Let $V \in \mathcal{C}^{\infty} ([0,T] \times [-R,R])$ such that : $$\partial _t^2 V - cV = F,$$ for some constant $c >0$ and some $F \in L^2( [0,T] \times [-R,R])$, with initial conditions $V(0,x)=V_0(x)$ and $\partial _t V(0,x)=V_1(x)$ with $V_0,V_1$ regular enough.
I am asked to prove that there is a constant $C$ (which may depend on $c$) such that :
$$\|V(t, \cdot )\|^2_{L^2([-R,R])} \leqslant C \left( \|V_0\|^2_{H^1([-R,R])} + \|V_1\|^2_{L^2([-R,R])} + \int_0 ^t \|F(s,\cdot)\|^2_{L^2([-R,R])} \, \mathrm{d}s \right)$$
The problem is that I don't see how to get that : if I multiply the equation by $\partial _t V$ I will only get an estimate on $\|\partial _t V (t, \cdot)\|^2_{L^2([-R,R])}$.
How I am supposed to solve this?