In this paper, the boundary value problem for the heat equation in the region which degenerates at the initial time is considered. Such problems arise in mathematical models of the processes occurring by opening of electric contacts, in particular, at the description of the heat transfer in a liquid metal bridge and electric arcing. The boundary value problem is reduced to a Volterra integral equation of the second kind which has a singular feature. The class of solutions for the integral equation is defined and the constructive method of its solution is developed.