How to prove uniqueness solution of the 3D wave

[lotusquantum]

Homework Statement

The three dimensional wave equation:
$c∂^{2}u/∂t^2 = ∇^2 u$

boundary conditions :
$u(x,y,z,t) = F(x,y,z,t)$ on S

initial conditions:
$u(x,y,z,0) = G(x,y,z)$

$∂u/∂t(x,y,z,0)=H(x,y,z)$

Homework Equations

how to prove the uniqueness solution of the above equation?

The Attempt at a Solution

Please recommend me some methods or examples to prove such problems. thansk

 

[mfb]

boundary conditions :
u(x,y,z,t)=F(x,y,z,t)

For all x,y,z,t? If F is a given boundary condition, the unique solution u=F is trivial. If F is something else, or that condition is true for some x,y,z,t only, please specify this.

Assuming u and u' are both solutions, what about u-u'?

 

[lotusquantum]

sorry! It's u(x,y,z,t) = F(x,y,z,t) on S

 

[mfb]

Where is S?

The initial conditions should be sufficient to get a unique solution.