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

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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

Hassan Ibrahim ^1, , Mustapha Jazar ^2, and Régis Monneau ^3,

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

Abstract
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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 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.

Keywords: Boundary value problems, parabolic systems, comparison principle..

Mathematics Subject Classification: Primary: 35K50, 35K40; Secondary: 35K5.

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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