I have a linear problem with double derivate of space and time, which has Dirichlet boundary condition in $(1)_{2}$ and Neumann's boundary condition in $(1)_{3}$:
\begin{equation} \frac{\delta^{2} u}{\delta t^{2}} - c^{2} \Delta u = f \text{ in } \Omega \end{equation}
\begin{equation} u = 0 \text{ on } \delta \Omega \text{ or} \end{equation}
\begin{equation} \nabla u \cdot n = 0 \text{ on } \delta \Omega \end{equation} with conditions
\begin{equation} u(0,x) = u_{0}(x) \text{ in } \Omega \end{equation} \begin{equation} \frac{\delta u}{\delta t}(0, x) = v_{0} (x) \text{ in } \Omega. \end{equation} Which is the weak form of the problem?
My attempt:
\begin{equation} \int (\frac{\delta^{2} u} {\delta t^{2}} - c^{2} \Delta u) \cdot v dx = \int fv \cdot dx, \end{equation}
for $\forall v \in V_{n}.$ Handling two different parts separately
\begin{equation} \int_{\Omega} \frac{\delta^{2} u} {\delta t^{2}}v dV + c^{2} \Big[ \int_{\Omega} \nabla u \cdot \nabla v dV - \int_{\delta \Omega} \nabla u \cdot \hat{n} \cdot v dA \Big] = \int fv \cdot dx. \end{equation}
I finally got the result which should be right one.
By integrating and multiplying Equation (1) with $v$ \begin{equation} \int v \ddot{u} dx + c^{2} v \Delta u = \int fv dx \end{equation} using Green I we get \begin{equation} \int v \ddot{u} dx - \int_{\partial} c^{2} v \frac{\partial u}{\partial n} dS + \int \nabla c^{2} v \cdot \nabla u dx = \int fv dx. \end{equation}
So the weak form is \begin{equation} (v,\ddot{u}) + c^{2}(\nabla v, \nabla u) - c^{2} (v, \frac{\partial u}{\partial n})_{\partial \Omega} = (f,v). \end{equation}