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
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