Energy identity ($E^{\prime }(t)+D(t)=F(t)$)

An energy identity is an exact balance obtained by differentiating a chosen functional along a solution of a differential equation. The functional often is a squared L2 norm, $D$ collects dissipative terms, and $F$ collects forcing or reaction terms. Dropping nonnegative terms gives an energy estimate.

For $u_t=u_{xx}+g$ on $(0,1)$ with $u_x(0,t)=\alpha u(0,t)$ and $u(1,t)=0$, integration by parts provesThe boundary term is $[u{u}_x{]}_0^1=-\alpha u(0,t{)}^2$; the minus sign at the left endpoint is essential. For $\alpha \ge 0$ it contributes dissipation. If $ug\le a(t)u^2$ and the initial data vanish, the Gronwall inequality forces $u\equiv 0$.