# How to prove uniqueness solution of the 3D wave

1. Apr 19, 2013

### lotusquantum

1. The problem statement, all variables and given/known data

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

$∂u/∂t(x,y,z,0)=H(x,y,z)$

2. Relevant equations
how to prove the uniqueness solution of the above equation?

3. The attempt at a solution
Please recommend me some methods or examples to prove such problems. thansk

Last edited: Apr 19, 2013
2. Apr 19, 2013

### Staff: Mentor

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'?

3. Apr 19, 2013

### lotusquantum

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

4. Apr 20, 2013

### Staff: Mentor

Where is S?

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