Abstract
In this paper we consider a model of loop dislocations which arise in rapid thermal annealing of impurities into a crystal. The loop is a torus D(R(t)) generated by rotation of a disc of unit radius about an axis, a distance R(t) from the center of the circle. The function R(t) varies according to the flux of the interstitial density c across the boundary ∂D(R(t)) of the torus. Outside D(R(t)) the function c satisfies a parabolic equation; on the boundary ∂D(R(t)), c is a function of the space variable and of R(t). The problem is to solve the system for c and R. This is a free-boundary problem where the free boundary is ∂D(R(t)). It is shown that a unique local solution does exist; however, for some class of data it “blows up” in finite time. For other classes of data we prove that a global solution exists, and we study its asymptotic behavior.
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Communicated by H. Brezis
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Friedman, A., Hu, B. & Velazquez, J.J.L. A free-boundary problem modeling loop dislocations in crystals. Arch. Rational Mech. Anal. 119, 229–291 (1992). https://doi.org/10.1007/BF00381671
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DOI: https://doi.org/10.1007/BF00381671