Proving uniqueness for the Heat, Wave, Poisson's equation

This is a quick memory guide for proving uniqueness for the Heat, Wave, and Posson's equation. For all three proofs, you need to use the identity, \[ u\nabla^2 u = \nabla \cdot (u\nabla u) - |\nabla u|^2 \]

Heat equation

The equation equation is \[ u_{tt} = \kappa \nabla^2 u. \]

Multiply by $u_t$ and integrate over the domain, $\Omega$. \[ \dd{}{t} \int_\Omega \frac{1}{2} (u_{t})^2 \, \de{V} = \kappa \int_\Omega u_t \nabla^2 u \, \de{V}. \]