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Phase transitions and generalized motion by mean curvature. (English) Zbl 0801.35045
Summary: We study the limiting behavior of solutions to appropriately rescaled versions of the Allen-Cahn equation, a simplified model for dynamic phase transitions. We rigorously establish the existence in the limit of a phase-antiphase interface evolving according to mean curvature motion. This assertion is valid for all positive time, the motion interpreted in the generalized sense of Evans-Spruck and Chen-Giga-Goto after the onset of geometric singularities.

MSC:
35K55 Nonlinear parabolic equations
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
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