= Insulated steady heat balance
{title2=$\int_RH\,dV=0$}
For a <steady state> <heat equation> $\nabla\cdot(k\nabla T)+H=0$ in a bounded region with zero outward <heat flux>, integration and the <divergence theorem> give the necessary compatibility condition
$$
\int_RH\,dV=0.
$$
In particular, a strictly positive <PDE source term> throughout a region of positive <volume> makes a <steady state> impossible. The time-dependent <conservation of energy> law instead gives $dE/dt=\int_RH\,dV>0$: stored thermal <energy> must grow. This is a compatibility condition on the total source, not a restriction requiring the source to vanish pointwise.
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