and the Lipschitz condition on $\bar{g}$ we get $$\begin{split}
 & \,\sum_{i=1}^d \varphi_n^{\prime\prime}(u_s-S'_s) \partial_i (u_s-S'_s)
 \, \bar{g}_i (s,x,u_s-S'_s,\nabla (u_s-S'_s)) \\&\leq \, \varphi_n^{\prime\prime}(u_s-S'_s)
 \, |\nabla (u_s-S'_s)|\, \big( |\bar{g}^0 (s,x)| + C|u_s-S'_s| +
 \alpha  |\nabla (u_s-S'_s)|  \, \big) \,   \\
 &   \leq \,  \epsilon \, \varphi_n^{\prime\prime} (u_s-S'_s)
 |\nabla (u_s-S'_s)|^2 + 2c_{\epsilon} \varphi_n^{\prime\prime} (u_s-S'_s)\, \big( K^2+
 C^2|u_s-S'_s|^2\, \big)+ \alpha \, \varphi_n^{\prime\prime} (u_s-S'_s)|\nabla (u_s-S'_s)|^2  \\
 & \leq   l(l-1)c_{\epsilon}K^2 + 2c_{\epsilon}(K^2 + C^2 ) l (l-1)
 |\varphi_n (u_s-S'_s)|+
 (\alpha + \epsilon) \, \varphi_n^{\prime\prime} (u_s-S'_s)
 |\nabla (u_s-S'_s)|^2.   \\
 \end{split}$$ In the same way as before $$\begin{split}&
\sum_{j=1}^{\infty}  \varphi_n^{\prime \prime } (u_s-S'_s )
\bar{h}_{j}^2(s, u_s-S'_s,\nabla (u_s-S'_s))\\& \leq \,
\varphi_n^{\prime \prime }  (u_s-S'_s)
 \, \big( \, c'_{\epsilon}(|\bar{h}^0 (s,x)|+ C|u_s-S'_s|)^2  \, + \,
 (1 + \epsilon)\beta^2 \,  |\nabla (u_s-S'_s)|^2  \, \big) \\
 &\leq \varphi_n^{\prime \prime }  (u_s-S'_s)
 \, \big(2c'_{\epsilon}K^2 \, +\, 2c'_{\epsilon}C^2 |u_s-S'_s |^2 + \,
 (1 + \epsilon)\beta^2 \,  |\nabla (u_s-S'_s)|^2  \, \big)\\
 & \leq \, 2c'_{\epsilon} l (l-1)K^2 +  2c'_{\epsilon}( K^2 +C^2 )l (l-1) \varphi_n(u_s -S'_s)
 + (1 + \epsilon) \, \beta^2\, \varphi_n^{\prime \prime }(u_s-S'_s)|\nabla (u_s-S'_s)|^2.
\end{split}$$ Thus taking the expectation, we deduce $$\label{e6}
\begin{split}
&E \, \int_{\mathcal{O}}\varphi_n (u_t (x)-S'_t(x)) \, dx + (\lambda
- \frac{1}{2}(1+ \epsilon)\beta^2 - (\alpha + 2 \epsilon) \, ) \, E
\, \int_0^t 