${B}_{0}$ is included in ${C}_{V}$.

The following lemma provides that there exist pullback-absorbing sets for the two processes mentioned above.

**Lemma 4.2** *Suppose that* (H1

^{′})-(H6

^{′})

*hold and the constants* *α*,

*d*,

${C}_{0}$,

$\overline{\gamma}$ *satisfy* $d-\frac{\sigma +d}{{\alpha}^{2}}-\frac{{C}_{0}}{{\alpha}^{2}}\overline{\gamma}>0$ *and* $\overline{\gamma}<1$.

- (1)
*Then*,

*for any* $t\in \mathbb{R}$ *and any family* $\stackrel{\u02c6}{B}=\{B(t):t\in \mathbb{R}\}$,

*there exits* $\tau (\stackrel{\u02c6}{B},t)\le t$ *such that any initial datum* $({u}_{0},\phi )\in {S}_{V}^{2q}$ *and any* $u\in D(\tau ,{u}_{0},\phi )$ *for any* $\tau \le \tau (\stackrel{\u02c6}{B},t)$ *satisfy that* $\parallel u(t)\parallel \le {R}_{V}(t)$,

*where the positive continuous function* ${R}_{V}(\cdot )$ *is given by* ${R}_{V}^{2}(t)=1+\frac{1}{{\alpha}^{2}}{\sigma}^{-1}{\int}_{\tau}^{t}{e}^{-{\kappa}_{\sigma}(t,s)}\beta (s)\phantom{\rule{0.2em}{0ex}}ds,\phantom{\rule{1em}{0ex}}\mathrm{\forall}t\in \mathbb{R}.$

- (2)
*Let*
${\stackrel{\u02c6}{D}}_{0}=\{{D}_{0}(t):t\in \mathbb{R}\}$
*be included*
$\mathcal{P}({C}_{V})$
*which is given by*
${D}_{0}(t)={\overline{B}}_{{C}_{V}}(0,{\tilde{R}}_{V}(t))\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}{\tilde{R}}_{V}(t)=\underset{t-h\le r\le t}{max}{R}_{V}(r),\phantom{\rule{1em}{0ex}}\mathrm{\forall}t\in \mathbb{R}.$

*Then the set* ${\stackrel{\u02c6}{D}}_{0}\in {\mathcal{D}}_{{C}_{V}}^{\sigma}$ *and is* ${\mathcal{D}}_{{S}_{V}^{2q}}^{\sigma}$-*pullback absorbing for the process* $({S}_{V}^{2q},U)$. *Therefore*, ${\stackrel{\u02c6}{D}}_{0}$ *is* ${\mathcal{D}}_{{C}_{V}}^{\sigma}$-*pullback absorbing for the process* $({C}_{V},U)$.

*Proof* Since the proof is a consequence of the definition of

${\mathcal{D}}_{{S}_{V}^{2q}}$, we only sketch it here. From Lemma 4.1, (H5

^{′}) and (H6

^{′}) , we have

$\begin{array}{rcl}{\parallel u(t)\parallel}^{2}& \le & \frac{1}{{\alpha}^{2}}{C}_{\tau}{e}^{-{\kappa}_{\sigma}(t,\tau )}+\frac{1}{{\alpha}^{2}}{\sigma}^{-1}{\int}_{\tau}^{t}{e}^{-{\kappa}_{\sigma}(t,s)}\beta (s)\phantom{\rule{0.2em}{0ex}}ds\\ =& \frac{1}{{\alpha}^{2}}{C}_{\tau}{e}^{{\kappa}_{\sigma}(0,t)}\cdot {e}^{-\kappa (0,\tau )}+\frac{1}{{\alpha}^{2}}{\sigma}^{-1}{\int}_{\tau}^{t}{e}^{-{\kappa}_{\sigma}(t,s)}\beta (s)\phantom{\rule{0.2em}{0ex}}ds\\ =& \frac{1}{{\alpha}^{2}}{C}_{\tau}{e}^{-{\kappa}_{\sigma}(0,t)}\cdot {e}^{(d-\frac{\sigma +d}{{\alpha}^{2}})\tau -\frac{{C}_{0}}{{\alpha}^{2}}{\int}_{0}^{\tau}\gamma (r)\phantom{\rule{0.2em}{0ex}}dr}+\frac{1}{{\alpha}^{2}}{\sigma}^{-1}{\int}_{\tau}^{t}{e}^{-{\kappa}_{\sigma}(t,s)}\beta (s)\phantom{\rule{0.2em}{0ex}}ds\\ \le & \frac{{C}_{\tau}}{{\alpha}^{2}}{e}^{{\kappa}_{\sigma}(0,t)}{e}^{(d-\frac{\sigma +d}{{\alpha}^{2}}-\frac{{C}_{0}}{{\alpha}^{2}})\tau}+\frac{1}{{\alpha}^{2}}{\sigma}^{-1}{\int}_{\tau}^{t}{e}^{-{\kappa}_{\sigma}(t,s)}\beta (s)\phantom{\rule{0.2em}{0ex}}ds\\ \le & 1+\frac{1}{{\alpha}^{2}}{\sigma}^{-1}{\int}_{-\mathrm{\infty}}^{t}{e}^{-{\kappa}_{\sigma}(t,s)}\beta (s)\phantom{\rule{0.2em}{0ex}}ds.\end{array}$

(4.9)

We complete the proof of (1). Estimate (2) is a consequence of (1). □

For the two processes $({S}_{V}^{2q},U)$ and $({C}_{V},U)$, they possess pullback-absorbing sets. In order to apply Lemma 2.3 to obtain the existence of pullback attractors, it is necessary to prove that the two multi-valued processes are asymptotically compact. This will be done in the following lemma.

**Lemma 4.3** *Suppose that the assumptions in Lemma * 4.2 *hold*. *The two processes* $({C}_{V},U)$ *and* $({S}_{V}^{2q},U)$ *are* ${\stackrel{\u02c6}{D}}_{0}$-*asymptotically compact*.

*Proof* For any

${t}_{0}\in \mathbb{R}$ , a sequence

$\{{\tau}_{n}\}\subset (-\mathrm{\infty},{t}_{0}-2h]$ with

${\tau}_{n}\to -\mathrm{\infty}$ and a sequence

$\{{u}^{n}\}$ with

${u}^{n}\in D({\tau}_{n},{\phi}_{n}(0),{\phi}_{n})$ with

${\phi}_{n}\in {D}_{0}({\tau}_{n})$, we shall prove that the sequence

$\{{u}_{{t}_{0}}^{n}\}$ is relatively compact in

${C}_{V}$. By the properties concerning operator

*b* mentioned in Section 3, we deduce that

${\parallel {\left({u}^{n}\right)}^{\prime}\parallel}_{{V}^{\prime}}+{\alpha}^{2}\parallel {\left({u}^{n}\right)}^{\prime}\parallel \le \nu \parallel {u}^{n}\parallel +{\parallel b({u}^{n},{u}^{n},\cdot )\parallel}_{{V}^{\prime}}+{\parallel g(t,{u}^{n}(t-\rho (t)))\parallel}_{{V}^{\prime}}.$

It is easy to get the above estimate which is independent of

*n*. The sequences of

$\{{u}^{n}\}$ and

$\{{({u}^{n})}^{\prime}\}$ possess their subsequence, relabeled the same in suitable spaces such that there exist

$u\in {L}^{\mathrm{\infty}}({t}_{0}-2h;V)$ and

${u}^{\prime}\in {L}^{2}({t}_{0}-h,{t}_{0};{V}^{\prime})$ satisfying

$\{\begin{array}{ll}{u}^{n}\stackrel{\ast}{\rightharpoonup}u& \text{weakly-star in}{L}^{\mathrm{\infty}}({t}_{0}-2h,{t}_{0};V),\\ {u}^{n}\rightharpoonup u& \text{weakly in}{L}^{2}({t}_{0}-2h,{t}_{0};V),\\ \frac{d}{dt}{u}^{n}\rightharpoonup \frac{d}{dt}u& \text{weakly in}{L}^{2}({t}_{0}-h,{t}_{0};{V}^{\prime}),\\ {u}^{n}(r)\to u(r)& \text{strongly in}V\text{a.e.}r\in ({t}_{0}-2h,{t}_{0}).\end{array}$

(4.10)

According to the assumptions on a function *g* and analogously as in Theorem 3.1, we deduce that $g(r,{u}^{n}(r-\rho (r)))\to g(r,u(r-\rho (r)))$ strongly in *V* a.e. $r\in ({t}_{0}-h,{t}_{0})$.

By the Lebesgue theorem and the uniform estimate of

${u}^{n}$ in

${L}^{\mathrm{\infty}}({t}_{0}-2h,{t}_{0};V)$, we deduce that the function

$g(r,{u}^{n}(r-\rho (r)))$ converges to the function

$g(r,u(r-\rho (r)))$ strongly. Therefore, for any

$t\in [{t}_{0}-h,{t}_{0}]$, we have

$u\in C([{t}_{0}-h,{t}_{0}];V)$ and

$\begin{array}{r}u(t)+{\alpha}^{2}\mathit{Au}(t)+{\int}_{{t}_{0}-h}^{t}(\nu \mathit{Au}(r)+B(u(r)))\phantom{\rule{0.2em}{0ex}}dr\\ \phantom{\rule{1em}{0ex}}=u({t}_{0}-h)+{\alpha}^{2}\mathit{Au}({t}_{0}-h)+{\int}_{{t}_{0}-h}^{t}g(r,u(r-\rho (r)))\phantom{\rule{0.2em}{0ex}}dr.\end{array}$

(4.11)

The uniform estimate of

$\{{({u}^{n})}^{\prime}\}$ in

${L}^{2}({t}_{0}-h,{t}_{0};{V}^{\prime})$ implies that the sequence

$\{{u}^{n}\}$ is equicontinuous in

${V}^{\prime}$ for any

${t}_{0}-h\le t\le {t}_{0}$. In addition, the sequence

$\{{u}^{n}\}$ is bounded, which is independent of

*n* in

$C([{t}_{0}-h,{t}_{0}];V)$. Using the Ascoli-Arzelà theorem, we can obtain

${u}^{n}\to u\phantom{\rule{1em}{0ex}}\text{strongly in}C([{t}_{0}-h,{t}_{0}];{V}^{\prime}).$

(4.12)

From the uniform boundedness of the sequence $\{{u}^{n}\}$ in $C([{t}_{0}-h,{t}_{0}];V)$, for any $r\in [{t}_{0}-h,{t}_{0}]$, we can also obtain ${u}^{n}(r)\rightharpoonup u(r)$, weakly in *V*.

By the analogous argument, for any compact sequence

$\{{r}_{n}\}\subset [{t}_{0}-h,{t}_{0}]$ and

$\{{r}_{n}\}\to r\in [{t}_{0}-h,{t}_{0}]$, we obtain that the sequence

$\{{u}^{n}({r}_{n})\}$ is convergent to

$u(r)$ weakly in

*V*. To achieve our result in Lemma 4.3, we only need to prove

${u}^{n}\to u\phantom{\rule{1em}{0ex}}\text{strongly in}C([{t}_{0}-h,{t}_{0}];V).$

The proof is slightly different from Proposition 6 in [20] or in [22]. We only sketch it here. We use a contradiction argument. Suppose that it is not true, then there would exist a value *ε*, a sequence (relabeled the same) $\{{r}_{n}\}\subset [{r}_{n}-h,{t}_{0}]$, and ${r}^{\prime}\in [{t}_{0}-h,{t}_{0}]$ with ${r}_{n}\to {r}^{\prime}$ satisfying $\parallel {u}^{n}({r}_{n})-u({r}^{\prime})\parallel \ge \epsilon $ for all $n\ge 1$. We shall see ${u}^{n}({r}_{n})\to u({r}^{\prime})$ in *V*. In order to achieve the last claim, because the sequence $\{{u}^{n}({r}_{n})\}$ is weakly convergent to $u(r)$ in *V*, we only need the convergence of the norms above. In other words, $\parallel {u}^{n}({r}_{n})\parallel \to \parallel u({r}^{\prime})\parallel $ as $n\to \mathrm{\infty}$.

From the weak convergence of

${u}^{n}({r}_{n})$ in

*V*, we get

$\parallel u\left({r}^{\prime}\right)\parallel \le \underset{n\to +\mathrm{\infty}}{lim}inf\parallel {u}^{n}\left({r}^{\prime}\right)\parallel .$

Therefore we have to check that

$\underset{n\to +\mathrm{\infty}}{lim}sup\parallel {u}^{n}({r}_{n})\parallel \le \parallel u\left({r}^{\prime}\right)\parallel .$

(4.13)

From the energy equality, for any

${t}_{0}-h\le r\le t\le {t}_{0}$, we obtain

$\begin{array}{r}\frac{1}{2}{|z(t)|}^{2}+\frac{1}{2}{\parallel z(t)\parallel}^{2}+\nu {\int}_{r}^{t}{\parallel z(s)\parallel}^{2}\phantom{\rule{0.2em}{0ex}}ds\\ \phantom{\rule{1em}{0ex}}=\frac{1}{2}{|z(r)|}^{2}+\frac{1}{2}{\parallel z(r)\parallel}^{2}+{\int}_{r}^{t}\left(g(s,z(s-\rho (s)))\right)\phantom{\rule{0.2em}{0ex}}ds,\end{array}$

(4.14)

where

$z=u$ or

$z={u}^{n}$. For any

$t\in [{t}_{0}-h,{t}_{0}]$, define the continuous functions

$J(t)$ and

${J}_{n}(t)$ as

$\begin{array}{c}J(t)=\frac{1}{2}{|u(t)|}^{2}+\frac{1}{2}{\parallel u(t)\parallel}^{2}-{\int}_{{t}_{0}-h}^{t}(g(s,u(s-\rho (s))),u(s))\phantom{\rule{0.2em}{0ex}}ds,\hfill \\ {J}_{n}(t)=\frac{1}{2}{|{u}^{n}(t)|}^{2}+\frac{1}{2}{\parallel {u}^{n}(t)\parallel}^{2}-{\int}_{{t}_{0}-h}^{t}(g(s,{u}^{n}(s-\rho (s))),{u}^{n}(s))\phantom{\rule{0.2em}{0ex}}ds.\hfill \end{array}$

By (4.14), it is clear that *J* and ${J}_{n}$ are non-increasing functions. By the convergence (4.10), for any $t\in ({t}_{0}-h,{t}_{0})$, we have that ${J}_{n}(t)\to J(t)$. Using the same analysis method as in [22], we can deduce that for $n\ge n({\kappa}_{\epsilon})$, ${J}_{n}({r}_{n})-J({r}^{\prime})\le \epsilon $, which gives (4.13) as desired. □

We can apply the technical method for any family in ${\mathcal{D}}_{{S}_{V}^{2q}}^{\sigma}$ in Lemma 4.3. Suppose that the assumptions in Lemma 4.3 hold. We can deduce that the processes $({C}_{V},U)$ and $({S}_{V}^{2q},U)$ are ${\mathcal{D}}_{{C}_{V}}^{\sigma}$-asymptotically compact and ${\mathcal{D}}_{{S}_{V}^{2q}}^{\sigma}$-asymptotically compact.

**Lemma 4.4** *Suppose that* (H1′)-(H6′) *hold*. *The two processes* $({C}_{V},U)$ *and* $({S}_{V}^{2q},U)$ *are semi*-*continuous and that* $U(t,\tau ):{S}_{V}^{2q}\to \mathcal{P}({S}_{V}^{2q})$ *and* $U(t,\tau ):{C}_{V}\to \mathcal{P}({C}_{V})$ *have compact values in their respective topologies*.

*Proof* In fact, the upper semi-continuity of the process $({S}_{V}^{2q},U)$ can be obtained by similar arguments to those used for the Galerkin sequence in Theorem 3.1.

As to the process $({C}_{V},U)$, applying the same energy-procedure in Lemma 4.3, we shall obtain that in $[\tau ,t]$ any set of solutions possesses a converging subsequence in this process, whence the assertion in Lemma 4.4 follows. □

According to the results in Section 2, the following two theorems shall be obtained, which are our result in this paper. Observing Lemmas 4.3 and 4.4, and applying Lemma 2.6, we obtain the following theorem.

**Theorem 4.1** *Suppose that* (H1′)-(H6′)

*hold*.

*For any* $t\in \mathbb{R}$,

*then there exist global pullback attractors* ${\mathcal{A}}_{{C}_{V}}=\{{\mathcal{A}}_{{C}_{V}}(t)\}$ *and* ${\mathcal{A}}_{{\mathcal{D}}_{{C}_{V}}^{\sigma}}=\{{\mathcal{A}}_{{\mathcal{D}}_{{C}_{V}}^{\sigma}}(t)\}$ *for the process* $({C}_{V},U)$ *in the universe of fixed bounded sets and in* ${\mathcal{D}}_{{C}_{V}}^{\sigma}$,

*respectively*.

*Moreover*,

*they are unique in the sense of Lemma * 2.5

*and negatively and strictly invariant for* *U* *respectively*,

*and the following holds*:

${\mathcal{A}}_{{C}_{V}}(t)\subset {\mathcal{A}}_{{\mathcal{D}}_{{C}_{V}}^{\sigma}}(t).$

The above theorem proves that there exist pullback attractors in the ${C}_{V}$ framework, while we shall prove that there exist pullback attractors in the ${S}_{V}^{2q}$ framework in the following theorem.

**Theorem 4.2** *Suppose that the assumptions in Theorem * 4.1

*hold*.

*For any* $t\in \mathbb{R}$,

*there exist global pullback attractors* ${\mathcal{A}}_{{S}_{V}^{2q}}=\{{\mathcal{A}}_{{S}_{V}^{2q}}(t)\}$ *and* ${\mathcal{A}}_{{\mathcal{D}}_{{S}_{V}^{2q}}^{\sigma}}=\{{\mathcal{A}}_{{\mathcal{D}}_{{S}_{V}^{2q}}^{\sigma}}(t)\}$ *for the process* $({S}_{V}^{2q},U)$ *in the universes of fixed bounded sets and in* ${\mathcal{D}}_{{S}_{V}^{2q}}^{\sigma}$.

*They are unique in the sense of Lemma * 2.5

*and negatively and strictly invariant for* *U*,

*respectively*,

*and we have* ${\mathcal{A}}_{{S}_{V}^{2q}}(t)\subset {\mathcal{A}}_{{\mathcal{D}}_{{S}_{V}^{2q}}^{\sigma}}(t)$.

*Moreover*,

*the relationship between the attractors for* $({S}_{V}^{2q},U)$ *and for* $({C}_{V},U)$ *is as follows*:

${\mathcal{A}}_{{S}_{V}^{2q}}(t)=f({\mathcal{A}}_{{C}_{V}}(t))\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}{\mathcal{A}}_{{\mathcal{D}}_{{S}_{V}^{2q}}^{\sigma}}(t)=f({\mathcal{A}}_{{\mathcal{D}}_{{C}_{V}}^{\sigma}}(t)),$

(4.15)

*where* $f:{C}_{V}\to {S}_{V}^{2q}$ *is the continuous mapping defined by* $f(\phi )=(\phi (0),\phi )$.

*Proof* The proof is rather similar to that of Theorem 5 in [20]. Since the regularity is different from [20], we only sketch the proof of (4.15) here.

By Theorem 3.1, we can conclude that $U(t,\tau )$ maps ${S}_{V}^{2q}$ into bounded sets in ${C}_{V}$ if $t\ge \tau +h$, and also maps bounded sets from ${S}_{V}^{2q}$ into bounded sets of ${C}_{V}$.

Noting that ${\mathcal{A}}_{{S}_{V}^{2q}}(t)$ is the minimal closed set, and using Lemma 2.5 and the above arguments, we deduce that $f({\mathcal{A}}_{{C}_{V}}(t))$ also attracts bounded sets in ${S}_{V}^{2q}$ in a pullback sense. Therefore the inclusion ${\mathcal{A}}_{{S}_{V}^{2q}}(t)\subset f({\mathcal{A}}_{{C}_{V}}(t))$ holds.

As to the opposite inclusion of the first identification in (4.15), for any bounded set *B*, it follows from the continuous injection $f({C}_{V})\subset {S}_{V}^{2q}$ and the attractor ${\mathcal{A}}_{{C}_{V}}(t)={\overline{{\bigcup}_{B}{\mathrm{\Lambda}}_{{C}_{V}}(B,t)}}^{{C}_{V}}$. Thus $f({\mathcal{A}}_{{C}_{V}}(t))={\overline{{\bigcup}_{B}f({\mathrm{\Lambda}}_{{C}_{V}}(B,t))}}^{{C}_{V}}$, whence the opposite inclusion of the first identification in (4.15) holds.

Analogously, it is obvious that the second relation in (4.15) holds. □