SOLUTION OF A TWO-DIMENSIONAL BOUNDARY VALUE PROBLEM OF HEAT CONDUCTION IN A DEGENERATING DOMAIN

Abstract

In the paper we consider the boundary value problem of heat conduction outside the cone, i.e. in the domain degenerating into a point at the initial moment of time. In this case, the boundary condition contain a derivative with respect to the time variable. The peculiarity of the problem under consideration consists precisely in the presence of a moving boundary and the degeneration of the solution domain into a point at the initial moment of time. The well-known classical methods are generally not applicable to this type of problems. By the method of heat potentials, such boundary value problems of heat conduction are reduced to the solution of singular Volterra type integral equations of the second kind A singular Volterra type equation is understood as an equation whose kernel has the following property: the integral of the kernel of the equation does not tend to zero as the upper limit tends to the lower one. Such integral equations cannot be solved by the method of successive approximations, and in most cases the corresponding homogeneous integral equations have nonzero solutions. We prove a theorem on the solvability of the considered boundary value problem in weighted spaces of essentially bounded functions. The issues of solvability of the singular Volterra integral equation of the second kind, to which the original problem is reduced, are studied. We found a nonzero solution of this singular integral equation.