  Using the Green's function of the 3D heat equation, we develop an analytical
account of the thermal behaviour of superconducting films subjected to
electrical currents larger than their critical current in the absence of an
applied magnetic field. Our model assumes homogeneity of films and current
density, and besides thermal coefficients employs parameters obtained by
fitting to experimental electrical field - current density characteristics at
constant bath temperature. We derive both a tractable dynamic equation for the
real temperature of the film up to the supercritical current density J^\ast
(the lowest current density inducing transition to the normal state), and a
thermal stability criterion that allows prediction of J^\ast . For two typical
YBCO films, J^\ast predictions agree with observations to within 5%. These
findings strongly support the hypothesis that a current-induced thermal
instability is generally the origin of the breakdown of superconductivity under
high electrical current densities, at least at temperatures not too far from
Tc.
