Say we have a blackbody sphere in a perfect vacuum with surface area of $1 m^2$ at temperature 5.1205 C = 278.2705 K. Let's say inside the sphere it has its own energy source that is outputting enough energy to keep the sphere at this temperature.
According to the Stefen-Boltzmann law, it will radiate $j^* = \sigma T^4$ = 340.00 W/$m^2$. Since the sphere has $1 m^2$ surface area it's radiating 340 W and thus it's at thermal equilibrium (i.e. its internal energy source must be emitting 340 W as well).
Now say we take a flashlight that outputs 100 W of light energy and we shine it on the sphere. Will this result in the sphere heating up past its 278.2705 K?
The answer seems to be obviously that yes, since more energy is going into the sphere. Now besides its own internal energy it has an extra +100 W, so it will at equilibrium net radiate 440 W across its $m^2$ and reach 296.8K = 23.65 C.
Or will this 100 W of energy be reflected and not absorbed because it's from a lower-energy source or because the energy coming in is lower than the energy it's emitting? This seems odd because, why wouldn't it just get absorbed?
However if it does get absorbed, it seems we can get a funny situation. Let's consider a similar sphere but without its own internal energy source. Place it in a perfectly insulating chamber (no heat escapes), except the flashlight energy can still get through (e.g. a chamber transparent to the flashlight's wavelengths but reflective to the wavelengths it would emit as a blackbody). Now we shine the light on it.
The system is net gaining energy at a rate of 100 W --- since anything the sphere is radiating is reflecting back. Does that mean it will get infinitely hot? If so isn't this odd, since it means we could basically melt steel with a flashlight? (i.e. once the sphere gets hot enough, take it out of the chamber and drop it on some steel and it will melt right through). Would that actually work? If not, where's the disconnect?