In the paper we consider the boundary value problem of heat conduction in a non-cylindrical domain, which is an inverted cone, i.e. in the domain degenerating into a point at the initial moment of time. In this case, the boundary conditions contain a derivative with respect to the time variable; in practice, problems of this kind arise in the presence of the condition of the concentrated heat capacity. We prove a theorem on the solvability of a 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 use the Carleman-Vekua method of equivalent regularization to solve the obtained singular Volterra integral equation.

In this paper, we consider a singular Volterra type integral equation of the second kind, to which some boundary value problems of heat conduction in domains with a boundary varying with time are reduced by the method of thermal potentials. The peculiarity of such problems is that the domain degenerates into a point at the initial moment of time. Accordingly, a distinctive feature of the integral equation under study is that the integral of the kernel, as the upper limit of integration tends to the lower one, is not equal to zero. This circumstance does not allow solving this equation by the method of successive approximations. We constructed the general solution of the corresponding characteristic equation and found the solution of the complete integral equation by the Carleman–Vekua method of equivalent regularization. It is shown that the corresponding homogeneous integral equation has a nonzero solution.

In this paper we consider the stabilization problem of the solution of a boundary value problem for the heat equation with a loaded two-dimensional Laplace operator. The loaded terms represent the values of the required function and traces of the first-order partial derivatives of the required function at fixed points. An algorithm for constructing boundary control functions is proposed.

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.