Consider the initial boundary value problem for wave equation:
$$\begin{align}
& {{u}_{tt}}-\Delta u+u=f\left( x,t \right)\text{ }\left( x,t \right)\in \Omega \times \left( 0,T \right], \\
& u=0\text{ }\left( x,t \right)\in \partial \Omega \times \left( 0,T \right], \\
& u\left( x,0 \right)={{u}_{t}}\left( x,0 \right)=0\text{ }x\in \Omega , \\
\end{align}$$
Derive the following estimates:
$$E\left( u \right)=\int_{\Omega }{\left( {{\left| {{u}_{t}} \right|}^{2}}+{{\left| \Delta u \right|}^{2}}+{{\left| u \right|}^{2}} \right)dx}\le C\left\| f \right\|_{{{L}^{2}}\left( \Omega \times \left[ 0,T \right] \right)}^{2}$$ where $C$ is a constant depending on $T$ and $\Omega$.
My attempt:
Multiply the given wave equation on ${{u}_{t}} $, integrating over $\Omega$, and using the Gauss-Green theorem together with the boundary condition, we obtain
$$\begin{align}
& \frac{d}{dt}E\left( t \right)=\frac{d}{dt}\int_{\Omega }{f{{u}_{t}}dx}, \\
& E\left( t \right)=\int_{\Omega }{\left( \frac{1}{2}{{\left| {{u}_{t}} \right|}^{2}}+\frac{1}{2}{{\left| \Delta u \right|}^{2}}+\frac{1}{2}{{\left| u \right|}^{2}} \right)dx}=\int_{\Omega }{f{{u}_{t}}dx}. \\
\end{align}$$
By Holder's inequality
$$\int_{\Omega }{f{{u}_{t}}dx}\le {{\left\| f \right\|}_{{{L}^{2}}}}{{\left\| {{u}_{t}} \right\|}_{{{L}^{2}}}}.$$
Need help to continue.