Use an energy argument for to show that the global Cauchy problem for the three-dimensional wave equation has a unique solution

Question

I am trying to use an energy argument for to show that the global Cauchy problem for the three-dimensional wave equation has a unique solution.

The wave equation is $$\partial^2u/\partial t^2=\nabla^2u$$

I looked up the energy functional and I want to use this $$E[u]=\frac1{2}\int_{R^3}((\partial u/\partial t)+\nabla u \nabla u))d^3 r$$

I have been reading what may be a similar question http://www.sgo.fi/~j/baylie/Partial%20Differential%20Equations%20in%20Action%20-%20From%20Modelling%20to%20Theory%20-%20S.%20Salsa%20(Springer,%202008)%20WW.pdf on page 263 that goes like the following but I don't know if this is the right track or not...:

Thanks for your help and time.