,0,0) :=h^0 = (h_1^0,...,h_i^0,...) .\\
\end{split}$$ In the sequel, $|\cdot|$ will always denote the underlying Euclidean or $l^2-$norm. For example$$|h(t,\omega,x,y,z)|^2=\sum_{i=1}^{+\infty}|h_i(t,\omega,x,y,z)|^2.$$

Let us note that this general setting of the SPDE we consider, encompasses the case of an SPDE driven by a space-time noise, colored in space and white in time as in [@SW] for example (see also Example 1 in [@DMZ12]).

**Assumption (H):** There exist non-negative constants $C,\
\alpha,\ \beta$ such that for almost all $\omega$, the following inequalities hold for all $(x,y,z,t)\in\mathcal{O}\times\mathbb{R}\times\mathbb{R}^d\times\mathbb{R}_+$:

1.  $|f(t,\omega,x,y,z)-f(t,\omega,x,y',z')|\leq C(|y-y'|+|z-z'|),$

2.  $|g(t,\omega,x,y,z)-g(t,\omega,x,y',z')|\leq
    C|y-y'|+\alpha|z-z'|,$

3.  $|h(t,\omega,x,y,z)-h(t,\omega,x,y',z')|\leq
    C|y-y'|+\beta|z-z'|,$

4.  the contraction property: $2\alpha+\beta^2<2\lambda.$

Moreover we introduce some integrability conditions on the coefficients $f^0, \;
g^0, \, h^0$ and the initial data $\xi$. Along this article, we fix a terminal time $T>0$.\
**Assumption (HI2)** $$E \left(\|\xi \|_2^2   +\left\| f^0\right\|_{2,2;t}^2+\left\|
\left|g^0\right|\right\| _{2,2;t}^2+\left\|\left| h^0\right|\right\|
_{2,2;t}^2\right) <\infty ,$$ for each $t\in[0,T]$.\
**Assumption (HIL)** $$E \int_K |\xi (x)|^2 dx + E \,
\int_{0}^t \int_K \big( |f_s^0(x)|^2 + |g_s^0 (x)|^2 + |h_s^0(x)|^2
\, \big) dx ds < \infty,$$for any compact set $K \subset \mathcal{O}$ and for any $t\in[0,T]$.

Weak solutions
--------------

We now introduce ${\ensuremath{\mathcal H}}_T$, the space of $H_0^1(\mathcal{O})$-valued predictable processes $(u_t)_{t \in[0,T]}$ such that $$\left( E \sup_{0\leq s\leq T} \left\| 