On a homogeneous parabolic problem in an infinite angular domain

In this paper we study a homogeneous boundary value problem for the heat equation in a noncylindrical domain with the special boundary conditions. The problem under consideration is useful for solving the single-phase Stefan problem. It has been shown that this homogeneous problem has a nontrivial solution up to constant factor in the weight class of essentially bounded functions. A class of functions in which this problem has only a trivial solution is found. Thus, a class of functions in which the corresponding inhomogeneous problem is uniquely solvable is defined.