Energy method

Remarks

Example: let $\Omega$ be bounded with smooth boundary and let $u \in C^2(\overline{\Omega} \times [0, T])$ solve the wave equation $u_{tt} = \Delta u$ with $u = 0$ on $\partial\Omega$. Take $E(t) = \frac{1}{2}\int_\Omega \big(u_t^2 + \abs{\nabla u}^2\big)\, dx$, where $\Delta$ is the Laplacian. Then $E'(t) = \int_\Omega (u_t u_{tt} + \nabla u \cdot \nabla u_t)\, dx = \int_\Omega u_t (u_{tt} - \Delta u)\, dx = 0$, since the boundary term contains $u_t = 0$ on $\partial\Omega$. So $E$ is constant, and two solutions with the same data and boundary values coincide. For the heat equation $u_t = \Delta u$ with $u = 0$ on $\partial\Omega$, the energy $E(t) = \frac{1}{2}\int_\Omega u^2\, dx$ satisfies $E'(t) = -\int_\Omega \abs{\nabla u}^2\, dx \le 0$. Non-example: for the backward heat equation $u_t = -\Delta u$ the same computation gives $E'(t) = \int_\Omega \abs{\nabla u}^2\, dx$, which is not bounded by $C E(t)$, so step 2 fails; indeed that problem is ill-posed.