Consider $u_t = \Delta u$ in $\Omega$ with $u = 0$ on $\partial\Omega$.

Define energy $E(t) = \frac{1}{2}\int_\Omega u^2\,dx$.

Taking time derivative: $\frac{dE}{dt} = \int_\Omega u \cdot u_t\,dx = \int_\Omega u\Delta u\,dx$

Integrating by parts (using $u = 0$ on $\partial\Omega$): $= -\int_\Omega |\nabla u|^2\,dx \leq 0$

By Poincaré inequality, $\int u^2 \leq C\int|\nabla u|^2$, so: $\frac{dE}{dt} \leq -\frac{1}{C}E(t)$

Gronwall's inequality gives $E(t) \leq E(0)e^{-t/C}$, proving exponential decay.

For $u_{tt} = c^2u_{xx}$ with compact support data, energy in ball $B_R$ satisfies: $\frac{d}{dt}\int_{B_R}(u_t^2 + c^2u_x^2) = \text{boundary flux}$

The flux travels at speed $c$, so energy cannot escape $B_R$ faster than $ct$. Thus support remains in $B_{R+ct}$. Full proof uses characteristic coordinates and careful estimation of boundary terms.