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

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.

• If I recall correctly, Salsa proved the energy inequality required for the Cauchy problem afterwards? – Chee Han Mar 20 '18 at 4:21
• Oh maybe... I will take a look. Thank you. – MathIsHard Mar 20 '18 at 18:51
• Oh I see where I might get some help. thank you! It does appear to go over an example proof of the problem on page 265-266. – MathIsHard Mar 20 '18 at 20:06