= Solution
{c}
Integrate the <steady state> equation over $R$ and use the <divergence theorem>:
$$
0=\iint_{\partial R}k\nabla T\cdot\mathbf n\,dA+\rho\iiint_R S(\mathbf x)\,dV.
$$
The zero <heat flux> condition makes the boundary term vanish. But $\rho>0$ and $S>0$ throughout a region of positive <volume>, so
$$
\rho\iiint_R S\,dV>0,
$$
a contradiction. Thus \b[no steady-state solution exists]. This is the <insulated steady heat balance>: positive internal <heat> generation continually increases the stored thermal <energy> when no <heat> can escape.
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