y(\theta_{t}\omega)}}{\sigma^{2}}\|\boldsymbol{f}(t+\tau)-\boldsymbol{f}_{\infty}\|^{2}_{\mathbb{H}}$
$\displaystyle\leq 2e^{\sigma
y(\theta_{t}\omega)}\|\nabla\bar{\boldsymbol{v}}(t)\|_{\mathbb{L}^{p}(\mathcal{O})}\|\mathscr{V}_{\tau}(t)\|^{\frac{2}{p-2}}_{\mathbb{L}^{p}(\mathcal{O})}\|\mathscr{V}_{\tau}(t)\|^{\frac{2(p-3)}{p-2}}_{\mathbb{H}}+2\sigma|y(\theta_{t}\omega)|\|\mathscr{V}_{\tau}(t)\|^{2}_{\mathbb{H}}$
$\displaystyle\quad+\frac{e^{-2\sigma
y(\theta_{t}\omega)}}{\sigma^{2}}\|\boldsymbol{f}(t+\tau)-\boldsymbol{f}_{\infty}\|^{2}_{\mathbb{H}}$
$\displaystyle\leq(Cp)^{\frac{1}{p-2}}\frac{Cp^{\frac{1}{p}}p^{2}}{p-1}e^{\sigma
y(\theta_{t}\omega)}\|\mathscr{V}_{\tau}(t)\|^{\frac{2(p-3)}{p-2}}_{\mathbb{H}}+2\sigma|y(\theta_{t}\omega)|\|\mathscr{V}_{\tau}(t)\|^{2}_{\mathbb{H}}+\frac{e^{-2\sigma
y(\theta_{t}\omega)}}{\sigma^{2}}\|\boldsymbol{f}(t+\tau)-\boldsymbol{f}_{\infty}\|^{2}_{\mathbb{H}},$
(5.1)

for a.e. $t\geq 0$, where we have used the bounds from (3.47), and the
constant $C$ is independent of $p$ and $\tau$. By applying [27, Theorem 21] to
(5), we arrive at

$\displaystyle\|\mathscr{V}_{\tau}(T)\|^{2}_{\mathbb{H}}\leq\biggl{\\{}[\widehat{M}(\tau)]^{\frac{1}{p-2}}+\frac{(Cp)^{\frac{1}{p-2}}Cp^{\frac{1}{p}}p^{2}}{(p-1)(p-2)}\int_{0}^{T}e^{\sigma
y(\theta_{t}\omega)}\/\mathrm{d}\/t\biggr{\\}}^{p-2}e^{2\sigma\int_{0}^{T}|y(\theta_{r}\omega)|\/\mathrm{d}\/r},$
(5.2)

where

$\displaystyle\widehat{M}(\tau):=\|\boldsymbol{v}_{\tau}-\bar{\boldsymbol{v}}_{0}\|^{2}_{\mathbb{H}}+\int_{0}^{T}\frac{e^{-2\sigma
y(\theta_{t}\omega)}}{\sigma^{2}}\|\boldsymbol{f}(t+\tau)-\boldsymbol{f}_{\infty}\|^{2}_{\mathbb{H}}.$

By Hypothesis 1.1 and
$\lim\limits_{\tau\to+\infty}\|\boldsymbol{v}_{\tau}-\bar{\boldsymbol{v}}_{0}\|_{\mathbb{H}}=0$,
we have $\lim\limits_{\tau\to+\infty}\widehat{M}(\tau)=0$. This fact gives

$\displaystyle\lim_{\tau\to+\infty}\|\mathscr{V}_{\tau}(T)\|^{2}_{\mathbb{H}}\leq
Cp^{2}p^{-\frac{2}{p}}\bigg{[}\frac{p^{2}}{(p-1)(p-2)}\bigg{]}^{p-2}\bigg{[}C\int_{0}^{T}e^{\sigma
y(\theta_{t}\omega)}\/\mathrm{d}\/t\bigg{]}^{p-2}e^{2\sigma\int_{0}^{T}|y(\theta_{r}\omega)|\/\mathrm{d}\/r}.$
(5.3)

Let us now take $T$ sufficiently small such that $C\int_{0}^{T}e^{\sigma
y(\theta_{t}\omega)}\/\mathrm{d}\/t<1$. Further, it provides us that
$\lim\limits_{p\to+\infty}p^{2}[C\int_{0}^{T}e^{\sigma
y(\theta_{t}\omega)}\/\mathrm{d}\/t]^{p-2}e^{2\sigma\int_{0}^{T}|y(\theta_{r}\omega)|\/\mathrm{d}\/r}=0,$
whereas
$\lim\limits_{p\to+\infty}p^{-\frac{2}{p}}\mathopen{}\mathclose{{}\left[\frac{p^{2}}{(p-1)(p-2)}}\right]^{p-2}=e^{3}$.
Hence
$\lim\limits_{\tau\to+\infty}\|\mathscr{V}_{\tau}(T)\|^{2}_{\mathbb{H}}=0$ if
$C\int_{0}^{T}e^{\sigma y(\theta_{t}\omega)}\/\mathrm{d}\/t<1$. We repeat the
same arguments a finite number of times to complete the proof. ∎

The following lemma demonstrates that the property $\mathrm{(iii)}$ of Theorem
2.10 hold.

######  Lemma 5.3.

There exists a $t_{0}>0$ such that
$\mathcal{B}_{t_{0}}(\omega):=\overline{\bigcup\limits_{t\geq
t_{0}}\mathcal{B}(t,\omega)}^{\delta}$ is a
$\/\mathrm{d}\/_{\mathcal{W}}$-bounded set.

###### Proof.

It is clear from Lemma 4.5 that

$\displaystyle\mathcal{B}(t,\omega)=\\{\boldsymbol{u}\in\mathcal{W}:\|\boldsymbol{u}\|_{\mathbb{V}}\leq
L_{1}(t,\omega)\ \text{ and }\
\|\nabla\land\boldsymbol{u}\|_{\mathrm{L}^{\infty}(\Omega)}\leq
L_{2}(t,\omega)\\},$

where

$\displaystyle L_{1}(t,\omega)$
$\displaystyle:=2e^{-\frac{\sigma^{2}}{2}t+2\sigma\int_{0}^{t}y(\theta_{r}\omega)\/\mathrm{d}\/r+2\sigma
y(\theta_{t}\omega)}\int_{-\infty}^{t}e^{\frac{\sigma^{2}}{2}\xi+2\sigma\int_{\xi}^{0}y(\theta_{r}\omega)\/\mathrm{d}\/r-2\sigma
y(\theta_{\xi}\omega)}\big{\\{}\|\boldsymbol{f}(\xi)\|^{2}_{\mathbb{L}^{2}(\mathcal{O})}$
$\displaystyle\qquad+\|\nabla\land\boldsymbol{f}(\xi)\|_{\mathrm{L}^{2}(\mathcal{O})}^{2}\big{\\}}\/\mathrm{d}\/\xi,$

and

$\displaystyle L_{2}(t,\omega)$
$\displaystyle:=2e^{-\frac{\sigma^{2}}{2}t+2\sigma\int_{0}^{t}y(\theta_{r}\omega)\/\mathrm{d}\/r+\sigma
y(\theta_{t}\omega)}\int_{-\infty}^{t}e^{\frac{\sigma^{2}}{2}\xi+2\sigma\int_{\xi}^{0}y(\theta_{r}\omega)\/\mathrm{d}\/r-\sigma
y(\theta_{\xi}\omega)}\|\nabla\land\boldsymbol{f}(\xi)\|_{\mathrm{L}^{\infty}(\mathcal{O})}\/\mathrm{d}\/\xi.$

By the properties of $y(\cdot)$ given in (3.3), we can find $t_{1},t_{2}>0$
such that (see the proof of Lemma 4.5)

$\displaystyle-\frac{\sigma^{2}}{2}t+2\sigma\int_{0}^{t}y(\theta_{r}\omega)\/\mathrm{d}\/r+2\sigma
y(\theta_{t}\omega)\leq-\delta_{1}t,\ \ \ \text{ for all }\ \ \ t\geq t_{1},$

and

$\displaystyle-\frac{\sigma^{2}}{2}t+2\sigma\int_{0}^{t}y(\theta_{r}\omega)\/\mathrm{d}\/r+\sigma
y(\theta_{t}\omega)\leq-\delta_{2}t,\ \ \ \text{ for all }\ \ \ t\geq t_{2}.$

Taking $t_{0}=\max\\{t_{1},t_{2}\\}$, we have for $t\geq t_{0}$,

$\displaystyle L_{1}(t,\omega)$ $\displaystyle\leq
2e^{-\delta_{1}t}\bigg{[}\int_{t_{0}}^{t}e^{\delta_{1}\xi}\big{\\{}\|\boldsymbol{f}(\xi)\|^{2}_{\mathbb{L}^{2}(\mathcal{O})}+\|\nabla\land\boldsymbol{f}(\xi)\|_{\mathrm{L}^{2}(\mathcal{O})}^{2}\big{\\}}\/\mathrm{d}\/\xi$
$\displaystyle\qquad+\int_{-t_{0}}^{t_{0}}e^{\frac{\sigma^{2}}{2}\xi+2\sigma\int_{\xi}^{0}y(\theta_{r}\omega)\/\mathrm{d}\/r-2\sigma
y(\theta_{\xi}\omega)}\big{\\{}\|\boldsymbol{f}(\xi)\|^{2}_{\mathbb{L}^{2}(\mathcal{O})}+\|\nabla\land\boldsymbol{f}(\xi)\|_{\mathrm{L}^{2}(\mathcal{O})}^{2}\big{\\}}\/\mathrm{d}\/\xi$
$\displaystyle\qquad+\int_{-\infty}^{-t_{0}}e^{\delta_{1}\xi}\big{\\{}\|\boldsymbol{f}(\xi)\|^{2}_{\mathbb{L}^{2}(\mathcal{O})}+\|\nabla\land\boldsymbol{f}(\xi)\|_{\mathrm{L}^{2}(\mathcal{O})}^{2}\big{\\}}\/\mathrm{d}\/\xi\bigg{]}$
$\displaystyle\leq\bigg{[}4\int_{-\infty}^{0}e^{\delta_{1}\xi}\big{\\{}\|\boldsymbol{f}(\xi+t)\|^{2}_{\mathbb{L}^{2}(\mathcal{O})}+\|\nabla\land\boldsymbol{f}(\xi+t)\|_{\mathrm{L}^{2}(\mathcal{O})}^{2}\big{\\}}\/\mathrm{d}\/\xi$
$\displaystyle\qquad+2e^{-\delta_{1}t}\int_{-t_{0}}^{t_{0}}e^{\frac{\sigma^{2}}{2}\xi+2\sigma\int_{\xi}^{0}y(\theta_{r}\omega)\/\mathrm{d}\/r-2\sigma
y(\theta_{\xi}\omega)}\big{\\{}\|\boldsymbol{f}(\xi)\|^{2}_{\mathbb{L}^{2}(\mathcal{O})}+\|\nabla\land\boldsymbol{f}(\xi)\|_{\mathrm{L}^{2}(\mathcal{O})}^{2}\big{\\}}\/\mathrm{d}\/\xi\bigg{]},$

and

$\displaystyle L_{2}(t,\omega)$
$\displaystyle\leq\bigg{[}4\int_{-\infty}^{0}e^{\delta_{2}\xi}\|\nabla\land\boldsymbol{f}(\xi+t)\|_{\mathrm{L}^{\infty}(\mathcal{O})}\/\mathrm{d}\/\xi$
$\displaystyle\qquad+2e^{-\delta_{2}t}\int_{-t_{0}}^{t_{0}}e^{\frac{\sigma^{2}}{2}\xi+\sigma\int_{\xi}^{0}y(\theta_{r}\omega)\/\mathrm{d}\/r-2\sigma
y(\theta_{\xi}\omega)}\|\nabla\land\boldsymbol{f}(\xi)\|_{\mathrm{L}^{\infty}(\mathcal{O})}\/\mathrm{d}\/\xi\bigg{]}.$

We infer from Hypothesis 1.2 that $\sup\limits_{t\geq t_{0}}L_{1}(t,\omega)$
and $\sup\limits_{t\geq t_{0}}L_{2}(t,\omega)$ are bounded. Hence,
$\bigcup\limits_{t\geq t_{0}}\mathcal{B}(t,\omega)$ is a
$\/\mathrm{d}\/_{\mathcal{W}}$-bounded set. Since $\mathbb{V}$ is compactly
embedded in $\mathbb{H}$, then arguing similarly as in the proof of Lemma 5.1,
one can conclude that
$\mathcal{B}_{t_{0}}(\omega):=\overline{\bigcup\limits_{t\geq
t_{0}}\mathcal{B}(t,\omega)}^{\delta}$ is a
$\/\mathrm{d}\/_{\mathcal{W}}$-bounded set. ∎

Now, we are ready to reveal the main result of this section, that is, weak
asymptotic autonomy of pullback stochastic weak attractors.

######  Theorem 5.4.

Suppose that Hypotheses 1.1 and 1.2 are satisfied. Then the
$\/\mathrm{d}\/_{\mathcal{W}}$-bounded and $\delta$-compact pullback
stochastic weak attractor $\mathscr{A}(t,\omega)$ of the system (1.2) and
stochastic weak attractor $\mathscr{A}_{\infty}(\omega)$ of the system (4.23)
satisfy the weak asymptotic autonomy, that is,

$\displaystyle\lim_{t\to+\infty}\mathrm{dist}_{\mathcal{W}}^{\delta}(\mathscr{A}(t,\omega),\mathscr{A}_{\infty}(\omega))=0,\
\ \mathbb{P}\text{-a.e. }\omega\in\Omega,$

where $\mathrm{dist}_{\mathcal{W}}^{\delta}(\cdot,\cdot)$ denotes the
Hausdorff semi-distance between two non-empty subsets of some Banach space
$\mathcal{W}$ with respect to the metric $\delta$, that is, for non-empty sets
$A,B\subset\mathcal{W}$

$\mathrm{dist}_{\mathcal{W}}^{\delta}(A,B)=\sup_{a\in A}\inf_{b\in
B}\delta(a,b).$

###### Proof.

Lemma 5.1, Proposition 5.2 and Lemma 5.3 demonstrate that conditions
$\mathrm{(i)}-\mathrm{(iii)}$ of Theorem 2.10 are satisfied. Hence, an
application of Theorem 2.10 completes the proof. ∎

Acknowledgments: The first author would like to thank the Council of
Scientific $\&$ Industrial Research (CSIR), India for financial assistance
(File No. 09/143(0938)/2019-EMR-I). M. T. Mohan would like to thank the
Department of Science and Technology (DST), Govt of India for Innovation in
Science Pursuit for Inspired Research (INSPIRE) Faculty Award (IFA17-MA110).

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