= Energy identity
{title2=$E'(t)+D(t)=F(t)$}
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> proves
$$
\frac12\frac d{dt}\int_0^1u^2\,dx+\int_0^1u_x^2\,dx+\alpha u(0,t)^2=\int_0^1ug\,dx.
$$
The boundary term is $[u u_x]_0^1=-\alpha u(0,t)^2$; the minus sign at the left endpoint is essential. For $\alpha\geq0$ it contributes dissipation. If $ug\leq a(t)u^2$ and the initial data vanish, the <Gronwall inequality> forces $u\equiv0$.
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