(x)-S'_t(x)) \, dx \; + \;
\int_0^t \mathcal{E} \big(\varphi_n^{\prime} (u_s-S'_s), \, u_s-S'_s
\big) \, ds = \int_{\mathcal{O}} \varphi_n (\xi(x)-S'_0(x) ) \, dx
\\ & + \int_{0}^{t}\int_{\mathcal{O}}
\varphi_n^{\prime}(u_s(x)-S'_s(x)) \bar{f}  (s,x, u_s-S'_s,\nabla (u_s-S'_s)) \, dx ds \\
&  - \sum_{i=1}^{d} \int_{0}^{t} \int_{\mathcal{O}}
\varphi_n^{\prime \prime } (u_s(x)-S'_s(x) )
\partial_i (u_s(x)-S'_s(x)) \, \bar{g}_{i} (s,x,u_s-S'_s, \nabla (u_s-S'_s)) \, dx \, ds \\& +
\sum_{j=1}^{\infty} \int_{0}^{t} \int_{\mathcal{O}}
\varphi_n^{\prime} (u_s(x)-S'_s(x)) \,  \bar{h}_{j}( s,x, u_s-S'_s,\nabla (u_s-S'_s))\, dx dB_{s}^{j}\\
& + \frac{1}{2}\, \sum_{j=1}^{\infty} \int_{0}^{t}\int_{\mathcal{O}}
\varphi_n^{\prime \prime } (u_s(x)-S'_s(x)) \bar{h}_{j}^2(s,x,
u_s-S'_s,\nabla (u_s-S'_s)) \, dx\, ds\\&
+\int_0^t\int_{\ensuremath{\mathcal O}}\varphi_n^{\prime}(u_s(x)-S'_s(x))\nu(dxds) \, .\\
\end{split}$$ Therefore, passing to the limit as $ n \to \infty $, the convergences come from the Lemma \[estimateul\] and the dominated convergence theorem.

From now on, we assume the following stronger hypothesis:$$\label{alphabet}\alpha+\frac{1}{2}\beta^2+72\beta^2<\lambda.$$ At this stage, the idea is to adapt the Moser iteration technics to our setting. To this end, in order to control uniformly the $L^l$ norms and make $l$ tend to $+\infty$, we introduce for each $l\geq2$, the processes $v$ and $v'$ given by $$\begin{split}
\label{v} v_t :&=\sup _{s\leq t}\left( \int_{{\ensuremath{\mathcal O}}}\left|
u_s-S'_s\right|^l dx +\gamma l\left( l-1\right)
\int_0^s\int_{{\ensuremath{\mathcal O}}}\left| u_r-S'_r\right|
^{l-2}\left| \nabla( u_r-S'_r)\right| ^2\, dx\, dr\right),  \\
 v_t^{\prime } :&=\int_{{\ensuremath{\mathcal O}}}\left| \xi-S'_0 \right| ^l\, dx+l^2 c_1
\left\| \left| 