Solutions of the heat equation in domains with singularities
A lot of papers are devoted to studying boundary value problems for parabolic equation in noncylindrical domains with a singular point at which the boundary surface is tangent to the hyperplane perpendicular to the \(t\)-axis. As a rule, in a neighborhood of such a point the domain lies on one side of the tangent hyperplane. In the present paper, the Dirichlet problem for the one-dimensional heat equation and for singular points that do not satisfy the above condition is studied. The singular points considered in the paper may be described by equations \(t= \pm\alpha x^2\); \(t= \pm\beta x^2\). A new method is developed, in which the model problem is reduced to a degenerate operator pencil on a one-dimensional manifold and then the resolvent of the pencil is studied. It is shown that the smoothness properties and the asymptotics of the solution are determined by the arrangement of roots of parabolic cylinder functions with noninteger indices. Properties of these roots were studied in the reviewed paper. A detail description of the behavior of the solution in the neighborhood of various singular point is given in the final part of the paper.
scientific article; zbMATH DE number 1274227