# American Institute of Mathematical Sciences

May  2010, 9(3): 703-719. doi: 10.3934/cpaa.2010.9.703

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

 1 Cermics, Paris-Est/ENPC, 6 et 8 avenue Blaise Pascal, Cité Descartes Champs-sur-Marne, 77455 Marne-la-Vallée Cedex 2, France 2 Lebanese University, LaMA-Liban, P.O. Box 826, Tripoli, Lebanon 3 CERMICS, Paris Est-ENPC, 6 & 8 avenue Blaise Pascal, Cité Descartes, Champs sur Marne, 77455 Marne la Vallée Cedex 2

Received  May 2009 Revised  November 2009 Published  January 2010

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 approx- imate 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 the global existence and uniqueness is done under the use of a certain comparison principle on the gradient of the solution.
Citation: Hassan Ibrahim, Mustapha Jazar, Régis Monneau. Dynamics of dislocation densities in a bounded channel. Part I: smooth solutions to a singular coupled parabolic system. Communications on Pure & Applied Analysis, 2010, 9 (3) : 703-719. doi: 10.3934/cpaa.2010.9.703
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