Electron. J. Diff. Equ., Vol. 2015 (2015), No. 39, pp. 1-14.

Existence and uniqueness for a dislocation model with short-range interactions and varying stress field

Hassan Ibrahim

Abstract:
In this article, we consider a coupled singular parabolic system, describing the dynamics of dislocation densities in a bounded domain. The model takes into consideration the short-range interactions between dislocations, which causes the singularity that appears under the form of dividing by a gradient term. We prove a long time existence and uniqueness under the assumption that the applied stresses on the domain is bounded in space and time. The proof relies on a comparison principle to avoid singularity, and on exponential gradient estimates for the long time existence.

Submitted January 9, 2015. Published February 12, 2015.
Math Subject Classifications: 35K50, 35K40, 35K55.
Key Words: Boundary value problem; parabolic system; comparison principle.

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Hassan Ibrahim
Lebanese University, Faculty of Sciences I
EDST, Equipe EDP-AN
Hadath-Beirut, Lebanon
email: ibrahim@cermics.enpc.fr

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