  The mathematical model describing the dynamics of closed contact heating
which involves vaporization of the metal when instantaneous explosion of
micro-asperity occurs is presented through a Stefan type problem. The
temperature field for metallic vaporization zone is introduced as heat
resistance that decreases linearity. Temperature fields for liquid and solid
phases of the metal described by spherical heat equations with nonlinear
thermal coefficients and Joule heat source have to be determined as well as the
free boundaries. Joule heating component depends on space and time variable
when alternating current is considered. Solution method of the problem based on
similarity variable transformation is applied which enables us to reduce the
problem to an ordinary differential equations and nonlinear integral equations.
Existence and uniqueness of the integral equations are proved by using fixed
point theorem in Banach space.
