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 space : \begin{equation} H_{\Gamma _{0}}^{1}(\Omega )=\left\{ u\in H^{1}(\Omega );\text{ }u=0\right\} , \end{equation} 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}

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}