- #1

fluidistic

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## Homework Statement

Demonstrate that if [tex]u_1[/tex] and [tex]u_2[/tex] are solutions of the wave equation [tex]\frac{\partial ^2 u}{\partial t^2} - \triangle u=0[/tex] such that [tex]u_1 (0,x)=u_2(0,x)[/tex], [tex]\partial _t u_1 (0,x)=\partial _t u_2(0,x)[/tex] and such that the difference "tends to 0 at infinity" sufficiently quickly, then [tex]u_1=u_2[/tex].

Hint: First prove that the following energy is conserved: [tex]E(t)=\int _{\mathbb{R}^3} \frac{1}{2} \left [ (\partial _t u)^2 +\nabla u \cdot \nabla u \right ] dV[/tex].

**2. The attempt at a solution**

Nothing concrete.

I don't understand the part "the difference tends to 0 at infinity". What difference?

Anyway, even assuming that the energy is conserved, I've absolutely no idea about what to do. I'm stuck on the this first exercise since a week. I don't ask for an answer, but rather any push/help.

I don't even know how to start proving that this energy is conserved.