 Insulated steady heat balance

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 conditionIn 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 $dE/dt={\int }_{R}HdV>0$: stored thermal energy must grow. This is a compatibility condition on the total source, not a restriction requiring the source to vanish pointwise.