An initial boundary-value problem with a noncoercive boundary condition in domains with edges

Abstract

We consider an initial boundary-value problem for a second order parabolic equation in a domain with edges. We assume that on a part of the boundary the unknown function satisfies a boundary condition of the type\(u_t + \overrightarrow b \cdot \nabla u = \varphi \) (where\(\overrightarrow b \cdot \overrightarrow n > 0,\overrightarrow n \) is the external normal, φ is a given function). In the case of more than one space variable the existence results of the general theory of parabolic initial boundary-value problems do not apply to the problems with boundary conditions of this type. Unique solvability of the problem is established in weighted Sobolev spaces where the weight multipliers are certain powers of the distance to the edge. Bibliography: 17 titles.