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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 ofthe gradient of the solution. Our system describes an approximate model of thedynamics of dislocation densities in a bounded channel submitted to an exteriorapplied stress. The system of equations is written on a bounded interval withDirichlet conditions and requires a special attention to the boundary. Theproof of existence and uniqueness is done under the use of two main tools: acertain comparison principle on the gradient of the solution, and a parabolicKozono-Taniuchi inequality

Dynamics of dislocation densities in a bounded channel. Part I: smooth solutions to a singular coupled parabolic system