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In this article we consider a mathematical model of an initial stage ofclosure electrical contact that involves a metallic vaporization afterinstantaneous exploding of contact due to arc ignition with power $P_0$ onfixed face $z=0$ and heat transfer in material with a variable cross section,when the radial component of the temperature gradient can be neglected incomparison with the axial component with heat flux and convective boundaryconditions prescribed at the known free boundary $z = \alpha (t)$. Thetemperature field in the liquid region of such kind of material can be modelledby Stefan problem for the generalized heat equation. The method of solution isbased on similarity variable, which enables us to reduce generalized heatequation to nonlinear ordinary differential equation. Moreover, we have todetermine temperature solution for the liquid phase and location of meltinginterface. Existence and uniqueness of the solution is proved by using thefixed point Banach theorem. The solution for two cases of thermal coefficients,in particular, constant and linear thermal conductivity are represented,existence and uniqueness for each type of solution is proved.

Nonlinear Stefan Problem for one-phase generalized heat equation with heat flux and convective boundary condition