On the Solvability of Heat Boundary Value Problems in Sobolev Spaces

In a domain, degenerating into a point at the initial moment of time, we consider a boundary value problem (BVP) of heat conduction that is a two-dimensional BVP with respect to space variables. The problem is studied for solvability in Sobolev Hilbert spaces. The correctness of the set problem is shown. The solvability of the heat BVP in a truncated cone is preliminary proved. Next, a system of embedded truncated cones is constructed, the union of which in the limit gives the domain (the cone) of the solution of the original boundary value problem. For each cone, a boundary value problem of heat conduction is posed, similar to the original problem, and its unique solvability is shown. Then, from the solutions of these problems, a sequence is compiled, and the terms of the sequence are continued by zero to complement the domain of these solutions up to the original cone. Using the methods of functional analysis, it is proved that the limit of this sequence is the only solution to the boundary problem under study.