Lp-solubility of the Dirichlet problem for them heat equation in non-cylindrical domains

The Dirichlet problem for the heat equation is considered in bounded and unbounded domains of paraboloid type with isolated characteristic points at the boundary. Necessary and sufficient conditions in terms of the weight ensuring the unique solubility of this problem in weighted Sobolev L-p-spaces are found. In particular, a criterion for the solubility of,the problem in the classical Sobolev space W-p,0(2,1) is established in the case when the domain is a ball.