Existence and uniqueness of the solution to a degenerate third boundary value problem for a multidimensional parabolic equation

We study the well-posedness of a third boundary value problem for a multidimensional parabolic equation in the case when the coefficient of the conormal derivative vanishes at some points. We show that under some conditions on the sign of this coefficient there exists nonexistence or nonuniqueness of a solution in the conventional anisotropic Sobolev space. Using the regularization method, we prove existence and uniqueness theorems for the regular solution in suitable weighted spaces.