Sci Advisor P: 5,366 Hm, it seems you want to calculate the net energy flow with $$dE_- = -p(T)\,dt = -\sigma\,A\,T^4\,dt$$ where p(T) is the integrated density from Stefan-Boltzmann, and from an external radiation source with $$dE_+ = p_0\,dt$$ Then you are looking for $$dE_- + dE_+ = 0$$ Correct?
 Sci Advisor P: 5,366 Maximum temperature under radiation OK, you have $$dE_+ + dE_- = 0$$ $$p_0\,dt - \sigma\,A\,T^4\,dt = 0$$ $$p_0 = \sigma\,A\,T^4$$ So the temperature is constant iff the black body radiation is balanced by radiation received from the external source.