\int_{\mathcal{O}} \varphi_n^{\prime \prime} (u_s -S'_s)
\,
|\nabla( u_s-S'_s) |^2  \, dx \, ds \\
& \, \leq \, l(l-1)c''_{\epsilon} K^2 \, + \,  c''_{\epsilon} l
(l-1)\big( K^2 +C^2+C+ c_{\epsilon}\big)
E \, \int_0^t \int_{\mathcal{O}} \varphi_n (u_s(x) -S'_s(x))\, dx \, ds. \\
\end{split}$$ On account of the contraction condition, one can choose $ \epsilon
> 0 $ small enough such that $$\lambda - \frac{1}{2}(1+
\epsilon)\beta^2 - (\alpha + 2 \epsilon)>0$$ and then $$\begin{split}
E\, \int_{\mathcal{O}} \varphi_n (u_t (x)-S'_t(x)) \, dx    &
 \leq \, c K^2 l (l-1) \, + \,  cl (l-1)
E \, \int_0^t \int_{\mathcal{O}} \varphi_n (u_s (x)-S'_s(x))\, dx \, ds \, .\\
\end{split}$$ We obtain by Gronwall’s Lemma, that $$\label{varphi1}
\begin{split}
E\, \int_{\mathcal{O}} \varphi_n (u_t (x)-S'_t(x)) \, dx    &
 \leq \, c\,K^2  l (l-1) \, \exp{ \big( c \, l (l-1)\,t \big) }\\
\end{split}$$ and so it is now easy from (\[e6\]) to get $$\label{varphi2}
\begin{split}
  E \, \int_0^t \int_{\mathcal{O}} \varphi_n^{\prime \prime} (u_s (x)-S'_s(x))  \,
|\nabla (u_s-S'_s) |^2  \, dx \, ds  \, \leq \,
 c^{\prime}\, K^2 l \, (l-1) \,  \exp{ \big( c l (l-1)\,t \big)}.\\
\end{split}$$ Finally, letting $ n \to \infty $ by Fatou’s lemma we deduce (\[uS’l\]) and (\[nablauS’l\]).\
Then with (\[uS’Itô’s\]), we know that $$-\int_0^t\int_{\ensuremath{\mathcal 