On the homogeneous boundary value problem for the heat equation in the degenerate domain

We consider the boundary problem of the theory of heat conduction in a domain with a moving boundary and the corresponding singular integral equation of Volterra, to which it is reduced. Feature of this integral equation is expressed in the fact that the inhomogeneous equation can not be solved by successive approximations. We solve the homogeneous equation corresponding to linear law of motion of the boundary alpha (t) = t. In the result we obtain an Eigen function of a singular integral equation under consideration and the nontrivial solution of the homogeneous boundary value problem.