  We study a coupled system of two parabolic equations in one space dimension.
This system is singular because of the presence of one term with the inverse of
the gradient of the solution. Our system describes an approximate model of the
dynamics of dislocation densities in a bounded channel submitted to an exterior
applied stress. The system of equations is written on a bounded interval with
Dirichlet conditions and requires a special attention to the boundary. The
proof of existence and uniqueness is done under the use of two main tools: a
certain comparison principle on the gradient of the solution, and a parabolic
Kozono-Taniuchi inequality
