Parabolic linear PDE with non-homogenous Neumann boundary condition.

Question

Let $\Omega \subset \mathbb{R}^n$ be an open bounded set with lipschitz boundary, $T>0$, $f \in L^2(0,T;L^2(\Omega))$, $g \in L^2(0,T;H^{\frac{1}{2}}(\Omega))$ and $h \in H^1(\Omega).$

I'm trying to prove the existence of weak solutions to the following parabolic problem $$\begin{cases} u'-\Delta u=f \; \;\ in \;\ \Omega \times]0,T[,\\ \dfrac{\partial u}{\partial n}:= \nabla u \cdot \overrightarrow{n}=g\;\; on\;\partial \Omega \times]0,T[,\;\;\ \textbf{(boundary condition)}\\ u=h\;\;\; on\; \{0\}\times\Omega \;\;\ \textbf{(initial condition)} \end{cases} $$ I proceeded by using the Faedo-Galerkin method. Consider: $$V:=\{u\;\; |\;\ u \in L^2(0,T;H^1(\Omega))\;\text{and}\; u' \in L^2(0,T;L^2(\Omega))\}$$
The weak formulation I obtained is the following (note that the test function is considered on space only). $$\begin{cases} \text{Finding } u \in V\; \text{satisftying}\; \forall v\in H^1(\Omega):\\ \displaystyle \int_{\Omega} u' v \;dx + \displaystyle \int_{\Omega} \nabla u . \nabla v \;dx=\displaystyle \int_{\Omega} f v \;dx + \displaystyle \int_{\partial \Omega} g v \;d\tau. \end{cases} $$ Let $$V_n:=Vect\{e_1,\cdots,e_n\}$$ where $\{e_n\}_{n \in \mathbb{N}^*}$ is the eigenbasis generated by the following eigenvalue problem: $$ \begin{cases} - \Delta u= \alpha u\;\ in \; \Omega, \\ \dfrac{\partial u}{\partial n}=0\;\;\; on \;\partial \Omega. \end{cases} $$ This basis is orthonormal in $L^2(\Omega)$ and orthogonal in $H^1(\Omega).$
I have established the following:

$\bullet$ The existence of the solution to the finite dimensionnal problem (i.e in the space $V_n$), which is of the form $$ u_n(x,t)=\sum_{i=1}^n d_k(t) e_k(x). $$ $\bullet$ The boundedness of $u_n$ in the space $L^2(0,T;H^1(\Omega)).$

I still have to prove the boundedness of $u'_n$ in $L^2(0,T;L^2(\Omega):$
Here is my attempt:

Since $u_n$ is a solution to the finite-dimensional problem, we have the following: $$ \forall e_l \in V_n: \displaystyle \int_{\Omega} u_n' e_l \;dx + \displaystyle \int_{\Omega} \nabla u_n . \nabla e_l \;dx=\displaystyle \int_{\Omega} f e_l \;dx + \displaystyle \int_{\partial \Omega} g e_l \;d\tau. $$ Multiplying by $d_l'(t)$ and summing over $l$, we obtain: $$ \Vert u_n'(t) \Vert_{L^2(\Omega)}^2 + \dfrac{1}{2} \dfrac{d}{dt} \Vert \nabla u_n \Vert_{L^2(\Omega)^n}^2=\int_{\Omega} f u'_n \; dx+ \int_{\partial \Omega} u_n' g \;d\tau. $$
By applying the Cauchy-Schwartz inequality and by continuity of the trace with $C>0$, we get
$$ \Vert u_n'(t) \Vert_{L^2(\Omega)}^2 + \dfrac{1}{2} \dfrac{d}{dt} \Vert \nabla u_n(t) \Vert_{L^2(\Omega)^n}^2 \leq \Vert f \Vert_{L^2(\Omega)} \Vert u'_n(t) \Vert_{L^2(\Omega)} + C \Vert u_n'(t) \Vert_{H^1(\Omega)} \Vert g \Vert_{L^2(\partial \Omega)} $$ Which yields, $$ \Vert u_n'(t) \Vert_{L^2(\Omega)}^2 + \dfrac{1}{2} \dfrac{d}{dt} \Vert \nabla u_n(t) \Vert_{L^2(\Omega)^n}^2 \leq \dfrac{1}{2} \Vert f \Vert_{L^2(\Omega)}^2 + \dfrac{1}{2} \Vert u'_n(t) \Vert_{L^2(\Omega)}^2 + C \Vert u_n'(t) \Vert_{H^1(\Omega)} \Vert g \Vert_{L^2(\partial \Omega)} $$ I stopped here, the term $\Vert u_n'(t) \Vert_{H^1(\Omega)}$ is causing me problems, in the homogenous Neumann boundary condtitions' case it doesn't show and we can conclude by Gronwall lemma. Is there a way arround it in the non-homogenous case?