I want to prove the existence of the solution of this system by using the Faedo-Galerkin approximation method, I have to choose a basis for working on and I don't know how to do it in this case, I suggest the basis of the following spaces : 
\begin{equation}
H_{\Gamma _{0}}^{1}(\Omega )=\left\{ u\in H^{1}(\Omega );\text{ }u=0\right\}
,
\end{equation}
and 
$$H_0(\Gamma_1)$$ to deal with the internal and boudary termes.
Is this write? thanx.
\begin{equation}
\left\{ 
\begin{array}{rrrr}
u_{tt}-\Delta u=0,&\text{in} & 
\Omega \times ]0,T[ & \left( 1.1\right)  \\ 
u=0, & \text{on
} & \Gamma _{0}\times ]0,T[ & \left( 1.2\right)  \\ 
u-w=0, & \text{on} & 
\Gamma _{1}\times ]0,T[ & \left( 1.3\right)  \\ 
w_{tt}-\Delta _{T}w+\partial _{\nu }u+w_{t}=0, & \text{on} & \Gamma
_{1}\times ]0,T[ & \left( 1.4\right)  \\ 
w=0,& \text{on
} & \partial \Gamma _{1}\times ]0,T[ & \left( 1.5\right)  \\ 
u(.,0)=u_{0},\text{ \ }u_{t}(.,0)=u_{1} & \text{in} & \Omega  & 
\left( 1.6\right)  \\ 
w(.,0)=w_{0},\text{ \ }w_{t}(.,0)=w_{1}\text{\ } & \text{on} & \Gamma _{1} & 
\left( 1.7\right) 
\end{array}%
\right.   \label{E1}
\end{equation}