We have the equation
\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 have proved that this equation possesses a solution in some space with the condition $${u_0} \in V \cap {H^2}(\Omega )$$ where $$V = \left\{ {v \in {H^1}(\Omega ),v = 0{\text{ on }}{\Gamma _0}} \right\}$$. My quetion is : Have I $${\partial _\nu }{u_0} \in {L^2}({\Gamma _1})?$$ Thanks.
