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

E. V. Frolova

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 25, Tome 213 (1994), pp. 206-223

Résumé

We consider an initial-boundary value problem for the second order parabolic equation in a domain with edges. We assume that on a part of the boundary an unknown function satisfies the boundary condition of the type $u_t+\vec b\cdot\nabla u=\varphi$ (where $\vec b\cdot\vec n>0$, $n$ is the external normal vector, $\varphi$ is a given function). In the case of more than one space variable the existence results of general theory of parabolic initial-boundary value problems can't be applied to problems with such a boundary condition. Unique solvability of the problem under condition is established in weighted Sobolev spaces where the weight multiplies is a certain power of a distance to the edge. Bibliography: 17 titles.