I have found it that in elementary discussions of black-body radiation, other than justifying electromagnetic waves should exist from Maxwell's equations, those equations are not used for anything further. It seems to be a mostly thermodynamic problem to which later a quantum-mechanical explanation was found.
My question is, were there any electromagnetic consideration for the ultraviolet catastrophe? If so, how did those considerations go along when quantum mechanics was proposed?
Ultraviolet catastrophe is a divergence that appears when considering a thermodynamic equilibrium of electromagnetic radiation. In fact, in modern (quantum) language, black body radiation is simply viewed as a photon gas in thermodynamic equilibrium (no black body is needed).
So the electromagnetic radiation, resulting from the Maxwell equations is the very essence of the subject. Furthermore, the expressions for the energy density of the electromagnetic field, which diverges, also come from from the Maxwell equations (aka Poynting theorem). However, another essential component is statistical mechanics, since we are talking about thermodynamic equilibrium.
Related
How can blackbody radiation be explained by quantization? In my answer to this question I provide the parallel calculation for the energy density of the electric field from classical and quantum perspective.
Does thermal energy include the energy of thermal radiation as part of its definition? - here the answer provides full derivation of Planck's formula.
