Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics V / 2 / d / ii / Solution

Integrate the steady state equation over $R$ and use the divergence theorem:The zero heat flux condition makes the boundary term vanish. But $\rho >0$ and $S>0$ throughout a region of positive volume, soa contradiction. Thus 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.